Worked example · Easy
Simplify a ratio
Simplify \(24:36\).
HCF\((24,36)=12\).
\[24:36=2:3\]Answer: \(\boxed{2:3}\)
Chapter 11
Ratio and proportion
Learn to compare quantities using ratios, scale ratios reliably, divide totals into unequal shares, and recognise whether a situation uses direct or inverse proportion.
By the end of this chapter, you should be able to
- simplify two-part and three-part ratios fully;
- convert units before simplifying measurement ratios;
- write ratios in the form \(1:n\), including scales;
- use equivalent ratios to find unknown quantities;
- divide a total quantity in a stated ratio;
- solve direct-proportion problems using a multiplier or unitary reasoning;
- recognise inverse proportion and divide by the multiplier when appropriate;
- check answers for sense and consistency with the ratio or proportion.
1
Subtopic 1
Ratio
A ratio compares quantities. The order matters: \(2:5\) does not mean the same as \(5:2\). Ratios can be simplified in the same way as fractions by dividing every part by a common factor.
6 : 4
Divide both parts by 2
3 : 2Simplifying a ratio
- If units are present, convert every quantity to the same unit.
- Find a common factor of all parts—ideally the HCF.
- Divide every part by the same factor.
- Check that no further common factor remains.
Worked example · Medium
Simplify a ratio with different units
Simplify 2.4 m : 60 cm.
Convert 2.4 m to centimetres:
\[2.4\text{ m}=240\text{ cm}\]\[240:60=4:1\]Answer: \(\boxed{4:1}\)
Worked example · Medium
Three-part ratio
Simplify \(42:63:105\).
The HCF of 42, 63 and 105 is 21.
\[42:63:105=2:3:5\]Writing a ratio in the form \(1:n\)
To make the left-hand part equal to 1, divide every part by the left-hand value.
Worked example · Medium
Write a scale as \(1:n\)
Write 8 mm : 2 km in the form \(1:n\).
Use the same unit:
\[2\text{ km}=2\,000\,000\text{ mm}\]\[8:2\,000\,000=1:250\,000\]Answer: \(\boxed{1:250\,000}\)
Map1 cm
↔
Ground50 000 cm = 500 m
Common mistake
Do not cancel quantities with different units. Convert first. In map scales, a surprisingly large value of \(n\) is normal because centimetres on a map represent metres or kilometres on the ground.
Additional exam-focused worked example
Worked example · Exam-style
Simplify the ratio \(\frac34:\frac58\).
Step-by-step solution
Multiply both quantities by 8 to remove denominators: \[\frac34:\frac58=6:5\] Answer \(\boxed{6:5}\).
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Simplify 35:49.
Show Answer
Steps: Divide both by 7: \(35:49=\boxed{5:7}\).
2Easy practice
Simplify 18:24:30.
Show Answer
Steps: Divide each part by 6: \(\boxed{3:4:5}\).
3Medium practice
Simplify 75 cm : 1.5 m.
Show Answer
Steps: Convert 1.5 m to 150 cm: \(75:150=\boxed{1:2}\).
4Medium practice
Write 24:40 in the form 1:n.
Show Answer
Steps: Divide both parts by 24: \(1:40/24=\boxed{1:5/3}\) or \(1:1.667\) approximately.
5Exam-style practice
The ratio of boys to girls is 8:12. What fraction of all students are girls?
Show Answer
Steps: Total parts \(8+12=20\). Fraction girls \(\frac{12}{20}=\boxed{\frac35}\).
2
Subtopic 2
Using ratio to find an unknown quantity
Equivalent ratios are produced by multiplying or dividing every part by the same number. If one actual quantity is known, use it to find the multiplier, then apply that multiplier to the other part.
| Ratio | 2 | 5 |
|---|---|---|
| × 4 | 8 | 20 |
Worked example · Easy
Scale a ratio
The ratio of red to white paint is \(1:3\). If 4 litres of red paint are used, how much white paint is needed?
The red quantity has been multiplied by 4, so multiply the white quantity by 4.
\[1:3\quad\to\quad4:12\]Answer: \(\boxed{12\text{ litres}}\)
Worked example · Medium
Map scale
A map uses 1 cm : 2.5 km. Two places are 6.8 cm apart on the map. Find the real distance.
\[6.8\times2.5=17\]
Answer: \(\boxed{17\text{ km}}\)
Worked example · Exam-style
Find a missing dimension
Two similar photographs have corresponding side lengths in the ratio \(3:7\). The smaller width is 12 cm. Find the corresponding larger width.
