IGCSE Mathematics | Worked Solutions | 25 Questions
Section A — Recall
Questions 1–10
1.
What is a ratio? Write a definition in your own words.
Answer
A ratio shows how two or more quantities compare to each other. It is written with a colon (:).
Example: A bag with 3 red balls and 5 blue balls has a red : blue ratio of 3 : 5.
2.
What does it mean for a ratio to be in its simplest form?
Answer
A ratio is in its simplest form when all the numbers share no common factor other than 1 — you cannot divide all parts by the same whole number any further.
3.
Write the ratio $15 : 20$ in its simplest form.
Answer
- HCF of 15 and 20: test 5 → $15 \div 5 = 3$ ✓ $20 \div 5 = 4$ ✓. HCF = 5
- $15 \div 5 : 20 \div 5$
$3 : 4$
4.
Write the ratio $30 : 45 : 60$ in its simplest form.
Answer
- HCF of 30, 45, 60: test 15 → $30 \div 15 = 2$ ✓ $45 \div 15 = 3$ ✓ $60 \div 15 = 4$ ✓. HCF = 15
- $30 \div 15 : 45 \div 15 : 60 \div 15$
$2 : 3 : 4$
5.
Before you can simplify a ratio where the two quantities have different units, what must you do first?
Answer
Convert both quantities to the same unit before writing or simplifying the ratio.
Example: To simplify 50 cm : 2 m, first convert: $2\text{ m} = 200\text{ cm}$, giving $50 : 200$, which simplifies to $1 : 4$.
6.
A quantity is shared in the ratio $2 : 5$. How many parts are there in total?
Answer
$2 + 5 = \mathbf{7}$ parts in total.
7.
A quantity is divided in the ratio $1 : 3$. What fraction of the total does the smaller share represent?
Answer
Total parts = $1 + 3 = 4$. The smaller share is 1 part out of 4, so $\dfrac{1}{4}$ of the total.
8.
A map has a scale of $1 : 25\,000$. What does this mean?
Answer
1 unit of distance on the map equals 25,000 of the same units in real life. For example, 1 cm on the map = 25,000 cm (= 250 m) in real life.
9.
Describe how you would decide which of two products is better value when they are different sizes.
Answer
Calculate the price per unit (e.g. price per gram or per ml) for each product by dividing the price by the quantity. The product with the lower price per unit is better value.
10.
When scaling a recipe up or down, all ingredient amounts are multiplied by the same number. What is this number called?
Answer
The scale factor.
How to find it: Scale factor = new number of people ÷ original number of people.
Section B — Application
Questions 11–20
11.
Write the ratio $75\text{ cm} : 2\text{ m}$ in its simplest form.
Answer
- Convert to the same unit: $2\text{ m} = 200\text{ cm}$
- Write the ratio: $75 : 200$
- HCF of 75 and 200: test 25 → $75 \div 25 = 3$ ✓ $200 \div 25 = 8$ ✓. HCF = 25
- $75 \div 25 : 200 \div 25$
$3 : 8$
12.
Share 63 in the ratio $2 : 7$. Find each share.
Answer
- Total parts: $2 + 7 = 9$ → place 63 in the Total column
- Multiplier: $63 \div 9 = 7$
- Multiply all parts by 7:
| Share A | Share B | Total | |
|---|---|---|---|
| Ratio | 2 | 7 | 9 |
| × 7 | 14 | 49 | 63 |
Share A = 14 Share B = 49
Check: $14 + 49 = 63$ ✓
13.
120 students choose one after-school activity. The ratio of sport : drama : art is $3 : 2 : 1$. How many students are in each group?
Answer
- Total parts: $3 + 2 + 1 = 6$ → place 120 in the Total column
- Multiplier: $120 \div 6 = 20$
- Multiply all parts by 20:
| Sport | Drama | Art | Total | |
|---|---|---|---|---|
| Ratio | 3 | 2 | 1 | 6 |
| × 20 | 60 | 40 | 20 | 120 |
Sport = 60 Drama = 40 Art = 20
Check: $60 + 40 + 20 = 120$ ✓
14.
