Mini-Lesson

Number & Algebra

26 missions. Guess, learn, then check. Your place is saved.

Mission 18 ÷ 2(2 + 2): is the answer 1 or 16?
Mission 2Two lights flash every 12 s and 18 s. When do they flash together?
Mission 3Half a cup of milk, then a third of a cup more: is that $\frac{2}{5}$ 2/5 of a cup?
Mission 4Is 0.999… (nines forever) exactly equal to 1?
Mission 5Fold a sheet of paper 42 times. Would it reach the Moon?
Mission 6Are you a billion seconds old yet?
Mission 7Eight '80 kg' people in a 640 kg lift. Could it be overloaded?
Mission 8Halve an A4 sheet and it keeps its shape. Why must that be $2$ √2?
Mission 9A 4-digit PIN: how many guesses to be certain of getting it?
Mission 10Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?
Mission 11A price goes up 20%, then down 20%. Is it back where it started?
Mission 12Money grows at 5% a year compound interest. When has it doubled?
Mission 134 builders take 6 days. How long would 8 builders take?
Mission 14A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?
Mission 15The pizza twice as wide costs three times as much. Good deal?
Mission 16A square grows 3 cm each way. Is its new area just $x^{2} + 9$ x² + 9?
Mission 17Is there one temperature that reads the same in °C and °F?
Mission 18When does the pricier phone plan become the cheaper one?
Mission 19Rise 1 for every 12 along: which of three ramps is legal?
Mission 20A thrown ball is 10 m up at two different times. When, and why?
Mission 213 coffees + 2 cakes = £13.50; 2 coffees + 3 cakes = £13. A cake?
Mission 22Four graphs, four equations: can you match them without plotting?
Mission 23Chessboard: 1 grain, then double each square. How many on square 64?
Mission 24Two runners' distance–time graphs cross. What does the crossing mean?
Mission 25Cycle out at 10 km/h, back at 30 km/h. Is your average 20 km/h?
Mission 26A sprinkler reaches 13 m. Does it wet a plant at (12, 6)?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

8 ÷ 2(2 + 2): is the answer 1 or 16?

Make a guess: 8 ÷ 2(2 + 2): is the answer 1 or 16?

Number · place value and the order of operations

Place value and ordering numbers

Each column is worth 10 times the one to its right. $\times 10$ × 10, $\times 100$ × 100, $\times 1000$ × 1000 move digits 1, 2, 3 places left: $3.7 \times 100 = 370$ 3.7 × 100 = 370, $52 \div 1000 = 0.052$ 52 ÷ 1000 = 0.052.
Compare decimals column by column from the left. On a number line, further right is bigger: $- 7 < - 2$ −7 < −2.
Symbols: $=$ =, $\neq =$ ≠ (not equal to), $<$ <, $>$ >, $\leq$ ≤ (less than or equal to), $\geq$ ≥ (greater than or equal to).

Quick check

?Which list is in order, smallest first?

- 0.6

,

- 0.65

,

0.06

,

\frac{1}{20}

Number · place value and the order of operations

Negative numbers and reciprocals

Adding a negative is the same as subtracting: $- 6 + (- 2) = - 8$ −6 + (−2) = −8. Subtracting a negative is the same as adding: $5 - (- 3) = 8$ 5 − (−3) = 8.
Multiplying or dividing: same signs give a positive, different signs give a negative. $(- 4) \times (- 5) = 20$ (−4) × (−5) = 20, $- 12 \div 3 = - 4$ −12 ÷ 3 = −4.
The reciprocal of a number is 1 divided by it: the reciprocal of 4 is $\frac{1}{4}$ 1/4, the reciprocal of $\frac{2}{3}$ 2/3 is $\frac{3}{2}$ 3/2.
A number times its reciprocal is 1. Zero has no reciprocal.

Calculate

?At 6 pm the temperature was 3 °C. It fell by 11 degrees by midnight, then rose by 5 degrees by 6 am. What was the temperature at 6 am?

°C

Number · place value and the order of operations

The order of operations

Order: Brackets, Indices (powers, roots), Division and Multiplication, Addition and Subtraction.
$\times$ × and $\div$ ÷ have equal priority: work them left to right. So do $+$ + and $-$ −.
A number next to a bracket means multiply: $3 (4 + 1) = 3 \times 5$ 3(4 + 1) = 3 × 5. A fraction line and a square root sign group like brackets.
Inverse operations undo each other ( $+$ + and $-$ −, $\times$ × and $\div$ ÷, squaring and square-rooting). Check $223 - 98 = 125$ 223 − 98 = 125 with $125 + 98 = 223$ 125 + 98 = 223.
Calculator: bracket a fraction's top and bottom; round only the final answer.

Worked example Number · place value and the order of operations

Using the order of operations

Question

Work out

30 - 2 \times (1 + 3)^{2} \div 4

1
Brackets first: $1 + 3 = 4$ 1 + 3 = 4, so the expression is $30 - 2 \times 4^{2} \div 4$ 30 − 2 × 4² ÷ 4
2
Then the power: $4^{2} = 16$ 4² = 16, giving $30 - 2 \times 16 \div 4$ 30 − 2 × 16 ÷ 4
3
Multiply and divide, left to right: $2 \times 16 = 32$ 2 × 16 = 32, then $32 \div 4 = 8$ 32 ÷ 4 = 8
4
Finally subtract: $30 - 8 = 22$ 30 − 8 = 22

Answer

22

⚠️ Watch out

working left to right from the start (

30 - 2 = 28

first) gives 112. Subtraction waits until the end.

Calculate

?Work out

\frac{( - 4 ) ^{2} - 3 \times 2 ^{3}}{- 2}

without a calculator.

Number · place value and the order of operations

Binary numbers

Our everyday numbers are base 10 (denary): column values 1, 10, 100, 1000, …
Binary is base 2: only the digits 0 and 1, with column values 1, 2, 4, 8, 16, 32, … (each double the last).
Binary to denary: add the column values where there is a 1. $110 1_{2} = 8 + 4 + 1 = 13$ 1101₂ = 8 + 4 + 1 = 13.
Denary to binary: take out the biggest power of 2 that fits, then the next. $25 = 16 + 8 + 1 = 1100 1_{2}$ 25 = 16 + 8 + 1 = 11001₂.

Calculate

?Convert the binary number

10110 1_{2}

to denary (base 10).

Mission check

?

8 \div 2 (2 + 2)

: using the order of operations, what is its value?

Mission 2

Two lights flash every 12 s and 18 s. When do they flash together?

Make a guess: Two lights flash every 12 s and 18 s. When do they flash together?

Number · factors, multiples and primes

Factors, multiples and primes

A factor (divisor) of a number divides into it exactly. Factors come in pairs: $36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6$ 36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6.
A multiple of a number is in its times table: the multiples of 12 are 12, 24, 36, 48, …
A prime number has exactly two factors, 1 and itself. 1 is not prime (it has only one factor), and 2 is the only even prime.
Common factors are shared factors; common multiples appear in both times tables.

Sort it

?Sort each number.

Tap a statement, then tap the group it belongs in. Then press Check.

Number · factors, multiples and primes

Prime factor decomposition

Every whole number above 1 can be written as a product of prime factors in exactly one way (the unique factorisation theorem).
Use a factor tree or repeated division by primes, and stop when every branch ends in a prime.
Write the answer in index form, smallest prime first: $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^{3} \times 3^{2} \times 5$ 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Check by multiplying back: $8 \times 9 \times 5 = 360$ 8 × 9 × 5 = 360.

Quick check

?Which is 600 written as a product of its prime factors?

Number · factors, multiples and primes

HCF and LCM

The highest common factor (HCF) is the biggest number that divides into both. The lowest common multiple (LCM) is the smallest number both divide into.
Listing: write out factors (HCF) or multiples (LCM) of each number until they match.
Prime factor method: put the prime factors in a Venn diagram, with shared primes in the overlap. HCF = product of the overlap; LCM = product of everything in the diagram.
Higher: for CCEA, the prime factor method is Higher (Unit M3).

Worked example Number · factors, multiples and primes

HCF and LCM using prime factors

Question

Find the HCF and the LCM of 84 and 90.

1
Write each as a product of primes: $84 = 2^{2} \times 3 \times 7$ 84 = 2² × 3 × 7, $90 = 2 \times 3^{2} \times 5$ 90 = 2 × 3² × 5
2
Place the shared primes (one 2 and one 3) in the overlap; left of it goes 2 and 7, right of it goes 3 and 5
3
HCF is the overlap: $2 \times 3 = 6$ 2 × 3 = 6
4
LCM is everything: $2 \times 7 \times 2 \times 3 \times 3 \times 5 = 1260$ 2 × 7 × 2 × 3 × 3 × 5 = 1260

Answer

HCF = 6, LCM = 1260

⚠️ Watch out

multiplying the two numbers (

84 \times 90 = 7560

) gives a common multiple, but not the lowest.

Calculate

?A teacher has 84 pencils and 120 rubbers. She puts them into identical packs, using every pencil and every rubber, with nothing left over. What is the greatest number of packs she can make?

Calculate

?

A = 2^{3} \times 3 \times 5^{2}

and

B = 2 \times 3^{2} \times 5

. Find the lowest common multiple of

A

and

B

.

Mission check

?Two lights flash together, then every 12 s and every 18 s. When do they next flash together?

Mission 3

Half a cup of milk, then a third of a cup more: is that $\frac{2}{5}$ 2/5 of a cup?

Make a guess: Half a cup of milk, then a third of a cup more: is that $\frac{2}{5}$ 2/5 of a cup?

Number · fractions

Equivalent fractions and mixed numbers

Equivalent fractions have the same value: multiply or divide the top and bottom by the same number. $\frac{2}{3} = \frac{8}{12}$ 2/3 = 8/12.
Simplest form: divide top and bottom by their HCF. $\frac{18}{24} = \frac{3}{4}$ 18/24 = 3/4 (divide by 6).
A mixed number has a whole part and a fraction part. Improper to mixed: $\frac{11}{4} = 2 \frac{3}{4}$ 11/4 = 23/4 (4 goes into 11 twice, remainder 3).
Mixed to improper: whole × denominator, plus numerator, over the same denominator.

Quick check

?Write

3 \frac{2}{5}

as an improper fraction.

Number · fractions

Fractions of amounts

To find a fraction of an amount, divide by the denominator, multiply by the numerator: $\frac{2}{5}$ 2/5 of 35 is $35 \div 5 \times 2 = 14$ 35 ÷ 5 × 2 = 14.
"Of" means multiply: $\frac{2}{5}$ 2/5 of 35 is $\frac{2}{5} \times 35$ 2/5 × 35.
To write one quantity as a fraction of another, put the first over the second (same units), then simplify: 45 minutes out of an hour is $\frac{45}{60} = \frac{3}{4}$ 45/60 = 3/4.
A fraction can be greater than 1: 90 minutes as a fraction of an hour is $\frac{90}{60} = \frac{3}{2}$ 90/60 = 3/2.

Calculate

?A school has 840 students.

\frac{3}{7}

of them walk to school.

\frac{1}{4}

of the rest come by bus. How many students come by bus?

Number · fractions

Adding and subtracting fractions

You can only add or subtract fractions when the pieces are the same size: rewrite them with a common denominator (the LCM of the denominators is the neatest).
Then add or subtract the numerators; the denominator stays the same. $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
Never add the denominators.
With mixed numbers, the safest route is to change to improper fractions first. Negative fractions follow the usual sign rules: $- \frac{1}{2} + \frac{1}{3} = - \frac{3}{6} + \frac{2}{6} = - \frac{1}{6}$ −1/2 + 1/3 = −3/6 + 2/6 = −1/6.

Worked example Number · fractions

Subtracting mixed numbers

Question

Work out

2 \frac{3}{4} - 1 \frac{5}{6}

1
Change to improper fractions: $\frac{11}{4} - \frac{11}{6}$ 11/4 − 11/6
2
Use the common denominator 12: $\frac{33}{12} - \frac{22}{12}$ 33/12 − 22/12
3
Subtract the numerators: $\frac{11}{12}$ 11/12

Answer

\frac{11}{12}

⚠️ Watch out

splitting into

2 - 1 = 1

and then working out

\frac{5}{6} - \frac{3}{4}

(because

\frac{3}{4} - \frac{5}{6}

looks impossible) gives

1 \frac{1}{12}

, which is wrong.

Quick check

?Work out

\frac{2}{3} + \frac{3}{4}

Number · fractions

Multiplying and dividing fractions

To multiply, multiply the numerators and multiply the denominators, cancelling first if you can: $\frac{2}{3} \times \frac{9}{10} = \frac{18}{30} = \frac{3}{5}$ 2/3 × 9/10 = 18/30 = 3/5.
Dividing by a number is the same as multiplying by its reciprocal: $3 \div 5 = 3 \times \frac{1}{5} = \frac{3}{5}$ 3 ÷ 5 = 3 × 1/5 = 3/5.
So to divide by a fraction, flip the second fraction and multiply: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
Change mixed numbers to improper fractions first. Signs as usual: $- \frac{1}{2} \times \frac{2}{3} = - \frac{1}{3}$ −1/2 × 2/3 = −1/3.

Calculate

?A plank is

3 \frac{3}{4}

m long. It is cut into pieces that are each

\frac{5}{8}

m long. How many pieces are there? Do not use a calculator.

Mission check

?Half a cup of milk, then a third of a cup more. How much milk is that?

Mission 4

Is 0.999… (nines forever) exactly equal to 1?

Make a guess: Is 0.999… (nines forever) exactly equal to 1?

Number · fractions, decimals and percentages

Converting between fractions, decimals and percentages

Percent means "number of parts per hundred": $60% = \frac{60}{100} = 0.6$ 60% = 60/100 = 0.6.
Fraction to decimal: divide the numerator by the denominator. $\frac{3}{5} = 3 \div 5 = 0.6$ 3/5 = 3 ÷ 5 = 0.6.
Decimal to fraction: use place value, then simplify. $0.35 = \frac{35}{100} = \frac{7}{20}$ 0.35 = 35/100 = 7/20.
Decimal to percentage: multiply by 100. $\frac{7}{4} = 1.75 = 175%$ 7/4 = 1.75 = 175%.
To order a mix of forms, change them all to decimals first.

Worked example Number · fractions, decimals and percentages

Converting both ways

Question

Write 0.45 as a fraction in its simplest form, and write

\frac{3}{8}

as a percentage.

1
0.45 is 45 hundredths: $\frac{45}{100}$ 45/100
2
Divide top and bottom by 5: $\frac{9}{20}$ 9/20
3
Divide 3 by 8 (short division, $3.000 \div 8$ 3.000 ÷ 8): $0.375$ 0.375
4
Multiply by 100: $37.5%$ 37.5%

Answer

0.45 = \frac{9}{20}

and

\frac{3}{8} = 37.5%

⚠️ Watch out

\frac{3}{8}

is not 3.8% or 38%. Divide the numerator by the denominator; don't just write the digits next to each other.