From 3 to 12, multiply by 4.
\[7\times4=28\]Answer: \(\boxed{28\text{ cm}}\)
Quick check
If the ratio is \(2:5\), the second quantity must be \(\frac52=2.5\) times the first. Use that relationship to check your answer.
Additional exam-focused worked example
Worked example · Exam-style
In a mixture, red:blue is 5:8. If there are 3.2 litres of blue, find the red paint needed.
Step-by-step solution
One ratio part represents \(3.2\div8=0.4\) L. Red uses 5 parts: \(5\times0.4=\boxed{2.0\text{ L}}\).
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
The ratio of pencils to pens is 4:3. There are 20 pencils. How many pens?
Show Answer
Steps: A factor of 5 turns 4 into 20, so \(3\times5=\boxed{15}\) pens.
2Easy practice
A drink uses juice:water = 2:5. How much water goes with 6 L of juice?
Show Answer
Steps: Juice was multiplied by 3: \(5\times3=\boxed{15\text{ L}}\).
3Medium practice
A map scale is 1 cm:4 km. A route measures 7.5 cm. Find its real distance.
Show Answer
Steps: \[7.5\times4=\boxed{30\text{ km}}\]
4Medium practice
Two photos have height ratio small:large = 3:8. The large photo is 40 cm tall. Find the small height.
Show Answer
Steps: One part is \(40\div8=5\) cm; small height \(3\times5=\boxed{15\text{ cm}}\).
5Exam-style practice
The ratio of adults to children is 5:7. If there are 84 children, how many people are there altogether?
Show Answer
Steps: One part is \(84\div7=12\). Adults \(5\times12=60\). Total \(60+84=\boxed{144}\).
3
Subtopic 3
Dividing a quantity in a given ratio
A ratio gives the number of equal shares. To divide a total, first find the total number of shares, then the value of one share.
Three-step method
- Add the ratio parts to find the total number of shares.
- Divide the total quantity by the number of shares to find one share.
- Multiply one share by each ratio part.
Worked example · Easy
Share money in a ratio
Share $360 in the ratio \(2:3\).
Total shares: \(2+3=5\).
\[360\div5=72\]Amounts:
\[2\times72=144,\qquad3\times72=216\]Answer: \(\boxed{\$144\text{ and }\$216}\)
Check: \(144+216=360\).
Worked example · Medium
Three-way sharing
Share 1.8 kg in the ratio \(2:3:4\). Give each amount in grams.
Convert: \(1.8\text{ kg}=1800\text{ g}\).
Total shares: \(2+3+4=9\).
\[1800\div9=200\text{ g per share}\]\[2\times200=400,\quad3\times200=600,\quad4\times200=800\]Answer: \(\boxed{400\text{ g},600\text{ g},800\text{ g}}\)
Worked example · Exam-style
Use one known share to find the total
Sweets are shared between A and B in the ratio \(5:2\). A receives 35 sweets. How many sweets are shared altogether?
5 shares correspond to 35 sweets.
\[35\div5=7\text{ sweets per share}\]Total shares: \(5+2=7\).
\[7\times7=\boxed{49}\]Additional exam-focused worked example
Worked example · Exam-style
Three friends divide $1 680 in the ratio 2:5:7. Find the difference between the largest and smallest shares.
Step-by-step solution
Total parts \(2+5+7=14\); one part \(1680\div14=120\). Largest \(7\times120=840\); smallest \(2\times120=240\). Difference \(840-240=\boxed{\$600}\).
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Share $240 in the ratio 1:3.
Show Answer
Steps: Four parts; one part \(240\div4=60\). Shares \(\boxed{\$60,\ \$180}\).
2Easy practice
Divide 72 sweets in the ratio 4:5.
Show Answer
Steps: Nine parts; one part \(72\div9=8\). Shares \(\boxed{32,40}\).
3Medium practice
Share 650 g in the ratio 2:3:5.
Show Answer
Steps: Ten parts; one part 65 g. Shares \(\boxed{130\text{ g},195\text{ g},325\text{ g}}\).
4Medium practice
A total of 54 pupils has boys:girls = 4:5. Find numbers of each.
Show Answer
Steps: Nine parts; one part 6 pupils. Boys \(4\times6=\boxed{24}\), girls \(5\times6=\boxed{30}\).
5Exam-style practice
A prize of $875 is shared in ratio 3:4. By how much does the larger share exceed the smaller?
Show Answer
Steps: Seven parts; one part $125. Larger $500, smaller $375; difference \(\boxed{\$125}\).
