Mini-Lesson

Number

9 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?

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.

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.

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.

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.

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.

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.
Higher: estimate roots from nearby squares: $51$ √51 is just over 7, as $7^{2} = 49$ 7² = 49.

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.

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.

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.
This is the error interval. A truncated value is cut off: 2.1 truncated to 1 d.p. gives $2.1 \leq x < 2.2$ 2.1 ≤ x < 2.2.
Higher: for counts (discrete data), 200 cars to the nearest hundred means $150 \leq n \leq 249$ 150 ≤ n ≤ 249.

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.
For $a + b$ a + √b in the denominator, multiply top and bottom by $a - b$ a − √b; the surds cancel as on screen 5.

Worked example Number · surds

Rationalising with a bracket (Higher)

Question

Rationalise the denominator of

\frac{4}{3 - 5}

. Simplify your answer.

1
Multiply top and bottom by $3 + 5$ 3 + √5: $\frac{4 ( 3 + 5 )}{( 3 - 5 ) ( 3 + 5 )}$ (4(3 + √5))/((3 − √5)(3 + √5))
2
The denominator: $9 + 35 - 35 - 5 = 4$ 9 + 3√5 − 3√5 − 5 = 4
3
The numerator: $12 + 45$ 12 + 4√5
4
Divide by 4: $\frac{12 + 4 5}{4} = 3 + 5$ (12 + 4√5)/4 = 3 + √5

Answer

3 + 5

⚠️ Watch out

multiplying by

3 - 5

(the same bracket) leaves a surd in the denominator; change the sign.

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?

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.
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.
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 by $\frac{a}{a}$ (√a)/(√a); for $a + b$ a + √b, multiply by $a - b$ a − √b.
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.

🏆

Mini-lesson complete!

⭐⭐⭐

You've worked through Number for OCR GCSE Maths (J560). 🎉

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?

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Next: test yourself in the Verify stage.

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→ Back to GCSE Mathematics · OCR

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

Number

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

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)

Rationalising with a bracket (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)

Big ideas

Mini-lesson complete!

📣 Smashed it? Share your score

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?