Calculate

?400 people were asked how they travel to work.

\frac{3}{8}

cycle, 0.3 walk, 22% take the bus and the rest drive. How many drive?

Number · fractions, decimals and percentages

Terminating and recurring decimals

A terminating decimal stops: $\frac{3}{40} = 0.075$ 3/40 = 0.075. A recurring decimal repeats for ever: $\frac{1}{3} = 0.333 \dots$ 1/3 = 0.333…, $\frac{1}{6} = 0.1666 \dots$ 1/6 = 0.1666…, $\frac{5}{11} = 0.4545 \dots$ 5/11 = 0.4545…
A fraction in its simplest form terminates exactly when its denominator has no prime factors other than 2 and 5 (because $10 = 2 \times 5$ 10 = 2 × 5). Otherwise it recurs.
Every recurring decimal is an exact fraction, even though its digits never end.
Higher: Rational: can be written as $\frac{a}{b}$ a/b, $a$ a and $b$ b integers ( $b \neq = 0$ b ≠ 0). Irrational ( $2$ √2, $π$ π): the decimal never ends or repeats.

Sort it

?Does each fraction give a terminating or a recurring decimal? Simplify first.

Tap a statement, then tap the group it belongs in. Then press Check.

Number · fractions, decimals and percentages

Recurring decimals to fractions (Higher)

Dot notation: a dot over a digit shows it repeats. $0. 4 ˙ = 0.444 \dots$ 0.4 = 0.444…, $0.1 7 ˙ = 0.1777 \dots$ 0.17 = 0.1777…, $0. 1 ˙ 2 ˙ = 0.121212 \dots$ 0.12 = 0.121212…, $0. 1 ˙ 2 3 ˙ = 0.123123 \dots$ 0.123 = 0.123123… (dots on the first and last digit of the repeating block).
Method: call the decimal $x$ x. Multiply by 10, 100, … so the repeating parts line up, then subtract to remove them.
One repeating digit: multiply by 10. Two: multiply by 100. Then solve for $x$ x and simplify.
Fraction to recurring decimal: divide, as on screen 4.

Worked example Number · fractions, decimals and percentages

Converting $0.2 7 ˙$ 0.27 to a fraction (Higher)

Question

Write

0.2 7 ˙

as a fraction in its simplest form.

1
Let $x = 0.2777 \dots$ x = 0.2777…
2
Multiply by 10 and by 100: $10 x = 2.777 \dots$ 10x = 2.777… and $100 x = 27.777 \dots$ 100x = 27.777… (the recurring parts now line up)
3
Subtract: $100 x - 10 x = 27.777 \dots - 2.777 \dots$ 100x − 10x = 27.777… − 2.777…, so $90 x = 25$ 90x = 25
4
Solve and simplify: $x = \frac{25}{90} = \frac{5}{18}$ x = 25/90 = 5/18

Answer

\frac{5}{18}

⚠️ Watch out

subtracting

x

from

10 x

here doesn't work, because

2.777 \dots - 0.2777 \dots

doesn't cancel the recurring part. Line up the repeating digits first.

Calculate

?(Higher) Write

0. 4 ˙ 5 ˙

as a fraction in its simplest form. Enter the denominator of your fraction.

Sort it

?Rational or irrational?

Tap a statement, then tap the group it belongs in. Then press Check.

Mission check

?Is 0.999… (nines for ever) exactly equal to 1?

Mission 5

Fold a sheet of paper 42 times. Would it reach the Moon?

Make a guess: Fold a sheet of paper 42 times. Would it reach the Moon?

Number · powers and indices

Powers, roots and repeated doubling

$5^{3} = 5 \times 5 \times 5 = 125$ 5³ = 5 × 5 × 5 = 125: the base is 5 and the index (power) is 3.
Learn the squares to $1 5^{2} = 225$ 15² = 225, the cubes of 1 to 5 and 10, and $2^{5} = 32$ 2⁵ = 32, $3^{4} = 81$ 3⁴ = 81, $1 0^{6}$ 10⁶ (a million).
A root undoes a power: $169 = 13$ √169 = 13, $364 = 4$ ∛64 = 4. A positive number has two square roots, e.g. $\pm 13$ ± 13.
Each fold doubles the thickness, so $n$ n folds multiply it by $2^{n}$ 2ⁿ.

Number · powers and indices

The three laws of indices

Multiplying powers of the same base: add the indices. $a^{m} \times a^{n} = a^{m + n}$ a^m × aⁿ = a^(m+n)
Dividing: subtract the indices. $a^{m} \div a^{n} = a^{m - n}$ a^m ÷ aⁿ = a^(m−n)
A power of a power: multiply the indices. $(a^{m})^{n} = a^{mn}$ (a^m)ⁿ = a^mn
The laws only work when the bases are the same: $2^{3} \times 5^{2}$ 2³ × 5² cannot be combined.
Shortcut: $2^{10} = 1024 \approx 1000$ 2¹⁰ = 1024 ≈ 1000, so $2^{40} = (2^{10})^{4} \approx 1 0^{12}$ 2⁴⁰ = (2¹⁰)⁴ ≈ 10¹².

Worked example Number · powers and indices

Using the laws on an algebraic expression

Question

Simplify

(2 x^{3})^{2} \times 3 x^{4} \div 6 x^{5}

.

1
Square everything in the bracket, numbers too: $(2 x^{3})^{2} = 4 x^{6}$ (2x³)² = 4x⁶
2
Multiply numbers, add indices: $4 x^{6} \times 3 x^{4} = 12 x^{10}$ 4x⁶ × 3x⁴ = 12x¹⁰
3
Divide numbers, subtract indices: $12 x^{10} \div 6 x^{5} = 2 x^{5}$ 12x¹⁰ ÷ 6x⁵ = 2x⁵

Answer

2 x^{5}

⚠️ Watch out

writing

(2 x^{3})^{2} = 2 x^{6}

forgets to square the 2, and leads to

x^{5}

.

Calculate

?A sheet of paper is 0.1 mm thick. Each fold doubles the thickness. What is the smallest number of folds that makes it thicker than 1 metre?

Number · powers and indices

Zero and negative indices

$2^{3} \div 2^{3} = 2^{0}$ 2³ ÷ 2³ = 2⁰, but any number divided by itself is 1. So $a^{0} = 1$ a⁰ = 1 (for $a \neq = 0$ a ≠ 0).
A negative index means a reciprocal: $a^{- n} = \frac{1}{a ^{n}}$ a⁻ⁿ = 1/aⁿ
$2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ 2⁻³ = 1/2³ = 1/8 and $5^{2} \div 5^{4} = 5^{- 2} = \frac{1}{25}$ 5² ÷ 5⁴ = 5⁻² = 1/25
A negative index never makes the answer negative.
Higher: on CCEA, zero and negative indices are Higher tier (Unit M7).

Quick check

?What is the value of

4^{- 2}

?

Number · powers and indices

Fractional indices (Higher)

The denominator of a fractional index is a root: $a^{\frac{1}{n}} = n a$ a^(1/n) = n√a, so $1 6^{\frac{1}{2}} = 4$ 16^(1/2) = 4 and $8 1^{\frac{1}{4}} = 3$ 81^(1/4) = 3.
The numerator is a power: $a^{\frac{m}{n}} = (n a)^{m}$ a^(m/n) = (n√a)^m. Take the root first to keep numbers small: $8^{\frac{2}{3}} = 2^{2} = 4$ 8^(2/3) = 2² = 4.
A negative fractional index: reciprocal, then root, then power. $3 2^{- \frac{1}{5}} = \frac{1}{2}$ 32^(−1/5) = 1/2
For a fraction, flip it to deal with the minus sign: $(\frac{a}{b})^{- n} = (\frac{b}{a})^{n}$ (a/b)⁻ⁿ = (b/a)ⁿ

Calculate

?Work out

(\frac{8}{125})^{- \frac{2}{3}}

. Give your answer as a decimal. (Higher)

Mission check

?Fold a 0.1 mm sheet of paper 42 times. Would it reach the Moon, about 384,400 km away?

Mission 6

Are you a billion seconds old yet?

Make a guess: Are you a billion seconds old yet?

Number · standard form

Writing a number in standard form

Standard form is $A \times 1 0^{n}$ A × 10ⁿ, where $1 \leq A < 10$ 1 ≤ A < 10 and $n$ n is an integer.
Big numbers have a positive power: $3450000 = 3.45 \times 1 0^{6}$ 3 450 000 = 3.45 × 10⁶ (the point moves 6 places).
Small numbers have a negative power: $0.000 72 = 7.2 \times 1 0^{- 4}$ 0.000 72 = 7.2 × 10⁻⁴.
One billion is $1 \times 1 0^{9}$ 1 × 10⁹. A calculator may show $3.2 \times 1 0^{- 5}$ 3.2 × 10⁻⁵ as 3.2E-05.
Higher: on CCEA this is called standard index form and is Higher tier (Unit M7).

Worked example Number · standard form

Converting in both directions

Question

(a) Write 0.000 305 in standard form. (b) Write

4.06 \times 1 0^{5}

as an ordinary number.

1
(a) Place the point after the first non-zero digit: $A = 3.05$ A = 3.05
2
(a) The point moves 4 places right, so the power is $- 4$ −4: $3.05 \times 1 0^{- 4}$ 3.05 × 10⁻⁴
3
(b) Multiplying by $1 0^{5}$ 10⁵ moves the point 5 places right: $406000$ 406 000

Answer

(a)

3.05 \times 1 0^{- 4}

(b) 406,000

⚠️ Watch out

counting zeros instead of places moved. 0.000 305 has five zeros in total, but the point moves only 4 places.

Sort it

?Is each number written in standard form?

Tap a statement, then tap the group it belongs in. Then press Check.

Number · standard form

Ordering and calculating

To order, compare the powers first, then $A$ A: $6.1 \times 1 0^{- 3} < 2.4 \times 1 0^{- 2}$ 6.1 × 10⁻³ < 2.4 × 10⁻².
Multiply: multiply the $A$ A values and add the powers.
Divide: divide the $A$ A values and subtract the powers.
Add or subtract: first rewrite both numbers with the same power of 10 (or as ordinary numbers).
Finally adjust so that $1 \leq A < 10$ 1 ≤ A < 10.

Worked example Number · standard form

Multiplying without a calculator

Question

Work out

(6 \times 1 0^{4}) \times (5 \times 1 0^{- 7})

. Give your answer in standard form.

1
Multiply the numbers: $6 \times 5 = 30$ 6 × 5 = 30
2
Add the powers: $1 0^{4} \times 1 0^{- 7} = 1 0^{- 3}$ 10⁴ × 10⁻⁷ = 10⁻³
3
Combine: $30 \times 1 0^{- 3}$ 30 × 10⁻³, which is not yet standard form
4
Adjust: $30 = 3 \times 1 0^{1}$ 30 = 3 × 10¹, so $3 \times 1 0^{1} \times 1 0^{- 3} = 3 \times 1 0^{- 2}$ 3 × 10¹ × 10⁻³ = 3 × 10⁻²

Answer

3 \times 1 0^{- 2}

⚠️ Watch out

stopping at

30 \times 1 0^{- 3}

loses the final mark because

A

is not less than 10.

Quick check

?Work out

(8 \times 1 0^{5}) \div (2 \times 1 0^{- 3})

.

Calculate

?Work out

(4.5 \times 1 0^{4}) + (6 \times 1 0^{3})

. Give your answer as an ordinary number.

Calculate

?A type of cell is

8 \times 1 0^{- 6}

m across. How many of these cells, placed end to end, make a line 1 cm long?

Mission check

?Are you a billion seconds old yet? (A year is about

3.16 \times 1 0^{7}

seconds.)

Mission 7

Eight '80 kg' people in a 640 kg lift. Could it be overloaded?

Make a guess: Eight '80 kg' people in a 640 kg lift. Could it be overloaded?

Number · accuracy and bounds

Decimal places and significant figures

Decimal places (d.p.) count digits after the decimal point: 3.14159 to 2 d.p. is 3.14.
Significant figures (s.f.) count from the first non-zero digit: 0.047 38 to 2 s.f. is 0.047.
Look at the next digit: 5 or more rounds up. Keep place-value zeros: 52,861 to 2 s.f. is 53,000.
Never round in the middle of a calculation: keep full calculator values and round only the final answer.

Quick check

?Round 0.030 96 to 2 significant figures.

Number · accuracy and bounds

Estimating by rounding

To estimate, round every number to 1 significant figure, then work it out by hand.
Use $\approx$ ≈ to show the answer is approximate: $\frac{612 \times 4.87}{0.193} \approx \frac{600 \times 5}{0.2} = 15000$ (612 × 4.87)/0.193 ≈ (600 × 5)/0.2 = 15 000
Dividing by a decimal such as 0.2 makes the answer bigger: $3000 \div 0.2 = 15000$ 3000 ÷ 0.2 = 15 000.
An estimate checks a calculator answer: the exact value here is about 15,400, so a result of 1540 would be wrong.

Calculate

?A school orders 289 books at £5.85 each. Estimate the total cost by rounding each number to 1 significant figure.

£

Number · accuracy and bounds

The bounds of a rounded measurement

80 kg to the nearest 10 kg means anything from 75 kg up to, but not including, 85 kg.
Go half a unit either way: lower bound 75 kg, upper bound 85 kg.
As an inequality: $75 \leq m < 85$ 75 ≤ m < 85. 85 is excluded because it rounds up to 90.
Higher: on CCEA, upper and lower bounds are Higher tier (Units M3 and M4).

Quick check

?A length

l

is 6.4 cm, correct to 1 decimal place. Which shows the possible values of

l

?

Number · accuracy and bounds

Bounds in calculations (Higher)

To find the biggest or smallest possible result, choose the right bound for each value.
Add: upper + upper gives the upper bound. Multiply: upper × upper.
Subtract: upper − lower gives the upper bound (take away as little as possible).
Divide: upper ÷ lower gives the upper bound (divide by as little as possible).
For lower bounds, swap every choice. Then round only at the end, to a sensible accuracy.

Worked example Number · accuracy and bounds

The slowest possible speed (Higher)

Question

A runner covers 100 m, to the nearest metre, in 12.3 s, to 1 d.p. Find the lower bound of the speed, to 3 significant figures.

1
Bounds of the distance: $99.5 \leq d < 100.5$ 99.5 ≤ d < 100.5
2
Bounds of the time: $12.25 \leq t < 12.35$ 12.25 ≤ t < 12.35
3
Smallest speed = smallest distance ÷ largest time: $\frac{99.5}{12.35} = 8.0566 \dots$ 99.5/12.35 = 8.0566…
4
Round at the end: 8.06 m/s

Answer

8.06 m/s

⚠️ Watch out

using both lower bounds,

99.5 \div 12.25

, gives 8.12 m/s, which is not the smallest speed.