4
Subtopic 4
Proportion
Proportion describes how two quantities change together. In this chapter, the key distinction is between direct proportion and inverse proportion.
Direct proportion
Quantity A ↑Quantity B ↑
If one quantity is multiplied by a factor, the other is multiplied by the same factor.
Example: more cupcakes → more flour.
Inverse proportion
Quantity A ↑Quantity B ↓
If one quantity is multiplied by a factor, the other is divided by that factor.
Example: more workers → less time for the same job.
Direct proportion
Worked example · Easy
Use a multiplier
5 notebooks cost $12. How much do 15 notebooks cost at the same rate?
15 is 3 times 5.
\[12\times3=\boxed{\$36}\]Worked example · Medium
Direct proportion with a non-integer multiplier
A car uses 18 litres of fuel for 144 km. Estimate the fuel needed for a similar 360 km journey.
Multiplier:
\[\frac{360}{144}=2.5\]\[18\times2.5=\boxed{45\text{ litres}}\]Inverse proportion
Worked example · Medium
Workers and time
6 workers can complete a job in 10 days. How long would 15 workers take, assuming they all work at the same rate?
Workers are multiplied by
\[\frac{15}{6}=2.5\]Time is divided by 2.5:
\[10\div2.5=\boxed{4\text{ days}}\]Worked example · Exam-style
Speed and time for a fixed journey
A journey takes 54 minutes at 40 km/h. How long would the same journey take at 60 km/h?
Speed multiplier:
\[\frac{60}{40}=1.5\]For the same distance, time is inversely proportional to speed:
\[54\div1.5=\boxed{36\text{ minutes}}\]Worked example · Exam-style
Decide direct or inverse
8 taps fill a tank in 15 hours. How long would 12 identical taps take?
More taps means less time, so this is inverse proportion.
Multiplier in number of taps:
\[\frac{12}{8}=1.5\]\[15\div1.5=\boxed{10\text{ hours}}\]How to decide
Ask what should happen physically. If one quantity increases, should the other also increase? That suggests direct proportion. If it should decrease for the same job, distance or supply, that suggests inverse proportion.
Additional exam-focused worked example
Worked example · Exam-style
Eight pumps empty a pool in 15 h, all working at the same rate. How long do 12 pumps need?
Step-by-step solution
For fixed work, pumps × time stays constant (inverse proportion). \[8\times15=12\times t\Rightarrow t=\frac{120}{12}=\boxed{10\text{ h}}\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
3 notebooks cost $7.50. How much do 6 cost at the same price?
Show Answer
Steps: Twice as many notebooks: \(7.50\times2=\boxed{\$15.00}\).
2Easy practice
4 kg of fruit cost $18. Find the cost of 10 kg.
Show Answer
Steps: Cost per kilogram \(18\div4=\$4.50\). For 10 kg: \(4.50\times10=\boxed{\$45}\).
3Medium practice
A recipe for 5 people needs 350 g of rice. How much is needed for 8?
Show Answer
Steps: Direct proportion: \(350\times8/5=\boxed{560\text{ g}}\).
4Medium practice
6 workers complete a job in 9 days. How long for 3 equally efficient workers?
Show Answer
Steps: Inverse proportion: \(6\times9=3\times t\Rightarrow t=\boxed{18\text{ days}}\).
5Exam-style practice
A car takes 4 h at 60 km/h for a fixed journey. What constant speed is required to finish in 3 h?
Show Answer
Steps: Distance \(60\times4=240\) km. Required speed \(240\div3=\boxed{80\text{ km/h}}\).
Chapter 11 practice
Start with straightforward ratio fluency, then move into scale, sharing, direct proportion and inverse-proportion problems. Keep each answer closed until you have attempted the question.
A. Build confidence
1Simplify a ratio[2 marks]
Simplify \(42:56\).
Show Answer
HCF\((42,56)=14\).
\[42:56=3:4\]\(\boxed{3:4}\)
2Different units[3 marks]
Simplify 1.5 kg : 450 g.
Show Answer
Convert 1.5 kg to 1500 g.
\[1500:450=10:3\]\(\boxed{10:3}\)
3Three-part ratio[2 marks]
Simplify \(18:30:42\).
Show Answer
Divide every part by 6.
\[18:30:42=\boxed{3:5:7}\]4Form \(1:n\)[3 marks]
Write \(4:15\) in the form \(1:n\).
