Mini-Lesson

Ratio, Proportion & Rates of Change

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

Mission 1Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?
Mission 2A price goes up 20%, then down 20%. Is it back where it started?
Mission 3Money grows at 5% a year compound interest. When has it doubled?
Mission 44 builders take 6 days. How long would 8 builders take?
Mission 5A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?
Mission 6The pizza twice as wide costs three times as much. Good deal?

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

Mission 1

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 2

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

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

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.

£

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 3

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

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?

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.

Worked example Ratio and proportion · growth and decay

An iterative process (Higher)

Question

A lake has 1,000 fish. Each year the number grows by 2%, then 50 fish are removed. So

u_{n + 1} = 1.02 u_{n} - 50

with

u_{0} = 1000

. Find

u_{3}

, the number of fish after 3 years.

1
Year 1: $u_{1} = 1.02 \times 1000 - 50 = 970$ u₁ = 1.02 × 1000 − 50 = 970
2
Year 2: $u_{2} = 1.02 \times 970 - 50 = 939.4$ u₂ = 1.02 × 970 − 50 = 939.4
3
Year 3: $u_{3} = 1.02 \times 939.4 - 50 = 908.188$ u₃ = 1.02 × 939.4 − 50 = 908.188
4
Interpret: fish are whole, so about 908 fish

Answer

about 908 fish

⚠️ Watch out

each step uses the previous answer, not the starting value of 1,000. Keep the unrounded value on your calculator between steps.

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 4

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.
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.
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 5

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.
Imperial–metric conversions (such as 5 miles ≈ 8 km) are given in the question.

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).
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

?A crate with a weight of 600 N rests on a face measuring 0.4 m by 0.5 m. Find the pressure it exerts on the floor, in N/m². Use pressure

=

force ÷ area.

N/m²

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 6

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?

Recap

Big ideas

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.
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.
Higher: an iterative rule such as $u_{n + 1} = 1.02 u_{n} - 50$ uₙ₊₁ = 1.02 uₙ − 50 is applied one step at a time.
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.

🏆

Mini-lesson complete!

⭐⭐⭐

You've worked through Ratio, Proportion & Rates of Change for AQA GCSE Maths (8300). 🎉

Mission 1Paint mixed 3 blue : 2 yellow. 8 L of blue makes how much paint?
Mission 2A price goes up 20%, then down 20%. Is it back where it started?
Mission 3Money grows at 5% a year compound interest. When has it doubled?
Mission 44 builders take 6 days. How long would 8 builders take?
Mission 5A 1 m by 1 m by 1 m tank is full of water. Could two people lift it?
Mission 6The pizza twice as wide costs three times as much. Good deal?

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