Calculate

?A floor tile measures 30 cm by 20 cm, each correct to the nearest centimetre. Work out the upper bound of its area. Give your answer exactly. (Higher)

cm²

Mission check

?A lift can take 640 kg. Eight people step in, each 80 kg to the nearest 10 kg. Could the lift be overloaded?

Mission 8

Halve an A4 sheet and it keeps its shape. Why must that be $2$ √2?

Make a guess: Halve an A4 sheet and it keeps its shape. Why must that be $2$ √2?

Number · surds

What a surd is (Higher)

A surd is a root that cannot be simplified to a whole number or fraction, such as $2$ √2 or $7$ √7. It is irrational, so leaving it as $2$ √2 keeps an answer exact.
$9 = 3$ √9 = 3 is not a surd.
Multiply and divide inside one root: $a \times b = ab$ √a × √b = √ab and $\frac{a}{b} = \frac{a}{b}$ (√a)/(√b) = √(a/b)
So $a \times a = a$ √a × √a = a: $2 \times 2 = 2$ √2 × √2 = 2.
You cannot add inside a root: $9 + 16 = 7$ √9 + √16 = 7, but $25 = 5$ √25 = 5.

Number · surds

Simplifying surds (Higher)

Split off the largest square factor: $12 = 4 \times 3 = 4 \times 3 = 23$ √12 = √(4 × 3) = √4 × √3 = 2√3
Useful squares to look for: 4, 9, 16, 25, 36, 49, 64, 100.
Simplify first, then collect like surds as you would like terms:

200 - 32 = 102 - 42 = 62

Different surds such as $2$ √2 and $3$ √3 cannot be combined by adding.

Quick check

?Simplify

75 + 12

.

Worked example Number · surds

Expanding a bracket with a surd (Higher)

Question

Expand and simplify

(4 - 3)^{2}

.

1
Write it as two brackets: $(4 - 3) (4 - 3)$ (4 − √3)(4 − √3)
2
Multiply every term: $16 - 43 - 43 + 33$ 16 − 4√3 − 4√3 + √3√3
3
Use $33 = 3$ √3√3 = 3: $16 - 83 + 3$ 16 − 8√3 + 3
4
Collect the whole numbers: $19 - 83$ 19 − 8√3

Answer

19 - 83

⚠️ Watch out

(4 - 3)^{2}

is not

16 - 3

; squaring a bracket always gives a middle term.

Calculate

?Expand and simplify

(5 + 2) (5 - 2)

. (Higher)

Number · surds

Rationalising the denominator (Higher)

Rationalise means rewrite a fraction so there is no surd in the denominator.
Multiply top and bottom by the surd: $\frac{10}{5} = \frac{10 5}{5} = 25$ 10/(√5) = (10√5)/5 = 2√5
With a number in front, still multiply by the surd only: $\frac{5}{3 2} = \frac{5 2}{6}$ 5/(3√2) = (5√2)/6
In the same way, $\frac{2}{2} = \frac{2 2}{2} = 2$ 2/(√2) = (2√2)/2 = √2.

Calculate

?A rectangle has area 18 cm² and width

3

cm. Its length is

a 3

cm. Find

a

. (Higher)

Mission check

?An A-size sheet has sides in the ratio

r : 1

. Half a sheet is the same shape. Why must

r = 2

?

Mission 9

A 4-digit PIN: how many guesses to be certain of getting it?

Make a guess: A 4-digit PIN: how many guesses to be certain of getting it?

Number · counting strategies

Listing systematically

Systematic listing means following a fixed order so that nothing is missed and nothing is repeated.
Fix the first choice, run through every option for the second, then move the first choice on.
A table or grid works well for two choices: rows for one, columns for the other.
Look for a structure you can count without writing everything out: the 2-digit codes 00, 01, …, 99 go from 0 to 99, so there are 100 of them.

Worked example Number · counting strategies

Listing arrangements

Question

Cards show the digits 1, 4 and 7. Each card is used once to make a 3-digit number. List all the numbers. How many are greater than 500?

1
First digit 1, then the other two both ways: 147, 174
2
First digit 4: 417, 471
3
First digit 7: 714, 741
4
That is 6 numbers. Those above 500 start with 7: 714 and 741

Answer

6 numbers; 2 are greater than 500

⚠️ Watch out

listing at random (147, 741, 417, …) is how numbers get missed or written twice. Fix the first digit each time.

Calculate

?A bag holds four counters numbered 1, 1, 2 and 5. Two counters are taken out and their numbers are added. How many different totals are possible?

Number · counting strategies

The product rule for counting (Higher)

If there are $m$ m ways to do one thing and, for each of those, $n$ n ways to do another, there are $m \times n$ m × n ways to do both.
It extends to more stages: multiply the number of choices at each stage. 3 starters, 4 mains and 2 desserts give $3 \times 4 \times 2 = 24$ 3 × 4 × 2 = 24 meals.
If an item can't be repeated, each stage has one fewer choice than the one before.
Deal with any restricted position first (for example, "the first digit can't be 0").

Worked example Number · counting strategies

Counting codes (Higher)

Question

A code is 2 letters (A to Z) followed by 3 digits (0 to 9). Letters and digits may repeat. How many different codes are there?

1
Choices for each letter: 26, and for each digit: 10
2
Multiply the choices: $26 \times 26 \times 10 \times 10 \times 10$ 26 × 26 × 10 × 10 × 10
3
Work it out: $676 \times 1000 = 676000$ 676 × 1000 = 676 000

Answer

676,000 codes

⚠️ Watch out

adding the choices (

26 + 26 + 10 + 10 + 10 = 82

) counts single options, not whole codes. Each stage multiplies.

Quick check

?(Higher) How many 3-digit numbers (100 to 999) have no repeated digits?

Calculate

?(Higher) In a league of 15 teams, every team plays every other team exactly once. How many matches are played?

Mission check

?A 4-digit PIN uses the digits 0–9, and digits can repeat. How many guesses make you certain to get it?

Mission 10

Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?

Make a guess: Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?

Ratio · simplifying and sharing

Ratio notation and simplest form

A ratio $a : b$ a : b compares the sizes of parts: 3 blue : 2 yellow means every 3 tins of blue go with 2 tins of yellow.
Simplify by dividing every part by the same number (the HCF): $12 : 18 = 2 : 3$ 12 : 18 = 2 : 3. Three-part ratios work the same way.
Write both quantities in the same units first, then drop the units.
The form $1 : n$ 1 : n: divide both parts by the first part. $n$ n may be a decimal.

Worked example Ratio · simplifying and sharing

Simplest form and the form 1 : n

Question

Write

50

cm

: 1.5

m in its simplest form. Then write

12 : 30

in the form

1 : n

.

1
Change to the same units: $1.5$ 1.5 m $= 150$ = 150 cm, so the ratio is $50 : 150$ 50 : 150
2
Divide both parts by the HCF, 50: $50 : 150 = 1 : 3$ 50 : 150 = 1 : 3
3
For $12 : 30$ 12 : 30, divide both parts by 12: $1 : \frac{30}{12}$ 1 : 30/12
4
Work out $n$ n: $\frac{30}{12} = 2.5$ 30/12 = 2.5

Answer

1 : 3

and

1 : 2.5

⚠️ Watch out

simplifying

50 : 1.5

without converting units gives a ratio that means nothing.

Quick check

?Write

45 : 60 : 75

in its simplest form.

Ratio · simplifying and sharing

Sharing in a ratio

To share an amount in the ratio $a : b$ a : b, add the parts to get the total number of parts.
One part = amount ÷ total parts. Multiply to get each share. Check the shares add back to the amount.
If you know one share instead (part : part), divide it by its own number of parts to find one part.
If you know the difference between two shares, divide it by the difference in their parts.

Worked example Ratio · simplifying and sharing

Sharing in a three-part ratio

Question

Share £416 in the ratio

4 : 3 : 1

.

1
Add the parts: $4 + 3 + 1 = 8$ 4 + 3 + 1 = 8 parts
2
Find one part: $416 \div 8 = 52$ 416 ÷ 8 = 52, so one part is £52
3
Multiply each part: $4 \times 52 = 208$ 4 × 52 = 208, $3 \times 52 = 156$ 3 × 52 = 156, $1 \times 52 = 52$ 1 × 52 = 52
4
Check the total: $208 + 156 + 52 = 416$ 208 + 156 + 52 = 416

Answer

£208, £156 and £52

⚠️ Watch out

dividing £416 by 4, 3 and 1 in turn (instead of by the 8 parts) gives shares that add to far more than £416.

Calculate

?In a school choir, the ratio of boys to girls is

3 : 5

. There are 24 more girls than boys. How many students are in the choir altogether?

Ratio · simplifying and sharing

Ratios, fractions and combining ratios

$3 : 2$ 3 : 2 makes 5 parts: blue is $\frac{3}{5}$ 3/5 of the paint and yellow is $\frac{2}{5}$ 2/5.
Yellow is $\frac{2}{3}$ 2/3 of blue.
Mixes in this ratio lie on $y = \frac{2}{3} x$ y = 2/3x (blue $x$ x, yellow $y$ y): a straight line through the origin.
To combine $a : b$ a : b and $b : c$ b : c, make the $b$ b parts equal: $2 : 3$ 2 : 3 and $4 : 5$ 4 : 5 become $8 : 12$ 8 : 12 and $12 : 15$ 12 : 15, so $8 : 12 : 15$ 8 : 12 : 15.

Quick check

?In a box of pens, red : blue

= 3 : 4

and blue : green

= 6 : 5

. Find red : green in its simplest form.

Calculate

?A concrete mix uses cement, sand and gravel in the ratio

1 : 2 : 4

by mass. A builder has 15 kg of sand and plenty of cement and gravel. What is the greatest mass of concrete mix she can make, using all the sand?

kg

Mission check

?Paint is mixed 3 blue : 2 yellow. You have 8 L of blue and plenty of yellow. How much paint can you make?

Mission 11

A price goes up 20%, then down 20%. Is it back where it started?

Make a guess: A price goes up 20%, then down 20%. Is it back where it started?

Ratio and proportion · percentages

Percentages as multipliers

Per cent means "parts per hundred": $35% = \frac{35}{100} = 0.35$ 35% = 35/100 = 0.35. Percentages can exceed 100%: $150% = 1.5$ 150% = 1.5.
To find 35% of an amount, multiply by 0.35.
To increase by 15%, multiply by $1 + 0.15 = 1.15$ 1 + 0.15 = 1.15. To decrease by 15%, multiply by $1 - 0.15 = 0.85$ 1 − 0.15 = 0.85.
Two changes in a row: multiply by each multiplier in turn. Each change is a percentage of the amount at that moment.

Worked example Ratio and proportion · percentages

Increasing with a multiplier

Question

A bike costs £240. Its price increases by 15%. Find the new price.

1
Write the multiplier: $100% + 15% = 115%$ 100% + 15% = 115%, so multiply by 1.15
2
Multiply: $240 \times 1.15 = 276$ 240 × 1.15 = 276
3
Check with the "find and add" method: $15%$ 15% of 240 is $36$ 36, and $240 + 36 = 276$ 240 + 36 = 276

Answer

£276

⚠️ Watch out

multiplying by 0.15 gives only the increase (£36), not the new price.

Quick check

?Which single multiplier decreases an amount by 7%?

Ratio and proportion · percentages

Percentage change and one quantity as a percentage of another

To write $A$ A as a percentage of $B$ B: $\frac{A}{B} \times 100$ A/B × 100. For example, 18 out of 24 is $\frac{18}{24} \times 100 = 75%$ 18/24 × 100 = 75%.
Percentage change $= \frac{change}{original} \times 100$ = change/original × 100. Always divide by the original value.
Percentage profit or loss uses the cost price as the original. To compare two scores out of different totals, turn both into percentages.
Simple interest: the same interest is paid each year, worked out on the original amount only.

Calculate

?A trader buys a bike for £350 and sells it for £294. Find her percentage loss.

Ratio and proportion · percentages

Reverse percentages (Higher)

In a reverse percentage (original value) problem you know the amount after the change.
Write the multiplier for the change, then divide by it: original $=$ = new amount $\div$ ÷ multiplier.
After a 30% reduction, the price is 70% of the original, so divide by 0.7.
Check by applying the change to your answer.

Worked example Ratio and proportion · percentages

Finding the original price (Higher)

Question

In a sale, prices are reduced by 30%. A coat costs £42 in the sale. Find its price before the sale.

1
Write the multiplier for the change: $100% - 30% = 70%$ 100% − 30% = 70%, so $\times 0.7$ × 0.7
2
Set up: original $\times 0.7 = 42$ × 0.7 = 42
3
Divide by the multiplier: $42 \div 0.7 = 60$ 42 ÷ 0.7 = 60
4
Check: $30%$ 30% of 60 is $18$ 18, and $60 - 18 = 42$ 60 − 18 = 42

Answer

£60

⚠️ Watch out

adding 30% to £42 gives £54.60. The 30% was taken off the original price, not the sale price.

Calculate

?After a 20% tax is added, the price of a printer is £186. Find the price before the tax was added. (Higher)

£

Calculate

?Rana invests £2,400 at 3.5% per year simple interest. What is the smallest whole number of years after which her total interest is at least £500?

Mission check

?A price goes up by 20%, then down by 20%. Is it back where it started?

Mission 12

Money grows at 5% a year compound interest. When has it doubled?

Make a guess: Money grows at 5% a year compound interest. When has it doubled?

Ratio and proportion · growth and decay

Compound interest

Simple interest is the same every year: it is always worked out on the original amount.
Compound interest is added to the balance, so next year's interest is worked out on a bigger amount: interest earns interest.
Each year at 5%, multiply by 1.05. After $n$ n years: amount $=$ = original $\times 1.0 5^{n}$ × 1.05ⁿ.
The interest earned is the final amount minus the original amount.

Worked example Ratio and proportion · growth and decay

Compound interest over three years

Question

£1,000 is invested at 5% per year compound interest. Find the amount after 3 years, and the interest earned.

1
Write the multiplier for a 5% increase: $1.05$ 1.05
2
Apply it once for each year: $1000 \times 1.0 5^{3}$ 1000 × 1.05³
3
Work it out: $1000 \times 1.157625 = 1157.625$ 1000 × 1.157625 = 1157.625, so £1,157.63 to the nearest penny
4
Subtract the original amount: $1157.63 - 1000 = 157.63$ 1157.63 − 1000 = 157.63

Answer

£1,157.63, so the interest earned is £157.63

⚠️ Watch out

1000 \times 0.05 \times 3 = £150

is simple interest. Compound interest is more, because years 2 and 3 earn interest on interest.

Calculate

?£2,500 is invested at 3.2% per year compound interest. Find the amount after 6 years. Give your answer to the nearest penny.

