IGCSE Ratio and Proportion: Exam Methods
A clear IGCSE lesson on simplifying ratios, sharing quantities, direct proportion, inverse proportion, and exam-style setup.
🎯 Key idea: Ratio and proportion questions are usually won by setting up the parts correctly before calculating.
You will meet these ideas in recipes, maps, scale drawings, rates, geometry, and word problems.
🔢 1. Simplifying Ratios
A ratio compares quantities in the same unit.
To simplify a ratio, divide every part by the highest common factor.
Worked example: Simplify $18:24$.
The highest common factor of $18$ and $24$ is $6$.
$18 [[div]] 6 = 3$
$24 [[div]] 6 = 4$
So $18:24 = 3:4$.
Common IGCSE trap: If the units are different, convert first.
For example, $1.5$ m is $150$ cm, so $1.5$ m : $60$ cm becomes $150:60 = 5:2$.
⚖️ 2. Sharing in a Ratio
When a total is shared in a ratio, add the ratio parts first.
Worked example: Share USD 560 in the ratio $3:5$.
Step 1
Add the parts.
$3+5=8$
Step 2
Find one part.
$560 [[div]] 8 = 70$
Step 3
Find each share.
$3 [[cdot]] 70 = 210$
$5 [[cdot]] 70 = 350$
The two amounts are USD 210 and USD 350.
Ratio share check:
For USD 560 in the ratio $3:5$, the one-part value is:
$$[[frac{560}{3+5}]] = 70$$
⚡ 3. Direct and Inverse Proportion
Direct proportion means both quantities change in the same direction.
If $y$ is directly proportional to $x$, write $y=kx$.
Direct proportion example: $y$ is directly proportional to $x$. When $x=6$, $y=18$. Find $y$ when $x=11$.
Step 1
Use $y=kx$.
$18=6k$
Step 2
Find $k$.
$k=3$
Step 3
Substitute $x=11$.
$y=3 [[cdot]] 11=33$
Inverse proportion means one quantity increases while the other decreases.
If $y$ is inversely proportional to $x$, write $y=[[frac{k}{x}]]$.
Inverse proportion example: $y$ is inversely proportional to $x$. When $x=4$, $y=15$. Find $y$ when $x=10$.
Step 1
Use $y=[[frac{k}{x}]]$.
$15=[[frac{k}{4}]]$
Step 2
Find $k$.
$k=60$
Step 3
Substitute $x=10$.
$y=[[frac{60}{10}]]=6$
✅ Exam check: Before calculating, ask whether the answer should get bigger or smaller.
If it should get bigger, direct proportion may fit. If it should get smaller, inverse proportion may fit.
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Practice Problems
Try these IGCSE-style questions. Open each answer only after you have attempted the problem.
1. Simplify $45:60$.
Show answer
Step 1
Find the highest common factor.
The highest common factor of $45$ and $60$ is $15$.
Step 2
Divide both parts by $15$.
$45 [[div]] 15 = 3$
$60 [[div]] 15 = 4$
Answer: $3:4$
2. Share £720 in the ratio $5:7$.
Show answer
Step 1
Add the ratio parts.
$5+7=12$
Step 2
Find one part.
$720 [[div]] 12=60$
Step 3
Find each share.
$5 [[cdot]] 60=300$
$7 [[cdot]] 60=420$
Answer: £300 and £420
3. The ratio of boys to girls is $4:5$. There are $36$ students in total. How many girls are there?
Show answer
Step 1
Add the parts.
$4+5=9$
Step 2
Find one part.
$36 [[div]] 9=4$
Step 3
Girls are $5$ parts.
$5 [[cdot]] 4=20$
Answer: $20$ girls
4. $y$ is directly proportional to $x$. When $x=8$, $y=20$. Find $y$ when $x=14$.
Show answer
Step 1
Use $y=kx$.
$20=8k$
Step 2
Find $k$.
$k=2.5$
Step 3
Substitute $x=14$.
$y=2.5 [[cdot]] 14=35$
Answer: $35$
5. $t$ is inversely proportional to $n$. When $n=6$, $t=12$. Find $t$ when $n=9$.
Show answer
Step 1
Use $t=[[frac{k}{n}]]$.
$12=[[frac{k}{6}]]$
Step 2
Find $k$.
$k=72$
Step 3
Substitute $n=9$.
$t=[[frac{72}{9}]]=8$
Answer: $8$
Interactive Quiz
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