Show Answer
Divide both parts by 4.
\[4:15=1:\frac{15}{4}=\boxed{1:3.75}\]B. Equivalent ratios and sharing
5Unknown quantity[3 marks]
The ratio of boys to girls is \(3:5\). If there are 18 boys, how many girls are there?
Show Answer
3 parts become 18, so the multiplier is 6.
\[5\times6=\boxed{30}\]6Map scale[4 marks]
A map scale is 1 cm : 4 km. Two towns are 7.5 cm apart on the map. Find the actual distance.
Show Answer
\[7.5\times4=\boxed{30\text{ km}}\]
7Share in a ratio[4 marks]
Share $540 in the ratio \(4:5\).
Show Answer
Total shares: \(4+5=9\).
\[540\div9=60\]\[4\times60=240,\qquad5\times60=300\]\(\boxed{\$240\text{ and }\$300}\)
8Three-way sharing[5 marks]
Divide 960 g in the ratio \(2:3:7\).
Show Answer
Total shares: \(2+3+7=12\).
\[960\div12=80\text{ g}\]Amounts:
\[160\text{ g},\ 240\text{ g},\ 560\text{ g}\]9Known share to total[4 marks]
A and B share money in the ratio \(7:3\). A receives $189. Find the total amount shared.
Show Answer
One share:
\[189\div7=27\]Total shares = 10.
\[10\times27=\boxed{\$270}\]C. Proportion
10Direct proportion[4 marks]
12 metres of cable has a mass of 7.8 kg. Find the mass of 30 metres of the same cable.
Show Answer
Multiplier:
\[30\div12=2.5\]\[7.8\times2.5=\boxed{19.5\text{ kg}}\]11Currency-style direct proportion[4 marks]
£40 is worth $52 at a fixed exchange rate. How many dollars is £150 worth?
Show Answer
Multiplier:
\[150\div40=3.75\]\[52\times3.75=\boxed{\$195}\]12Inverse proportion[4 marks]
10 workers take 18 days to complete a job. How long would 15 workers take at the same rate?
Show Answer
Worker multiplier:
\[15\div10=1.5\]Divide the time by 1.5:
\[18\div1.5=\boxed{12\text{ days}}\]13Inverse proportion with a supply[4 marks]
A supply of food feeds 24 animals for 15 days. How long will the same supply feed 30 animals?
Show Answer
Animal multiplier:
\[30\div24=1.25\]\[15\div1.25=\boxed{12\text{ days}}\]D. Exam-style challenge
14Scale in \(1:n\)[5 marks]
On a plan, 6 mm represents 1.8 m. Write the scale in the form \(1:n\).
Show Answer
Convert 1.8 m to 1800 mm.
\[6:1800=\boxed{1:300}\]15Ratio and total[6 marks]
The ratio of red, blue and green counters is \(4:5:3\). There are 24 green counters. Find the total number of counters.
Show Answer
3 shares correspond to 24, so one share is 8.
Total shares:
\[4+5+3=12\]\[12\times8=\boxed{96}\]16Mixture problem[6 marks]
A drink is made by mixing concentrate and water in the ratio \(2:7\). A bottle contains 1.35 litres of drink. Find the volume of concentrate in millilitres.
Show Answer
Total shares = 9.
Convert 1.35 L to 1350 ml.
\[1350\div9=150\text{ ml per share}\]\[2\times150=\boxed{300\text{ ml}}\]17Direct or inverse?[6 marks]
At a constant speed, a machine produces 840 components in 6 hours. a) How many components will it produce in 9 hours? b) State whether this is direct or inverse proportion.
Show Answer
a) Time multiplier:
\[9\div6=1.5\]\[840\times1.5=\boxed{1260}\]b) More time gives more output at the same rate, so it is \(\boxed{\text{direct proportion}}\).
18Fixed-distance journey[7 marks]
A journey takes 72 minutes at an average speed of 50 km/h. The same distance is travelled at 80 km/h. Find the new journey time.
Show Answer
For a fixed distance, speed and time are inversely proportional.
Speed multiplier:
\[80\div50=1.6\]\[72\div1.6=\boxed{45\text{ minutes}}\]Chapter 11 high-grade checklist
Units first
I convert measurement ratios to the same units before simplifying.
Simplest form
I divide every part by the same common factor and check that no further simplification is possible.
Equivalent ratios
I apply exactly the same multiplier to every part of a ratio.
Sharing
I add the parts, find one share, then multiply back—and check the shares sum to the total.
Direct proportion
I multiply both quantities by the same factor.
Inverse proportion
For the same job, distance or supply, if one quantity is multiplied by a factor, I divide the other by that factor.