£

Ratio and proportion · growth and decay

Depreciation and repeated changes

Depreciation is a loss in value each year. A 15% loss each year means multiplying by $0.85$ 0.85 each year.
If the rate changes from year to year, use each year's multiplier in turn: $\times 0.7$ × 0.7, then $\times 0.8$ × 0.8, then $\times 0.85$ × 0.85.
Never add the percentages: each one is a percentage of the value at the start of that year.
A value that keeps falling by a percentage gets smaller every year but never reaches zero.

Quick check

?A car is worth £15,000 when new. It loses 30% of its value in the first year, 20% in the second year and 15% in the third year. What is it worth after 3 years?

Ratio and proportion · growth and decay

Growth and decay formula (Higher)

For a percentage change of $r %$ r% each time period, the multiplier is $m = 1 + \frac{r}{100}$ m = 1 + r/100 (growth) or $m = 1 - \frac{r}{100}$ m = 1 − r/100 (decay).
After $n$ n periods: $A = P \times m^{n}$ A = P × mⁿ where $P$ P is the starting amount.
A town of 12,000 people growing by 2% a year has about $12000 \times 1.0 2^{10} \approx 14, 628$ 12000 × 1.02¹⁰ ≈ 14,628 people after 10 years.
To find when a value passes a target, try whole values of $n$ n until it does, and interpret the answer in context.

Worked example Ratio and proportion · growth and decay

When does a population fall below a target? (Higher)

Question

A population of 8,000 birds decreases by 4% each year. After how many whole years is it first below 6,000?

1
Write the multiplier for a 4% decrease: $m = 0.96$ m = 0.96
2
Write the formula: population after $n$ n years $= 8000 \times 0.9 6^{n}$ = 8000 × 0.96ⁿ
3
Try $n = 7$ n = 7: $8000 \times 0.9 6^{7} = 6011.6$ 8000 × 0.96⁷ = 6011.6, which is not below 6,000
4
Try $n = 8$ n = 8: $8000 \times 0.9 6^{8} = 5771.1$ 8000 × 0.96⁸ = 5771.1, which is below 6,000

Answer

after 8 years

⚠️ Watch out

2000 \div (4% of 8000) = 6.25

assumes the same 320 birds are lost every year. The yearly loss shrinks as the population shrinks.

Calculate

?Lea has £4,000 to invest for 10 years. Account A pays 2.5% per year compound interest. Account B pays 3% per year simple interest. How much more is the better account worth after 10 years? Give your answer to the nearest penny.

£

Mission check

?£1,000 grows at 5% a year compound interest. When has it first doubled?

Mission 13

4 builders take 6 days. How long would 8 builders take?

Make a guess: 4 builders take 6 days. How long would 8 builders take?

Ratio and proportion · direct and inverse proportion

Direct proportion

Two quantities are in direct proportion if multiplying one by a number multiplies the other by the same number: twice the fabric costs twice as much.
Unitary method: find the value for one unit, then multiply.
The graph of direct proportion is a straight line through the origin.
Higher: in algebra, $y$ y is directly proportional to $x$ x, written $y \propto x$ y ∝ x, means $y = k x$ y = kx for a constant $k$ k.

Worked example Ratio and proportion · direct and inverse proportion

The unitary method

Question

5 m of fabric costs £32. Find the cost of 8 m of the same fabric.

1
Find the cost of one metre: $32 \div 5 = 6.40$ 32 ÷ 5 = 6.40, so £6.40 per metre
2
Multiply by the number of metres: $6.40 \times 8 = 51.20$ 6.40 × 8 = 51.20
3
Check with a scale factor: $8$ 8 m is $\frac{8}{5} = 1.6$ 8/5 = 1.6 times as much, and $32 \times 1.6 = 51.20$ 32 × 1.6 = 51.20

Answer

£51.20

⚠️ Watch out

writing £51.2 loses the mark for money. Give two decimal places.

Ratio and proportion · direct and inverse proportion

Inverse proportion

Two quantities are in inverse proportion if multiplying one by a number divides the other by it: double the speed, half the time.
Their product stays the same: 60 km/h for 3 hours or 90 km/h for $180 \div 90 = 2$ 180 ÷ 90 = 2 hours.
Method: multiply to find the fixed total (pump-hours, worker-days), then divide.
Higher: in algebra, $y \propto \frac{1}{x}$ y ∝ 1/x means $y = \frac{k}{x}$ y = k/x: " $y$ y is proportional to $\frac{1}{x}$ 1/x". Its graph is a curve that never meets the axes.

Quick check

?6 identical pumps empty a tank in 10 hours. How long would 4 of these pumps take?

Sort it

?Is

y

directly proportional to

x

, inversely proportional to

x

, or neither?

Tap a statement, then tap the group it belongs in. Then press Check.

Ratio and proportion · direct and inverse proportion

Proportion equations with powers and roots (Higher)

Write the statement as an equation with a constant $k$ k: $y \propto x^{2}$ y ∝ x² gives $y = k x^{2}$ y = kx²; $y \propto x$ y ∝ √x gives $y = k x$ y = k√x; $y \propto \frac{1}{x ^{2}}$ y ∝ 1/x² gives $y = \frac{k}{x ^{2}}$ y = k/x².
Substitute one known pair of values to find $k$ k.
Write the equation with that value of $k$ k, then use it to find the unknown.
For example, if $y \propto x$ y ∝ √x and $y = 12$ y = 12 when $x = 9$ x = 9: $12 = k \times 3$ 12 = k × 3, so $k = 4$ k = 4, and when $x = 25$ x = 25, $y = 4 \times 5 = 20$ y = 4 × 5 = 20.

Worked example Ratio and proportion · direct and inverse proportion

Direct proportion to a square (Higher)

Question

y

is directly proportional to

x^{2}

. When

x = 3

,

y = 45

. Find

y

when

x = 5

.

1
Write the equation: $y = k x^{2}$ y = kx²
2
Substitute $x = 3$ x = 3, $y = 45$ y = 45: $45 = k \times 9$ 45 = k × 9, so $k = 5$ k = 5
3
Write the equation with $k$ k: $y = 5 x^{2}$ y = 5x²
4
Substitute $x = 5$ x = 5: $y = 5 \times 25 = 125$ y = 5 × 25 = 125

Answer

y = 125

⚠️ Watch out

scaling

45

by

\frac{5}{3}

(to get 75) treats

y

as proportional to

x

, not to

x^{2}

.

Calculate

?The force

F

between two small charged spheres is inversely proportional to the square of the distance

d

between them. When

d = 3

,

F = 20

. Find

F

when

d = 4

. Give your answer as a decimal. (Higher)

Calculate

?A farmer has enough feed to last 12 horses for 15 days. She buys 3 more horses. Each horse eats the same amount each day. How many days will the feed now last?

days

Mission check

?4 builders take 6 days to build a wall. How long would 8 builders take?

Mission 14

A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?

Make a guess: A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?

Measures · compound measures and units

Converting area and volume units

Length: $1 m = 100 cm$ 1 m = 100 cm, $1 cm = 10 mm$ 1 cm = 10 mm, $1 km = 1000 m$ 1 km = 1000 m.
Area: square the length factor. A 1 m square is 100 cm by 100 cm, so $1 m^{2} = 10 0^{2} = 10, 000 cm^{2}$ 1 m² = 100² = 10,000 cm².
Volume: cube the length factor. $1 m^{3} = 10 0^{3} cm^{3}$ 1 m³ = 100³ cm³, and $1 cm^{3} = 1000 mm^{3}$ 1 cm³ = 1000 mm³.
Capacity: $1 litre = 1000 cm^{3}$ 1 litre = 1000 cm³ and $1 ml = 1 cm^{3}$ 1 ml = 1 cm³.
Know the common approximate equivalents: 1 inch ≈ 2.5 cm, 5 miles ≈ 8 km, 1 kg ≈ 2.2 pounds, 1 gallon ≈ 4.5 litres.

Quick check

?Convert

3.5 m^{2}

to

cm^{2}

.

Measures · compound measures and units

Speed, distance and time

$speed = \frac{distance}{time}$ speed = distance/time so distance $=$ = speed × time and time $=$ = distance ÷ speed. Average speed $=$ = total distance ÷ total time.
Minutes ÷ 60 gives hours: 2 h 15 min $= 2.25$ = 2.25 h. 0.4 h $= 0.4 \times 60 = 24$ = 0.4 × 60 = 24 min.
$72 km/h$ 72 km/h is $72, 000$ 72,000 m in 3,600 s, which is $20 m/s$ 20 m/s.
24-hour clock 15 15 is 3.15 pm on the 12-hour clock.

Worked example Measures · compound measures and units

Average speed with hours and minutes

Question

A cyclist rides 54 km in 2 hours 15 minutes. Find her average speed in km/h.

1
Change the time to hours: $15 \div 60 = 0.25$ 15 ÷ 60 = 0.25, so the time is $2.25$ 2.25 hours
2
Use speed $=$ = distance ÷ time: $54 \div 2.25$ 54 ÷ 2.25
3
Work it out: $54 \div 2.25 = 24$ 54 ÷ 2.25 = 24

Answer

24 km/h

⚠️ Watch out

typing 2 h 15 min as 2.15 gives

54 \div 2.15 \approx 25.12

km/h. 15 minutes is 0.25 of an hour, not 0.15.

Calculate

?A train leaves at 09 47 and travels 168 km at an average speed of 96 km/h. How many minutes does the journey take?

minutes

Measures · compound measures and units

Density, pressure and other rates

$density = \frac{mass}{volume}$ density = mass/volume in g/cm³ or kg/m³.
$pressure = \frac{force}{area}$ pressure = force/area in newtons per square metre (N/m²), also called pascals (Pa).
Higher: pressure is Higher tier only; density is on both tiers.
Other rates work the same way: pay per hour, cost per kg, litres per minute (flow rate), km per litre (fuel use), people per km² (population density).

Worked example Measures · compound measures and units

Finding a density

Question

A metal block measures 5 cm by 4 cm by 2 cm. Its mass is 316 g. Find the density of the metal.

1
Find the volume: $5 \times 4 \times 2 = 40 cm^{3}$ 5 × 4 × 2 = 40 cm³
2
Use density $=$ = mass ÷ volume: $316 \div 40$ 316 ÷ 40
3
Work it out: $316 \div 40 = 7.9$ 316 ÷ 40 = 7.9

Answer

7.9 g/cm^{3}

⚠️ Watch out

the units come from the division: grams ÷ cm³ gives g/cm³. Dividing volume by mass gives a number with no meaning here.

Calculate

?A water tank is a cuboid 1.2 m long, 0.8 m wide and 0.5 m deep. It is empty. Water flows in at 8 litres per minute. How many minutes does it take to fill the tank?

minutes

Calculate

?Oak has a density of

0.75 g/cm^{3}

. An oak plank is 2 m long, 20 cm wide and 3 cm thick. Find its mass in kg.

kg

Mission check

?A 1 m by 1 m by 1 m tank is full of water (1 cm³ of water has a mass of about 1 g). Could two people lift it?

Mission 15

The pizza twice as wide costs three times as much. Good deal?

Make a guess: The pizza twice as wide costs three times as much. Good deal?

Ratio · proportion in everyday life

Best buys

To find the best buy, compare the same unit: price per gram, or grams per penny.
Convert first so the units match: 1.2 kg = 1200 g; 500 ml = 0.5 L.
Compare what you actually get. For pizzas that is area, $π r^{2}$ π r², not diameter.
A 10 cm pizza has area $π \times 5^{2} \approx 79$ π × 5² ≈ 79 cm². A 30 cm pizza has $π \times 1 5^{2} \approx 707$ π × 15² ≈ 707 cm², 9 times as much: triple the width, nine times the pizza.

Worked example Ratio · proportion in everyday life

Which pack is better value?

Question

Cereal comes in 750 g for £2.85 or 1.2 kg for £4.20. Which is better value?

1
Use the same units: 1.2 kg = 1200 g, £2.85 = 285p, £4.20 = 420p
2
Price per gram, small pack: $285 \div 750 = 0.38$ 285 ÷ 750 = 0.38p per g
3
Price per gram, large pack: $420 \div 1200 = 0.35$ 420 ÷ 1200 = 0.35p per g
4
Compare: the lower price per gram is better value

Answer

The 1.2 kg pack is better value (0.35p per g against 0.38p per g).

⚠️ Watch out

say which unit you compared. If you work out grams per penny instead, the bigger number wins.

Quick check

?Juice is sold as 1.5 L for £1.80, 2 L for £2.30 or 500 ml for £0.65. Which is the best value?

Ratio · proportion in everyday life

Recipes and exchange rates

Quantities in direct proportion keep the same ratio. Scale them by one multiplier.
Unitary method: find one, then multiply. A recipe for 6 needs 450 g flour; for 1 person, 75 g; for 4 people, 300 g.
An exchange rate is a multiplier. If £1 = €1.15, multiply pounds by 1.15 to get euros, and divide euros by 1.15 to get pounds.
Sense-check: you should get more euros than pounds at this rate.

Calculate

?Exam-style. A recipe for 8 people needs 600 g of flour and 4 eggs. Ana has 1 kg of flour and 5 eggs. What is the greatest number of people she can make the recipe for?

Calculate

?Exam-style. Sam changes £400 to euros when £1 = €1.15. Sam spends €385 on holiday, then changes the euros left back to pounds when £1 = €1.25. How many pounds does Sam get back?

£

Ratio · proportion in everyday life

Map scales

A map scale 1 : 25,000 means 1 cm on the map is 25,000 cm in real life.
Change units: 25,000 cm = 250 m, so 4 cm on the map is 1 km.
Map to real: multiply by 25,000. Real to map: divide by 25,000. Change the units at the end (100 cm = 1 m, 100,000 cm = 1 km).
Higher: areas scale by the square. 1 cm² on this map is 250 m × 250 m = 62,500 m².

Calculate

?A walking map has scale 1 : 25,000. Two farms are 7.4 cm apart on the map. How far apart are they in real life, in kilometres?

km

Quick check

?A road is 3 km long. How long is it on a map with scale 1 : 50,000?

Mission check

?A 20 cm pizza costs £6 and a 40 cm pizza costs £18. Is the large pizza a good deal?

Mission 16

A square grows 3 cm each way. Is its new area just $x^{2} + 9$ x² + 9?

Make a guess: A square grows 3 cm each way. Is its new area just $x^{2} + 9$ x² + 9?

Algebra · expressions

Terms, expressions and like terms

An expression is a collection of terms such as $5 a$ 5a, $- 9 b$ −9b or $x^{2}$ x². The number in front of a letter is its coefficient.
Algebra shorthand: $ab$ ab means $a \times b$ a × b, $3 y$ 3y means $y + y + y$ y + y + y, $a^{2} b$ a²b means $a \times a \times b$ a × a × b, $\frac{a}{b}$ a/b means $a \div b$ a ÷ b.
Like terms have exactly the same letters and powers. Collect them by adding their coefficients: $2 a + 3 b + 5 a - 9 b = 7 a - 6 b$ 2a + 3b + 5a − 9b = 7a − 6b.
$x^{2}$ x² and $x$ x are not like terms: $3 x^{2} + 2 x$ 3x² + 2x cannot be simplified.
To expand a single bracket, multiply every term inside by the term outside: $3 x (2 x - 4 y) = 6 x^{2} - 12 x y$ 3x(2x − 4y) = 6x² − 12xy and $- 4 (x - 3) = - 4 x + 12$ −4(x − 3) = −4x + 12.

Quick check

?Expand and simplify

3 (2 x - 1) - 2 (x - 4)

.

Algebra · expressions

Expanding two brackets

Multiply every term in the first bracket by every term in the second. A grid keeps track: four boxes, four products.
$(x + 5) (x + 2)$ (x + 5)(x + 2): the boxes are $x^{2}$ x², $2 x$ 2x, $5 x$ 5x and $10$ 10. Collect the like terms: $x^{2} + 7 x + 10$ x² + 7x + 10.
The grid is also an area picture: a rectangle $(x + 5)$ (x + 5) by $(x + 2)$ (x + 2) splits into four smaller rectangles.
Higher: for CCEA, multiplying two linear expressions is Higher tier (Unit M3).

Worked example Algebra · expressions

Expanding and simplifying two brackets (Higher)

Question

Expand and simplify

(2 x + 1) (x - 4)

.

1
Multiply the first terms: $2 x \times x = 2 x^{2}$ 2x × x = 2x²
2
Multiply the outer and inner pairs: $2 x \times (- 4) = - 8 x$ 2x × (−4) = −8x and $1 \times x = x$ 1 × x = x
3
Multiply the last terms: $1 \times (- 4) = - 4$ 1 × (−4) = −4
4
Collect the like terms: $- 8 x + x = - 7 x$ −8x + x = −7x

Answer

2 x^{2} - 7 x - 4

⚠️ Watch out

the sign belongs to the term after it. The

- 4

in the second bracket makes two of the four products negative.

Calculate

?Expand

(x - 6)^{2}

and write it in the form

x^{2} + b x + c

. Enter the value of

b

(exact).

Algebra · expressions

Factorising: expanding in reverse

Factorise means write as a product. Take out the highest common factor of every term: $9 x^{2} + 15 x y = 3 x (3 x + 5 y)$ 9x² + 15xy = 3x(3x + 5y). "Factorise fully" means nothing more can come out.
A quadratic $x^{2} + b x + c$ x² + bx + c factorises to $(x + p) (x + q)$ (x + p)(x + q) where $p$ p and $q$ q multiply to $c$ c and add to $b$ b: $x^{2} + 7 x + 10 = (x + 5) (x + 2)$ x² + 7x + 10 = (x + 5)(x + 2).
Difference of two squares: $x^{2} - a^{2} = (x - a) (x + a)$ x² − a² = (x − a)(x + a), so $x^{2} - 16 = (x - 4) (x + 4)$ x² − 16 = (x − 4)(x + 4).
Check any factorisation by expanding it again.
Higher: for CCEA, factorising quadratics and the difference of two squares are Higher tier (Units M3 and M4).

Quick check

?Factorise

x^{2} + 2 x - 24

.

Worked example Algebra · expressions

Factorising $a x^{2} + b x + c$ ax² + bx + c (Higher)

Question

Factorise

6 x^{2} - 5 x - 6

.

1
Multiply $a$ a by $c$ c: $6 \times (- 6) = - 36$ 6 × (−6) = −36
2
Find two numbers that multiply to $- 36$ −36 and add to $- 5$ −5: $- 9$ −9 and $4$ 4
3
Split the middle term with them: $6 x^{2} - 9 x + 4 x - 6$ 6x² − 9x + 4x − 6
4
Factorise each pair: $3 x (2 x - 3) + 2 (2 x - 3)$ 3x(2x − 3) + 2(2x − 3)
5
Take out the common bracket: $(2 x - 3) (3 x + 2)$ (2x − 3)(3x + 2)

Answer

(2 x - 3) (3 x + 2)

⚠️ Watch out

check by expanding.

(2 x - 3) (3 x + 2) = 6 x^{2} + 4 x - 9 x - 6 = 6 x^{2} - 5 x - 6

.

Calculate

?A square photo of side

x

cm sits in a square frame of side

(x + 4)

cm. The visible frame (the outer square minus the photo) has an area of 200 cm². Form an expression for the frame's area, simplify it, and find

x

(exact).

Mission check

?A square photo of side

x

cm is made 3 cm wider and 3 cm taller. Which expression gives its new area in cm²?

Mission 17

Is there one temperature that reads the same in °C and °F?

Make a guess: Is there one temperature that reads the same in °C and °F?

Algebra · formulae

Expressions, equations, formulae

An expression has no equals sign: $3 x + 5$ 3x + 5.
An equation is true only for particular values of the letter: $3 x + 5 = 20$ 3x + 5 = 20 is true only when $x = 5$ x = 5.
A formula links two or more quantities, so you can work one out from the others: $v = u + a t$ v = u + at.
Higher: an identity is true for every value of the letter. It is often written with $\equiv$ ≡: $2 (x + 3) \equiv 2 x + 6$ 2(x + 3) ≡ 2x + 6.

Sort it

?Sort each line into the right group.

Tap a statement, then tap the group it belongs in. Then press Check.

Algebra · formulae

Substituting into formulae

Substitute means replace each letter by its value, then follow the order of operations: brackets, powers, then $\times$ × and $\div$ ÷, then $+$ + and $-$ −.
Put negative numbers in brackets: if $x = - 3$ x = −3, then $x^{2} = (- 3)^{2} = 9$ x² = (−3)² = 9 and $2 x^{2} = 2 \times 9 = 18$ 2x² = 2 × 9 = 18.
Watch the minus: $5 - 2 x = 5 - 2 \times (- 3) = 5 + 6 = 11$ 5 − 2x = 5 − 2 × (−3) = 5 + 6 = 11.
A formula in words works the same way: write it with letters first, then substitute.

Worked example Algebra · formulae

Substituting into a kinematics formula

Question

Use

s = u t + \frac{1}{2} a t^{2}

to find

s

when

u = 12

,

t = 3

and

a = - 4

.

1
Substitute, keeping the negative in brackets: $s = 12 \times 3 + \frac{1}{2} \times (- 4) \times 3^{2}$ s = 12 × 3 + 1/2 × (−4) × 3²
2
Work out the power first: $3^{2} = 9$ 3² = 9
3
Work out each product: $12 \times 3 = 36$ 12 × 3 = 36 and $\frac{1}{2} \times (- 4) \times 9 = - 18$ 1/2 × (−4) × 9 = −18
4
Add: $36 + (- 18) = 18$ 36 + (−18) = 18

Answer

s = 18

⚠️ Watch out

only

t

is squared. Squaring

a t

gives

\frac{1}{2} \times 144 = 72

and the wrong answer 108.

Calculate

?The formula

F = 1.8 C + 32

converts a temperature

C

in °C to

F

in °F. Find

F

when

C = - 25

(exact).

Algebra · formulae

Changing the subject

The subject is the letter on its own on one side, like $v$ v in $v = u + a t$ v = u + at.
To change the subject, undo the operations on the letter you want, in reverse order, doing the same to both sides.
$v = u + a t$ v = u + at: subtract $u$ u, then divide by $a$ a, so $t = \frac{v - u}{a}$ t = (v − u)/a.
$y = 3 x - 2$ y = 3x − 2: add 2, then divide by 3, so $x = \frac{y + 2}{3}$ x = (y + 2)/3.
Higher: if the subject is squared, undo the square last with a square root: $A = π r^{2}$ A = π r² gives $r = \frac{A}{π}$ r = √(A/π).

Worked example Algebra · formulae

Rearranging the temperature formula

Question

Make

C

the subject of

F = \frac{9 C}{5} + 32

.

1
Subtract 32 from both sides: $F - 32 = \frac{9 C}{5}$ F − 32 = 9C/5
2
Multiply both sides by 5: $5 (F - 32) = 9 C$ 5(F − 32) = 9C
3
Divide both sides by 9: $C = \frac{5 ( F - 32 )}{9}$ C = (5(F − 32))/9
4
Check with a known value: $F = 212$ F = 212 gives $C = \frac{5 \times 180}{9} = 100$ C = (5 × 180)/9 = 100

Answer

C = \frac{5 ( F - 32 )}{9}

⚠️ Watch out

the bracket matters.

\frac{5 F - 32}{9}

subtracts 32 after multiplying by 5, which is a different formula.

Worked example Algebra · formulae

When the subject appears twice (Higher)

Question

Make

x

the subject of

a x + 5 = 3 x + b

.

1
Get every $x$ x term on one side, everything else on the other: $a x - 3 x = b - 5$ ax − 3x = b − 5
2
Factorise out $x$ x: $x (a - 3) = b - 5$ x(a − 3) = b − 5
3
Divide by the bracket: $x = \frac{b - 5}{a - 3}$ x = (b − 5)/(a − 3)
4
Check with numbers, e.g. $a = 7$ a = 7, $b = 13$ b = 13: $x = 2$ x = 2, and $7 \times 2 + 5 = 19 = 3 \times 2 + 13$ 7 × 2 + 5 = 19 = 3 × 2 + 13

Answer

x = \frac{b - 5}{a - 3}

⚠️ Watch out

you can't divide by

a

and by 3 separately. Collect the

x

terms, then factorise: that is the only way to get one

x

.

Calculate

?A taxi charges a fixed £3.50 plus £1.80 for every mile. Write a formula for the cost

C

(in £) of a journey of

m

miles. A journey costs £26. How many miles was it? Give your answer as a decimal.

miles

Mission check

?Is there one temperature that reads the same in °C and °F? Use

F = 1.8 C + 32

.

Mission 18

When does the pricier phone plan become the cheaper one?

Make a guess: When does the pricier phone plan become the cheaper one?

Algebra · linear equations and inequalities

Solving linear equations

Keep the equation balanced: whatever you do to one side, do to the other.
Brackets: expand them first. $7 (x + 3) = 5 x - 8$ 7(x + 3) = 5x − 8 becomes $7 x + 21 = 5 x - 8$ 7x + 21 = 5x − 8.
Unknown on both sides: subtract the smaller $x$ x term from both sides so $x$ x appears once: $2 x + 21 = - 8$ 2x + 21 = −8, so $x = - 14.5$ x = −14.5.
Fractions: multiply every term by the denominator: $\frac{4 x + 5}{2} = 3$ (4x + 5)/2 = 3 gives $4 x + 5 = 6$ 4x + 5 = 6, so $x = 0.25$ x = 0.25.

Worked example Algebra · linear equations and inequalities

Brackets and the unknown on both sides

Question

Solve

5 (x - 1) = 4 - x

.

1
Expand the bracket: $5 x - 5 = 4 - x$ 5x − 5 = 4 − x
2
Add $x$ x to both sides: $6 x - 5 = 4$ 6x − 5 = 4
3
Add 5 to both sides: $6 x = 9$ 6x = 9
4
Divide both sides by 6: $x = 1.5$ x = 1.5

Answer

x = 1.5

⚠️ Watch out

check by substituting.

5 (1.5 - 1) = 2.5

and

4 - 1.5 = 2.5

, so both sides agree.

Algebra · linear equations and inequalities

Forming an equation from words

Choose a letter for the unknown and say what it stands for: "let $m$ m be the number of months".
Write each quantity as an expression in that letter. A fixed charge plus a rate per unit is "fixed + rate × units".
When two quantities are equal, write expression = expression and solve.
Finish by answering the question asked, in context and with units.

Worked example Algebra · linear equations and inequalities

When do two costs match?

Question

Gym A charges £60 to join plus £15 a month. Gym B has no joining fee and charges £27 a month. After how many months is the total cost the same?

1
Let $m$ m be the number of months. Gym A costs $60 + 15 m$ 60 + 15m and gym B costs $27 m$ 27m.
2
Set the costs equal: $60 + 15 m = 27 m$ 60 + 15m = 27m
3
Subtract $15 m$ 15m from both sides: $60 = 12 m$ 60 = 12m
4
Divide by 12: $m = 5$ m = 5

Answer

after 5 months (both cost £135). After that, gym A is cheaper, because it adds less each month.

⚠️ Watch out

the question asks for months, so "

m = 5

" alone is not the full answer. Say what it means.

Calculate

?A rectangle has length

(3 x + 2)

cm and width

(x - 1)

cm. Its perimeter is 50 cm. Form and solve an equation, then find the length of the rectangle (exact).

cm

Algebra · linear equations and inequalities

Linear inequalities

$<$ < less than, $>$ > greater than, $\leq$ ≤ at most, $\geq$ ≥ at least.
Number line: open circle for $<$ < or $>$ > (end value not included), closed circle for $\leq$ ≤ or $\geq$ ≥.
Solve an inequality like an equation: $3 x - 2 < 10$ 3x − 2 < 10 gives $3 x < 12$ 3x < 12, so $x < 4$ x < 4.
If you multiply or divide by a negative number, reverse the inequality sign: $- 2 x < 6$ −2x < 6 gives $x > - 3$ x > −3.

Calculate

?Solve

5 - 2 x < 13

. What is the smallest integer

x

that satisfies it? (exact)

Algebra · linear equations and inequalities

Regions on a graph (Higher)

An inequality in $x$ x and $y$ y, such as $x + y \leq 5$ x + y ≤ 5, describes a region of the grid.
Draw the boundary line $x + y = 5$ x + y = 5: solid for $\leq$ ≤ or $\geq$ ≥, dashed for $<$ < or $>$ >.
Test a point not on the line, such as $(0, 0)$ (0, 0): $0 + 0 \leq 5$ 0 + 0 ≤ 5 is true, so the region is on the side containing $(0, 0)$ (0, 0).
Several inequalities together give the region where all are true.

Sort it

?The region R is defined by

x \geq 0

,

y \geq 1

and

x + y \leq 5

. Sort each point.

Tap a statement, then tap the group it belongs in. Then press Check.

Mission check

?Plan A costs £12 a month plus £0.08 a minute. Plan B costs £20 a month plus £0.03 a minute. When is Plan B cheaper?

Mission 19

Rise 1 for every 12 along: which of three ramps is legal?

Make a guess: Rise 1 for every 12 along: which of three ramps is legal?

Algebra · straight-line graphs

Gradient measures steepness

Gradient = change in $y$ y ÷ change in $x$ x ("rise over run").

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Uphill from left to right: positive. Downhill: negative. Horizontal: 0.
The bigger the gradient (ignoring its sign), the steeper the line.
"Rise 1 for every 12 along" is a gradient of $\frac{1}{12}$ 1/12.

Algebra · straight-line graphs

The equation $y = m x + c$ y = mx + c

In $y = m x + c$ y = mx + c, $m$ m is the gradient and $c$ c is the $y$ y-intercept: the line crosses the $y$ y-axis at $(0, c)$ (0, c).
Rearrange to $y = \dots$ y = … first: $2 y = 6 x - 4$ 2y = 6x − 4 gives $y = 3 x - 2$ y = 3x − 2, so $m = 3$ m = 3 and $c = - 2$ c = −2.
To plot: make a table of values (three or more $x$ x values), plot the points, rule the line.
$y = k$ y = k is a horizontal line; $x = k$ x = k is a vertical line.
CCEA: gradients and intercepts from a graph are Foundation (M2); $y = m x + c$ y = mx + c and line equations are Higher (M3).

Quick check

?What is the gradient of the line

2 y = 12 - 3 x

?

Worked example Algebra · straight-line graphs

Equation of the line through two points

Question

Find the equation of the line through

(2, 7)

and

(5, 13)

.

1
Find the gradient: $m = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2$ m = (13 − 7)/(5 − 2) = 6/3 = 2
2
Substitute one point into $y = 2 x + c$ y = 2x + c: $7 = 2 \times 2 + c$ 7 = 2 × 2 + c
3
Solve for $c$ c: $c = 3$ c = 3
4
Check with the other point: $2 \times 5 + 3 = 13$ 2 × 5 + 3 = 13 ✓

Answer

y = 2 x + 3

⚠️ Watch out

subtract the coordinates in the same order on the top and the bottom, or the sign of the gradient flips.

Calculate

?A taxi fare is a fixed charge plus a set amount per mile, so fare against miles is a straight line. A 3-mile trip costs £8.40 and an 8-mile trip costs £15.40. Find the fixed charge. Give your answer as a decimal.

£

Algebra · straight-line graphs

Parallel and perpendicular lines

Parallel lines have equal gradients. The line parallel to $y = 4 x - 1$ y = 4x − 1 through $(1, - 3)$ (1, −3): $- 3 = 4 \times 1 + c$ −3 = 4 × 1 + c, so $y = 4 x - 7$ y = 4x − 7.
Higher: perpendicular lines have gradients that multiply to $- 1$ −1. Use the negative reciprocal: gradient 2 becomes $- \frac{1}{2}$ −1/2.
Higher: $2 y = 3 x + 1$ 2y = 3x + 1 has gradient $\frac{3}{2}$ 3/2, so a perpendicular line has gradient $- \frac{2}{3}$ −2/3.
For CCEA (Unit M3) and Edexcel IGCSE, parallel lines are Higher tier content too.

Calculate

?(Higher) Line

L

has equation

y = 2 x + 5

. Line

M

is perpendicular to

L

and passes through

(4, 1)

. Find the

y

-intercept of line

M

.

Algebra · straight-line graphs

Midpoint and length of a line segment

The midpoint averages the coordinates:

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

The length comes from Pythagoras: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ √((x₂ − x₁)² + (y₂ − y₁)²).
For $A (1, 2)$ A(1, 2) and $B (7, 10)$ B(7, 10): the midpoint is $(4, 6)$ (4, 6) and the length is $6^{2} + 8^{2} = 10$ √(6² + 8²) = 10.

Calculate

?Point

P

is

(- 3, 8)

and point

Q

is

(9, - 2)

. Find the length of

PQ

. Give your answer to 3 significant figures.

Mission check

?The rule: no steeper than a rise of 1 for every 12 along. Ramp P rises 0.6 m over 9 m, ramp Q is the line

y = 0.1 x

, ramp R rises 0.25 m over 3 m. Which ramps are legal?

Mission 20

A thrown ball is 10 m up at two different times. When, and why?

Make a guess: A thrown ball is 10 m up at two different times. When, and why?

Algebra · quadratic equations

Solving by factorising

Make one side 0: write the equation as $a x^{2} + b x + c = 0$ ax² + bx + c = 0.
Factorise, then use: if $A B = 0$ AB = 0, then $A = 0$ A = 0 or $B = 0$ B = 0.
Each bracket gives one root, so there are usually two solutions.
The roots are where the graph of $y = a x^{2} + b x + c$ y = ax² + bx + c crosses the $x$ x-axis; a graph gives approximate solutions.
CCEA: solving by factorising is Higher (Unit M3). Foundation finds approximate solutions from a graph (Unit M6).

Worked example Algebra · quadratic equations

Solving $x^{2} + 2 x - 15 = 0$ x² + 2x − 15 = 0 (Higher)

Question

Solve

x^{2} + 2 x - 15 = 0

.

1
Find two numbers that multiply to $- 15$ −15 and add to $2$ 2: they are $5$ 5 and $- 3$ −3
2
Factorise: $(x + 5) (x - 3) = 0$ (x + 5)(x − 3) = 0
3
Set each bracket equal to zero: $x + 5 = 0$ x + 5 = 0 or $x - 3 = 0$ x − 3 = 0

Answer

x = - 5

or

x = 3

⚠️ Watch out

x + 5 = 0

gives

x = - 5

, not

5

. Each root has the opposite sign to the number in its bracket.

Quick check

?Solve

x^{2} - x - 12 = 0

.

Calculate

?A rectangle is

x

cm wide and

(x + 3)

cm long. Its area is 40 cm², which gives the equation

x^{2} + 3 x - 40 = 0

. Find the perimeter of the rectangle.

cm

Algebra · quadratic equations

The quadratic formula (Higher)

For $a x^{2} + b x + c = 0$ ax² + bx + c = 0:

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Use it when the quadratic does not factorise, or when you need decimals or surds.
Rearrange to $\dots = 0$ … = 0 first, and take each of $a$ a, $b$ b, $c$ c with its sign.
If $b^{2} - 4 a c$ b² − 4ac is negative there are no solutions: the graph does not cross the $x$ x-axis.
CCEA uses it when the coefficient of $x^{2}$ x² is not 1 (Unit M4). Check your board's formula sheet to see whether it is printed; if not, learn it.

Worked example Algebra · quadratic equations

Using the formula (Higher)

Question

Solve

2 x^{2} - 3 x - 4 = 0

. Give your answers to 2 decimal places.

1
Identify the coefficients: $a = 2$ a = 2, $b = - 3$ b = −3, $c = - 4$ c = −4
2
Work out $b^{2} - 4 a c = (- 3)^{2} - 4 \times 2 \times (- 4) = 9 + 32 = 41$ b² − 4ac = (−3)² − 4 × 2 × (−4) = 9 + 32 = 41
3
Substitute: $x = \frac{3 \pm 41}{4}$ x = (3 ± √41)/4
4
Evaluate each sign: $\frac{3 + 6.403 \dots}{4}$ (3 + 6.403…)/4 and $\frac{3 - 6.403 \dots}{4}$ (3 − 6.403…)/4

Answer

x = 2.35

or

x = - 0.85

⚠️ Watch out

- b

is

+ 3

here because

b = - 3

; and

(- 3)^{2} = 9

, not

- 9

.

Calculate

?(Higher) Solve

2 x^{2} + 7 x - 3 = 0

. Give the positive solution to 3 significant figures.

Mission check

?A ball's height after

t

seconds is

h = 15 t - 5 t^{2}

metres. It is 10 m up when

5 t^{2} - 15 t + 10 = 0

. When is the ball 10 m up?

Mission 21

3 coffees + 2 cakes = £13.50; 2 coffees + 3 cakes = £13. A cake?

Make a guess: 3 coffees + 2 cakes = £13.50; 2 coffees + 3 cakes = £13. A cake?

Algebra · simultaneous equations

Two unknowns, two equations

One equation in $x$ x and $y$ y has endless solutions. Simultaneous equations are two equations that must both be true at once.
The solution is the pair $x$ x, $y$ y that fits both.
Graphically, each linear equation is a line; the solution is where the lines intersect. $y = 2 x - 1$ y = 2x − 1 and $x + y = 5$ x + y = 5 meet at $(2, 3)$ (2, 3).
Always check your pair in both original equations.
CCEA: Foundation solves these graphically (Unit M6); solving algebraically is Higher (Unit M7).

Worked example Algebra · simultaneous equations

Solving by elimination (Higher)

Question

Solve

3 x + 2 y = 16

and

5 x - 3 y = 14

.

1
Match the $y$ y coefficients: multiply the first by 3 and the second by 2: $9 x + 6 y = 48$ 9x + 6y = 48 and $10 x - 6 y = 28$ 10x − 6y = 28
2
The signs of $6 y$ 6y differ, so add: $19 x = 76$ 19x = 76
3
Divide: $x = 4$ x = 4
4
Substitute into $3 x + 2 y = 16$ 3x + 2y = 16: $12 + 2 y = 16$ 12 + 2y = 16, so $y = 2$ y = 2
5
Check in the other equation: $5 \times 4 - 3 \times 2 = 14$ 5 × 4 − 3 × 2 = 14 ✓

Answer

x = 4

,

y = 2

⚠️ Watch out

add when the matched terms have different signs and subtract when they have the same sign. Multiply every term, including the number on the right.

Quick check

?Solve

5 x + 2 y = 20

and

3 x - 2 y = 4

.

Algebra · simultaneous equations

Solving by substitution (Higher)

Use substitution when one equation already says $y = \dots$ y = … or $x = \dots$ x = ….
Replace that letter in the other equation, using brackets: $y = 2 x - 1$ y = 2x − 1 into $x + y = 5$ x + y = 5 gives $x + (2 x - 1) = 5$ x + (2x − 1) = 5.
Solve for one letter: $3 x = 6$ 3x = 6, so $x = 2$ x = 2; then $y = 2 \times 2 - 1 = 3$ y = 2 × 2 − 1 = 3.
Elimination and substitution give the same answer; pick whichever is quicker.

Calculate

?Solve the simultaneous equations

y = 2 x - 3

and

3 x + 2 y = 22

. Give the value of

y

.

Calculate

?At a cinema, 3 adult tickets and 2 child tickets cost £34. One adult ticket and 4 child tickets cost £23. Find the cost of one child ticket. Give your answer as a decimal.

£

Algebra · simultaneous equations

One linear, one quadratic (Higher)

Rearrange the linear equation to $y = \dots$ y = … (or $x = \dots$ x = …) and substitute it into the quadratic one.
You get a quadratic equation in one letter: make it $= 0$ = 0 and solve.
Find the partner value for each root using the linear equation. Give the answers as pairs.
A line can meet a curve at two points, one point (it touches) or no points.
The curve might be a circle such as $x^{2} + y^{2} = 25$ x² + y² = 25: the same method works.

Worked example Algebra · simultaneous equations

A line meeting a curve (Higher)

Question

Solve

y = 2 x + 1

and

y = x^{2} - 2

.

1
Set the two expressions for $y$ y equal: $x^{2} - 2 = 2 x + 1$ x² − 2 = 2x + 1
2
Make one side zero: $x^{2} - 2 x - 3 = 0$ x² − 2x − 3 = 0
3
Factorise: $(x - 3) (x + 1) = 0$ (x − 3)(x + 1) = 0, so $x = 3$ x = 3 or $x = - 1$ x = −1
4
Use $y = 2 x + 1$ y = 2x + 1 for each: $y = 7$ y = 7 or $y = - 1$ y = −1

Answer

x = 3, y = 7

and

x = - 1, y = - 1

⚠️ Watch out

two

x

values need two matching

y

values. Stopping at

x

loses marks.

Quick check

?Higher. Solve

y = x + 4

and

y = x^{2} - 2 x

.

Mission check

?3 coffees and 2 cakes cost £13.50. 2 coffees and 3 cakes cost £13.00. How much is one cake?

Mission 22

Four graphs, four equations: can you match them without plotting?

Make a guess: Four graphs, four equations: can you match them without plotting?

Algebra · graphs of functions

Lines and parabolas

A linear graph $y = m x + c$ y=mx+c is a straight line: gradient $m$ m, crossing the $y$ y-axis at $(0, c)$ (0, c).
A quadratic graph $y = a x^{2} + b x + c$ y=ax²+bx+c is a parabola: U-shaped if $a > 0$ a>0, ∩-shaped if $a < 0$ a<0.
The roots are where the curve crosses the $x$ x-axis ( $y = 0$ y=0). The $y$ y-intercept is at $(0, c)$ (0, c).
The turning point lies on the line of symmetry, halfway between the roots.

Worked example Algebra · graphs of functions

Reading a quadratic graph

Question

The graph of

y = x^{2} - 2 x - 3

is drawn. Find its roots, its turning point, and use it to solve

x^{2} - 2 x - 3 = 5

.

1
Read where the curve crosses the $x$ x-axis: $x = - 1$ x = −1 and $x = 3$ x = 3
2
The line of symmetry is halfway between: $x = \frac{- 1 + 3}{2} = 1$ x = (−1+3)/2 = 1
3
Substitute $x = 1$ x = 1: $y = 1 - 2 - 3 = - 4$ y = 1 − 2 − 3 = −4, so the turning point is $(1, - 4)$ (1, −4)
4
Draw the line $y = 5$ y = 5 and read where it meets the curve: $x = - 2$ x = −2 and $x = 4$ x = 4

Answer

roots

x = - 1

and

x = 3

; turning point

(1, - 4)

; solutions

x = - 2

or

x = 4

⚠️ Watch out

to solve

x^{2} - 2 x - 3 = 5

you read off at

y = 5

, not at

y = 0

.

Calculate

?The graph of

y = x^{2} + 2 x - 8

is drawn for

- 6 \leq x \leq 4

. Use the graph to solve

x^{2} + 2 x - 8 = 7

. Enter the positive solution.

Algebra · graphs of functions

Cubics and reciprocals

A cubic $y = a x^{3} + \dots$ y=ax³+… is S-shaped: rising left to right if $a > 0$ a>0, falling if $a < 0$ a<0. $y = x^{3} - 4 x$ y = x³ − 4x has roots $- 2$ −2, 0 and 2.
The reciprocal graph $y = \frac{a}{x}$ y=a/x has two branches that never touch the axes (the asymptotes); $x \neq = 0$ x ≠ 0.
To match an equation to a graph: check the highest power of $x$ x and its sign, then the intercepts.
On your course cubic and reciprocal graphs are Higher tier.

Sort it

?Sort each equation by the shape of its graph.

Tap a statement, then tap the group it belongs in. Then press Check.

Calculate

?Higher. The graphs of

y = \frac{6}{x}

and

y = x + 1

are drawn on the same axes. Use the graphs to find the

x

-coordinate of the intersection point in the first quadrant.

Algebra · graphs of functions

Exponential graphs (Higher)

An exponential graph $y = k^{x}$ y = k^x ( $k > 0$ k > 0) always passes through $(0, 1)$ (0, 1), because $k^{0} = 1$ k⁰ = 1.
If $k > 1$ k > 1 it shows growth: $y = 2^{x}$ y = 2^x goes 1, 2, 4, 8 as $x$ x goes 0, 1, 2, 3.
If $0 < k < 1$ 0 < k < 1 it shows decay: $y = (\frac{1}{2})^{x}$ y = (1/2)^x falls towards 0.
It never touches the $x$ x-axis: $y = 0$ y=0 is an asymptote, and $y$ y is always positive.

Mission check

?Four graphs, four equations. Graph P has two separate branches and never meets either axis. Which equation matches it, and why?

Mission 23

Chessboard: 1 grain, then double each square. How many on square 64?

Make a guess: Chessboard: 1 grain, then double each square. How many on square 64?

Algebra · sequences

Two kinds of rule

A term-to-term rule gets each term from the one before: 5, 9, 13, 17, … is "add 4".
A position-to-term rule (the nth term) gets any term from its position $n$ n: here $4 n + 1$ 4n + 1.
Substitute $n$ n to find a term: the 10th term is $4 \times 10 + 1 = 41$ 4 × 10 + 1 = 41.
The chessboard has term-to-term rule "double".

Worked example Algebra · sequences

Finding the nth term of a linear sequence

Question

Find the nth term of 7, 10, 13, 16, … and its 100th term. Is 100 a term?

1
Find the common difference: $10 - 7 = 3$ 10 − 7 = 3, so the nth term starts $3 n$ 3n
2
Compare with $3 n$ 3n (3, 6, 9, 12): each term is 4 more, so add 4
3
Write the rule: nth term $= 3 n + 4$ = 3n + 4
4
Substitute $n = 100$ n = 100: $3 \times 100 + 4 = 304$ 3 × 100 + 4 = 304
5
Solve $3 n + 4 = 100$ 3n + 4 = 100: $n = 32$ n = 32, a whole number, so 100 is the 32nd term

Answer

nth term

3 n + 4

; 100th term 304; 100 is a term (the 32nd).

⚠️ Watch out

the nth term is not "

n + 3

". The difference 3 multiplies

n

.

Calculate

?The sequence 40, 37, 34, 31, … is linear. Find its 20th term.

Calculate

?Exam-style. Pattern

n

is a row of

n

hexagons made from matchsticks. Pattern 1 uses 6 sticks, pattern 2 uses 11 and pattern 3 uses 16. Which pattern uses exactly 101 matchsticks?

Algebra · sequences

Special sequences

Square numbers $n^{2}$ n²: 1, 4, 9, 16, … and cube numbers $n^{3}$ n³: 1, 8, 27, 64, ….
Triangular numbers: 1, 3, 6, 10, 15, … (add 2, then 3, then 4, …).
A geometric sequence multiplies by the same number $r$ r each time: 3, 6, 12, 24 ( $r = 2$ r = 2). The chessboard 1, 2, 4, 8, … is geometric: square $n$ n holds $2^{n - 1}$ 2ⁿ⁻¹.

Sort it

?Sort each sequence.

Tap a statement, then tap the group it belongs in. Then press Check.

Algebra · sequences

The nth term of a quadratic sequence (Higher)

If the second differences are constant, the nth term is $a n^{2} + bn + c$ an² + bn + c.
The coefficient of $n^{2}$ n² is half the second difference.
Subtract $a n^{2}$ an² from each term. What is left is linear: find its nth term and add it on.

Worked example Algebra · sequences

Quadratic nth term (Higher)

Question

Find the nth term of 4, 11, 22, 37, ….

1
First differences: 7, 11, 15. Second differences: 4, 4, so it is quadratic
2
Halve the second difference: $\frac{4}{2} = 2$ 4/2 = 2, so it starts $2 n^{2}$ 2n²
3
Subtract $2 n^{2}$ 2n² (2, 8, 18, 32) from each term: 2, 3, 4, 5
4
The leftover 2, 3, 4, 5 has nth term $n + 1$ n + 1

Answer

nth term

= 2 n^{2} + n + 1

(check:

n = 1

gives

2 + 1 + 1 = 4

)

⚠️ Watch out

the coefficient of

n^{2}

is half the second difference, not the second difference itself.

Calculate

?Higher. Find the 10th term of the quadratic sequence 3, 9, 19, 33, 51, ….

Mission check

?On the chessboard, square 1 has 1 grain and each square doubles the one before. How many grains are on square 64?

Mission 24

Two runners' distance–time graphs cross. What does the crossing mean?

Make a guess: Two runners' distance–time graphs cross. What does the crossing mean?

Algebra · real-life graphs

Distance–time graphs

The gradient of a distance–time graph is the speed: distance ÷ time.
Steeper means faster. A horizontal line means stopped.
Where two graphs cross, the two objects are at the same place at the same time. If the lines do not have the same gradient there, one is overtaking the other.
Use the units on the axes: metres per minute, km/h, m/s.

Worked example Algebra · real-life graphs

Reading two runners' graphs

Question

Runner A runs 2000 m in 10 minutes at a steady pace. Runner B covers 1200 m in the first 4 minutes. Find each speed for the first 4 minutes, in m/s, and say who is faster at the start.

1
Gradient of B's first section: $\frac{1200}{4} = 300$ 1200/4 = 300 m per minute
2
Convert to m/s: $\frac{1200}{4 \times 60} = 5$ 1200/(4 × 60) = 5 m/s
3
Gradient of A's line: $\frac{2000}{10 \times 60} = 3.33$ 2000/(10 × 60) = 3.33 m/s (3 s.f.)
4
Compare: B's line is steeper at the start

Answer

B runs at 5 m/s and A at 3.33 m/s, so B is faster at the start.

⚠️ Watch out

change minutes to seconds before dividing, or the answer is in metres per minute.

Calculate

?Exam-style. A cyclist rides 15 km in 45 minutes, stops for 15 minutes, then rides 15 km back home in 30 minutes. Find the average speed for the whole journey.

km/h

Algebra · real-life graphs

Conversion graphs and rates

A conversion graph is a straight line through $(0, 0)$ (0, 0): the two quantities are in direct proportion.
The gradient is the rate: kilometres per mile, euros per pound.
A line that starts above $(0, 0)$ (0, 0) has a fixed charge. A plumber charging £40 call-out plus £30 per hour has intercept 40 and gradient 30 (£ per hour).
Read across from one axis to the line, then down (or across) to the other axis.

Calculate

?The conversion graph is a straight line through

(0, 0)

and (50 miles, 80 km). Use it to convert 28 miles to kilometres.

km

Algebra · real-life graphs

Gradient of a curve (Higher)

On a curve the gradient changes. Draw a tangent: a straight line that just touches the curve at the point.
The gradient of the tangent is the instantaneous rate of change there, e.g. the speed at that moment on a distance–time graph.
Pick two points far apart on the tangent: gradient = change in $y$ y ÷ change in $x$ x.

Calculate

?Higher. A tangent is drawn to a distance–time curve at

t = 3

s. It passes through (1, 2) and (5, 14), with time in seconds and distance in metres. Estimate the speed at

t = 3

s.

m/s

Mission check

?Two runners start together. Their distance–time graphs cross at (8 minutes, 1600 m). Runner B's line is steeper at the start. What does the crossing mean?

Mission 25

Cycle out at 10 km/h, back at 30 km/h. Is your average 20 km/h?

Make a guess: Cycle out at 10 km/h, back at 30 km/h. Is your average 20 km/h?

Algebra · algebraic fractions

Simplifying algebraic fractions (Higher)

An algebraic fraction has algebra in the numerator, the denominator or both.
To simplify, factorise the numerator and denominator fully, then cancel common factors (whole brackets or whole factors).
$\frac{x ^{2} - 2 x}{x ^{2} - 5 x + 6} = \frac{x ( x - 2 )}{( x - 3 ) ( x - 2 )} = \frac{x}{x - 3}$ (x² − 2x)/(x² − 5x + 6) = (x(x − 2))/((x − 3)(x − 2)) = x/(x − 3)
You can only cancel factors, never single terms: in $\frac{x + 3}{x}$ (x + 3)/x nothing cancels.

Quick check

?Simplify fully

\frac{2 x ^{2} + 3 x}{4 x ^{2} - 9}

.

Algebra · algebraic fractions

The four operations with algebraic fractions (Higher)

Multiply: factorise, cancel, then multiply numerators and denominators.
Divide: flip the second fraction and multiply: $\frac{3 a}{4} \div \frac{9 a}{10} = \frac{3 a}{4} \times \frac{10}{9 a} = \frac{5}{6}$ 3a/4 ÷ 9a/10 = 3a/4 × 10/9a = 5/6.
Add or subtract: use a common denominator, just as $\frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6}$ 1/2 + 1/3 = (3 + 2)/6.
$\frac{1}{x} + \frac{1}{x + 1} = \frac{( x + 1 ) + x}{x ( x + 1 )} = \frac{2 x + 1}{x ( x + 1 )}$ 1/x + 1/(x + 1) = ((x + 1) + x)/(x(x + 1)) = (2x + 1)/(x(x + 1))
Never add the denominators: $\frac{1}{2} + \frac{1}{3}$ 1/2 + 1/3 is not $\frac{2}{5}$ 2/5.

Calculate

?Write

\frac{2}{x} + \frac{3}{x + 1}

as a single fraction

\frac{a x + b}{x ( x + 1 )}

. Enter the value of

a

(exact).

Worked example Algebra · algebraic fractions

Subtracting algebraic fractions (Higher)

Question

Write

\frac{3 x + 1}{x + 2} - \frac{x - 2}{x - 1}

as a single fraction.

1
Use the common denominator $(x + 2) (x - 1)$ (x + 2)(x − 1): $\frac{( 3 x + 1 ) ( x - 1 ) - ( x - 2 ) ( x + 2 )}{( x + 2 ) ( x - 1 )}$ ((3x + 1)(x − 1) − (x − 2)(x + 2))/((x + 2)(x − 1))
2
Expand the first product: $(3 x + 1) (x - 1) = 3 x^{2} - 2 x - 1$ (3x + 1)(x − 1) = 3x² − 2x − 1
3
Expand the second product: $(x - 2) (x + 2) = x^{2} - 4$ (x − 2)(x + 2) = x² − 4
4
Subtract the whole second product: $3 x^{2} - 2 x - 1 - (x^{2} - 4) = 2 x^{2} - 2 x + 3$ 3x² − 2x − 1 − (x² − 4) = 2x² − 2x + 3

Answer

\frac{2 x ^{2} - 2 x + 3}{( x + 2 ) ( x - 1 )}

⚠️ Watch out

the minus applies to all of

x^{2} - 4

, so

- (- 4) = + 4

. Writing

- x^{2} - 4

gives the wrong numerator

2 x^{2} - 2 x - 5

.

Worked example Algebra · algebraic fractions

Solving an equation with algebraic fractions (Higher)

Question

Solve

\frac{2}{x + 2} + \frac{3}{2 x - 1} = 1

.

1
Multiply every term by $(x + 2) (2 x - 1)$ (x + 2)(2x − 1): $2 (2 x - 1) + 3 (x + 2) = (x + 2) (2 x - 1)$ 2(2x − 1) + 3(x + 2) = (x + 2)(2x − 1)
2
Expand both sides: $7 x + 4 = 2 x^{2} + 3 x - 2$ 7x + 4 = 2x² + 3x − 2
3
Rearrange to $= 0$ = 0: $2 x^{2} - 4 x - 6 = 0$ 2x² − 4x − 6 = 0, so $x^{2} - 2 x - 3 = 0$ x² − 2x − 3 = 0
4
Factorise: $(x - 3) (x + 1) = 0$ (x − 3)(x + 1) = 0
5
Check neither value makes a denominator zero: both are fine

Answer

x = 3

or

x = - 1

⚠️ Watch out

multiply every term, including the 1 on the right, by the common denominator.

Calculate

?A walker goes 12 km uphill at

x

km/h, then 12 km back down at

(x + 2)

km/h. The whole walk takes 5 hours. Form an equation and solve it to find the uphill speed (exact).

km/h

Calculate

?Some equations have no simple method of solving, so you use trial and improvement: try a value, see if the answer is too big or too small, and narrow it down.

x^{3} - 2 x = 30

has a solution between 3 and 4. Use trial and improvement to find it to 1 decimal place.

Mission check

?Cycle a route at 10 km/h and ride back along it at 30 km/h. What is your average speed for the whole trip?

Mission 26

A sprinkler reaches 13 m. Does it wet a plant at (12, 6)?

Make a guess: A sprinkler reaches 13 m. Does it wet a plant at (12, 6)?

Algebra · circles on a grid

The equation of a circle (Higher)

A point $(x, y)$ (x, y) on a circle with centre the origin and radius $r$ r makes a right-angled triangle with legs $x$ x and $y$ y.
By Pythagoras, every point on the circle satisfies

x^{2} + y^{2} = r^{2}

$x^{2} + y^{2} = 25$ x² + y² = 25 has radius 5: $(3, 4)$ (3, 4) is on it because $9 + 16 = 25$ 9 + 16 = 25.
If $x^{2} + y^{2} < r^{2}$ x² + y² < r² the point is inside; if $x^{2} + y^{2} > r^{2}$ x² + y² > r² it is outside.

Quick check

?What is the radius of the circle

x^{2} + y^{2} = 49

?

Worked example Algebra · circles on a grid

Finding a missing coordinate (Higher)

Question

The point

(p, 4)

lies on the circle

x^{2} + y^{2} = 41

, and

p < 0

. Find

p

.

1
Substitute the point: $p^{2} + 4^{2} = 41$ p² + 4² = 41
2
Simplify: $p^{2} = 25$ p² = 25
3
Square-root, keeping both signs: $p = 5$ p = 5 or $p = - 5$ p = −5
4
Use the condition $p < 0$ p < 0

Answer

p = - 5

⚠️ Watch out

the square root has two signs. Read the question to choose the right one.

Sort it

?Is each point inside, on or outside the circle

x^{2} + y^{2} = 100

?

Tap a statement, then tap the group it belongs in. Then press Check.

Algebra · circles on a grid

The tangent at a point (Higher)

A tangent touches the circle at one point and is perpendicular to the radius at that point.
The radius from $O (0, 0)$ O(0, 0) to $P (a, b)$ P(a, b) has gradient $\frac{b}{a}$ b/a.
So the tangent at $P$ P has gradient $- \frac{a}{b}$ −a/b, the negative reciprocal.
Substitute $P$ P into $y = m x + c$ y = mx + c to find $c$ c.

Worked example Algebra · circles on a grid

Equation of a tangent (Higher)

Question

Find the equation of the tangent to

x^{2} + y^{2} = 25

at the point

(3, 4)

.

1
Gradient of the radius to $(3, 4)$ (3, 4): $\frac{4}{3}$ 4/3
2
Tangent gradient, the negative reciprocal: $- \frac{3}{4}$ −3/4
3
Substitute $(3, 4)$ (3, 4) into $y = - \frac{3}{4} x + c$ y = −3/4x + c: $4 = - \frac{9}{4} + c$ 4 = −9/4 + c, so $c = \frac{25}{4}$ c = 25/4
4
Multiply by 4 to clear fractions: $4 y = - 3 x + 25$ 4y = −3x + 25

Answer

y = - \frac{3}{4} x + \frac{25}{4}

, or

3 x + 4 y = 25

⚠️ Watch out

the radius gradient

\frac{4}{3}

is not the tangent gradient. Flip it and change the sign.

Quick check

?The point

(- 2, 4)

lies on

x^{2} + y^{2} = 20

. What is the gradient of the tangent at

(- 2, 4)

?

Calculate

?The point

(1, 3)

lies on the circle

x^{2} + y^{2} = 10

. The tangent to the circle at

(1, 3)

crosses the

x

-axis at

Q

. Find the

x

-coordinate of

Q

.

Calculate

?The line

y = x + 1

meets the circle

x^{2} + y^{2} = 25

at two points. Find the positive

x

-coordinate of the points of intersection.

Mission check

?A sprinkler at the origin sprays a circle of radius 13 m. Is the plant at

(12, 6)

inside, on or outside the wet circle?

Recap

Big ideas

Place value: each column is 10 times the next; compare decimals column by column, and on a number line further right is bigger.
Negatives: same signs multiply or divide to a positive; subtracting a negative adds.
Brackets, then powers and roots, then $\times$ × and $\div$ ÷ left to right, then $+$ + and $-$ − left to right.
Binary uses column values 1, 2, 4, 8, 16, … and only the digits 0 and 1.
Factors divide into a number; multiples are in its times table; a prime has exactly two factors (so 1 is not prime).
Every whole number above 1 has one prime factor decomposition; write it in index form.
HCF: shared primes (the Venn overlap). LCM: all the primes in the Venn diagram. "Next time together" problems need the LCM; "biggest equal groups" problems need the HCF.
Equivalent fractions: multiply or divide top and bottom by the same number; simplify using the HCF.
Add and subtract with a common denominator; never add the denominators.
Multiply tops and bottoms; to divide, multiply by the reciprocal. Change mixed numbers to improper fractions first.
Fraction to decimal: divide. Decimal to percentage: multiply by 100. To order, change everything to decimals.
A fraction in simplest form terminates only if its denominator's prime factors are 2s and 5s; otherwise it recurs. Every recurring decimal is an exact fraction.
Higher: recurring decimal to fraction by multiplying by a power of 10 and subtracting.
Higher: Rational numbers can be written as $\frac{a}{b}$ a/b; irrational ones ( $2$ √2, $π$ π) cannot.
An index counts repeated multiplication; a root undoes a power.
Same base: multiply → add indices, divide → subtract, power of a power → multiply. $a^{0} = 1$ a⁰ = 1 and $a^{- n} = \frac{1}{a ^{n}}$ a⁻ⁿ = 1/aⁿ.
Higher: $a^{\frac{m}{n}} = (n a)^{m}$ a^(m/n) = (n√a)^m. Take the root first.
Standard form is $A \times 1 0^{n}$ A × 10ⁿ with $1 \leq A < 10$ 1 ≤ A < 10; the power counts how far the point moves.
Multiply: multiply the numbers, add the powers. Divide: divide, subtract. Add or subtract: match the powers first.
Always finish by adjusting $A$ A back into the range $1 \leq A < 10$ 1 ≤ A < 10.
Round to d.p. or s.f. only at the end; estimate by rounding every number to 1 s.f.
A rounded value stands for a range: half a unit below (included) up to half a unit above (not included).
Higher: for the largest result, pick upper bounds for what you add or multiply and lower bounds for what you subtract or divide by.
A surd such as $2$ √2 is an exact irrational root. $a \times b = ab$ √a × √b = √ab, but $a + b \neq = a + b$ √a + √b ≠ √(a+b).
Simplify by taking out the largest square factor, then collect like surds.
Rationalise $\frac{k}{a}$ k/(√a) by multiplying top and bottom by $a$ √a.
List systematically: fix one choice, run through the others, then move on; use a table or grid for two choices.
Higher: product rule: multiply the number of choices at each stage; reduce the choices when items can't repeat, and do restricted positions first.
When order doesn't matter (A v B is the same match as B v A), divide out the double counting.
Simplify a ratio by dividing every part by the HCF, after putting the quantities in the same units.
Sharing: add the parts, find one part, then multiply. If you know one share or a difference, start from that number of parts instead.
A ratio links to fractions of the whole ( $\frac{3}{5}$ 3/5 and $\frac{2}{5}$ 2/5) and to a straight line through the origin.
Every percentage change is a multiplier: $\times 1.15$ × 1.15 for a 15% increase, $\times 0.85$ × 0.85 for a 15% decrease. Two changes in a row multiply.
Percentage change is the change divided by the original value, times 100.
Higher: in a reverse percentage problem, divide the new amount by the multiplier.
Repeated percentage change means multiplying by the multiplier once per time period: $A = P \times m^{n}$ A = P × mⁿ.
Compound interest earns interest on interest; simple interest is the same every year.
Use each year's multiplier in turn when the rate changes, and never add the percentages.
Direct proportion: both multiply by the same factor; graph is a straight line through the origin; $y = k x$ y = kx.
Inverse proportion: the product stays the same; double one, halve the other; $y = \frac{k}{x}$ y = k/x.
Higher: for powers and roots, write $y = k x^{2}$ y = kx², $y = \frac{k}{x ^{2}}$ y = k/x², $y = k x$ y = k√x, …, find $k$ k from one pair of values, then use the equation.
Area units use the length factor squared; volume units use it cubed. $1 m^{3} = 1, 000, 000 cm^{3} = 1000$ 1 m³ = 1,000,000 cm³ = 1000 litres.
Compound measures are rates: speed $= \frac{distance}{time}$ = distance/time, density $= \frac{mass}{volume}$ = mass/volume, pressure $= \frac{force}{area}$ = force/area.
Change hours and minutes to decimal hours before dividing, and give the units of your answer.
Best value: compare the same unit (price per gram, per litre, per cm²) after converting units.
Recipes, exchange rates and map scales are all direct proportion: find the multiplier, and divide to go back.
When lengths double, areas multiply by 4: compare pizzas, maps and floor plans by area.
Collect only like terms (same letters, same powers); expand by multiplying every term in one bracket by every term in the other.
$(a + b)^{2} = a^{2} + 2 ab + b^{2}$ (a + b)² = a² + 2ab + b², never just $a^{2} + b^{2}$ a² + b²: the grid shows the two missing strips.
Factorising undoes expanding: common factor first, then look for $x^{2} + b x + c$ x² + bx + c or a difference of two squares, and check by expanding.
Substitute with brackets round negatives, then follow the order of operations.
Change the subject by undoing operations in reverse order; if the subject appears twice, collect those terms and factorise.
An equation is true for some values, an identity ( $\equiv$ ≡) for every value, and a formula links quantities.
Expand brackets, collect the unknown on one side, and keep the equation balanced; check by substituting.
To compare two deals, write each as an expression, set them equal, and interpret what happens either side of the solution.
Inequalities solve like equations, except that multiplying or dividing by a negative reverses the sign.
Gradient = change in $y$ y ÷ change in $x$ x; in $y = m x + c$ y = mx + c, $m$ m is the gradient and $(0, c)$ (0, c) the $y$ y-intercept.
To find a line's equation: gradient first, then substitute a point to find $c$ c.
Parallel lines share a gradient; perpendicular gradients multiply to $- 1$ −1 (Higher).
Midpoint: average the coordinates. Length: Pythagoras on the changes in $x$ x and $y$ y.
Make one side 0, factorise, and set each bracket to 0: each bracket gives a root.
Higher: if it will not factorise, use $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ x = (−b ± √(b² − 4ac))/2a.
A quadratic graph can cross a line twice, so a quadratic equation can have two solutions.
Two unknowns need two equations; the solution fits both and is where their graphs intersect.
Eliminate by matching coefficients and adding or subtracting; substitute when one equation gives $y = \dots$ y = ….
Higher: for a line and a curve, substitute to get a quadratic, then find a $y$ y for each $x$ x.
The highest power of $x$ x and its sign set the shape: line, parabola, S-shaped cubic, or two-branch reciprocal.
Roots are where $y = 0$ y = 0; to solve $f (x) = k$ f(x) = k from a graph, read off where the line $y = k$ y = k meets the curve.
Higher/Extended: $y = k^{x}$ y = k^x passes through $(0, 1)$ (0, 1) and never touches the $x$ x-axis.
A linear sequence adds the same amount each time; its nth term is (difference) $\times n$ × n + adjustment.
A geometric sequence multiplies by the same amount each time, so it grows much faster.
Higher/Extended: constant second differences mean a quadratic nth term; the $n^{2}$ n² coefficient is half the second difference.
Gradient of a distance–time graph = speed; steeper means faster; a crossing means the same place at the same time.
A straight-line graph's gradient is a rate (km per mile, £ per hour); its intercept is a fixed amount.
Higher/Extended: the tangent's gradient gives the rate of change at an instant.
Factorise first, then cancel whole factors; never cancel single terms.
Add or subtract algebraic fractions over a common denominator, and multiply every term by it to solve an equation.
A circle with centre the origin and radius $r$ r has equation $x^{2} + y^{2} = r^{2}$ x² + y² = r²; compare $x^{2} + y^{2}$ x² + y² with $r^{2}$ r² to test a point.
The tangent at $P$ P is perpendicular to the radius $OP$ OP: use the negative reciprocal of the radius gradient.
To find where a line meets the circle, substitute the line into $x^{2} + y^{2} = r^{2}$ x² + y² = r² and solve the quadratic.

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Mission 18 ÷ 2(2 + 2): is the answer 1 or 16?
Mission 2Two lights flash every 12 s and 18 s. When do they flash together?
Mission 3Half a cup of milk, then a third of a cup more: is that $\frac{2}{5}$ 2/5 of a cup?
Mission 4Is 0.999… (nines forever) exactly equal to 1?
Mission 5Fold a sheet of paper 42 times. Would it reach the Moon?
Mission 6Are you a billion seconds old yet?
Mission 7Eight '80 kg' people in a 640 kg lift. Could it be overloaded?
Mission 8Halve an A4 sheet and it keeps its shape. Why must that be $2$ √2?
Mission 9A 4-digit PIN: how many guesses to be certain of getting it?
Mission 10Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?
Mission 11A price goes up 20%, then down 20%. Is it back where it started?
Mission 12Money grows at 5% a year compound interest. When has it doubled?
Mission 134 builders take 6 days. How long would 8 builders take?
Mission 14A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?
Mission 15The pizza twice as wide costs three times as much. Good deal?
Mission 16A square grows 3 cm each way. Is its new area just $x^{2} + 9$ x² + 9?
Mission 17Is there one temperature that reads the same in °C and °F?
Mission 18When does the pricier phone plan become the cheaper one?
Mission 19Rise 1 for every 12 along: which of three ramps is legal?
Mission 20A thrown ball is 10 m up at two different times. When, and why?
Mission 213 coffees + 2 cakes = £13.50; 2 coffees + 3 cakes = £13. A cake?
Mission 22Four graphs, four equations: can you match them without plotting?
Mission 23Chessboard: 1 grain, then double each square. How many on square 64?
Mission 24Two runners' distance–time graphs cross. What does the crossing mean?
Mission 25Cycle out at 10 km/h, back at 30 km/h. Is your average 20 km/h?
Mission 26A sprinkler reaches 13 m. Does it wet a plant at (12, 6)?

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Specification points and sources for this lesson are listed at the bottom of the page.

Number & Algebra

8 ÷ 2(2 + 2): is the answer 1 or 16?

Place value and ordering numbers

Negative numbers and reciprocals

The order of operations

Using the order of operations

Binary numbers

Two lights flash every 12 s and 18 s. When do they flash together?

Factors, multiples and primes

Prime factor decomposition

HCF and LCM

HCF and LCM using prime factors

Half a cup of milk, then a third of a cup more: is that 52​2/5 of a cup?

Equivalent fractions and mixed numbers

Fractions of amounts

Adding and subtracting fractions

Subtracting mixed numbers

Multiplying and dividing fractions

Is 0.999… (nines forever) exactly equal to 1?

Converting between fractions, decimals and percentages

Converting both ways

Terminating and recurring decimals

Recurring decimals to fractions (Higher)

Converting 0.27˙0.27 to a fraction (Higher)

Fold a sheet of paper 42 times. Would it reach the Moon?

Powers, roots and repeated doubling

The three laws of indices

Using the laws on an algebraic expression

Zero and negative indices

Fractional indices (Higher)

Are you a billion seconds old yet?

Writing a number in standard form

Converting in both directions

Ordering and calculating

Multiplying without a calculator

Eight '80 kg' people in a 640 kg lift. Could it be overloaded?

Decimal places and significant figures

Estimating by rounding

The bounds of a rounded measurement

Bounds in calculations (Higher)

The slowest possible speed (Higher)

Halve an A4 sheet and it keeps its shape. Why must that be 2​√2?

What a surd is (Higher)

Simplifying surds (Higher)

Expanding a bracket with a surd (Higher)

Rationalising the denominator (Higher)

A 4-digit PIN: how many guesses to be certain of getting it?

Listing systematically

Listing arrangements

The product rule for counting (Higher)

Counting codes (Higher)

Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?

Ratio notation and simplest form

Simplest form and the form 1 : n

Sharing in a ratio

Sharing in a three-part ratio

Ratios, fractions and combining ratios

A price goes up 20%, then down 20%. Is it back where it started?

Percentages as multipliers

Increasing with a multiplier

Percentage change and one quantity as a percentage of another

Reverse percentages (Higher)

Finding the original price (Higher)

Money grows at 5% a year compound interest. When has it doubled?

Compound interest

Compound interest over three years

Depreciation and repeated changes

Growth and decay formula (Higher)

When does a population fall below a target? (Higher)

4 builders take 6 days. How long would 8 builders take?

Direct proportion

The unitary method

Inverse proportion

Proportion equations with powers and roots (Higher)

Direct proportion to a square (Higher)

A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?

Converting area and volume units

Speed, distance and time

Average speed with hours and minutes

Density, pressure and other rates

Half a cup of milk, then a third of a cup more: is that $\frac{2}{5}$ 2/5 of a cup?

Converting $0.2 7 ˙$ 0.27 to a fraction (Higher)

Halve an A4 sheet and it keeps its shape. Why must that be $2$ √2?

A square grows 3 cm each way. Is its new area just $x^{2} + 9$ x² + 9?

Factorising $a x^{2} + b x + c$ ax² + bx + c (Higher)

The equation $y = m x + c$ y = mx + c

Solving $x^{2} + 2 x - 15 = 0$ x² + 2x − 15 = 0 (Higher)