A bag contains blue and red counters. The ratio of blue : red is $4 : 3$. There are 28 blue counters. How many red counters are there?
Answer
The 28 is the blue quantity — place it in the Blue column, not the Total column.
- Multiplier: $28 \div 4 = 7$
- Multiply all parts by 7:
| Blue | Red | Total | |
|---|---|---|---|
| Ratio | 4 | 3 | 7 |
| × 7 | 28 | 21 | 49 |
Red counters = 21
15.
Ali and Ben share some money in the ratio $5 : 3$. Ali receives £40 more than Ben. Find how much each person receives.
Answer
The £40 is the difference between their shares. Use a Difference column. Subtract the ratio parts to find the difference column value.
- Difference in ratio parts: $5 – 3 = 2$ → place 40 in the Difference column
- Multiplier: $40 \div 2 = 20$
- Multiply all parts by 20:
| Ali | Ben | Difference | |
|---|---|---|---|
| Ratio | 5 | 3 | 2 |
| × 20 | 100 | 60 | 40 |
Ali = £100 Ben = £60
Check: $100 – 60 = 40$ ✓ Ratio: $100 : 60 = 5 : 3$ ✓
16.
A recipe for 6 people uses 450 ml of milk. How much milk is needed for 10 people?
Answer
- Scale factor: $10 \div 6 = \dfrac{5}{3}$
- Simplify before multiplying: 450 and 3 share a factor of 3 → $450 \div 3 = 150$
- Milk needed: $150 \times 5 = 750\text{ ml}$
$750\text{ ml}$
Alternative check: Milk per person $= 450 \div 6 = 75\text{ ml}$. For 10 people: $75 \times 10 = 750\text{ ml}$ ✓
17.
A map has a scale of $1 : 50\,000$. Two towns are $6\text{ cm}$ apart on the map. Find the real distance between the towns in km.
Answer
- Multiply map distance by scale factor: $6 \times 50\,000 = 300\,000\text{ cm}$
- Convert to metres: $300\,000 \div 100 = 3\,000\text{ m}$
- Convert to km: $3\,000 \div 1\,000 = 3\text{ km}$
Real distance = 3 km
18.
Pack A: 400 g for 160p. Pack B: 600 g for 222p. Which pack is better value?
Answer
- Pack A price per 100 g: $160 \div 4 = 40\text{p per 100 g}$
- Pack B price per 100 g: $222 \div 6 = 37\text{p per 100 g}$
- Compare: $37\text{p} < 40\text{p}$, so Pack B costs less per 100 g
Pack B is better value at 37p per 100 g, compared to 40p per 100 g for Pack A.
19.
A class has 35 students. The ratio of boys : girls is $3 : 4$. How many girls are in the class?
Answer
- Total parts: $3 + 4 = 7$ → place 35 in the Total column
- Multiplier: $35 \div 7 = 5$
- Multiply all parts by 5:
| Boys | Girls | Total | |
|---|---|---|---|
| Ratio | 3 | 4 | 7 |
| × 5 | 15 | 20 | 35 |
Girls = 20
20.
A drink is made by mixing water and juice in the ratio $7 : 2$. Mia has 350 ml of water and 80 ml of juice. Find the maximum amount of drink Mia can make.
Answer
- Divide each available amount by its ratio number:
Water: $350 \div 7 = 50$
Juice: $80 \div 2 = 40$ - Choose the lowest value as the multiplier: 40 (using 50 would require $2 \times 50 = 100\text{ ml}$ juice, but only 80 ml is available)
- Amounts used: Water $= 7 \times 40 = 280\text{ ml}$ Juice $= 2 \times 40 = 80\text{ ml}$
- Total drink: $280 + 80 = 360\text{ ml}$
Maximum drink = 360 ml
Note: All 80 ml of juice is used up. Mia has $350 – 280 = 70\text{ ml}$ of water left over.
Section C — Challenge
Questions 21–25
21.
(a) Simplify $48 : 72 : 120$
(b) A school has 360 students. Girls : boys = $7 : 5$. How many boys?
(c) 30 more girls join. New ratio girls : boys in simplest form.
Answer
(a) Simplify 48 : 72 : 120
- Test 24: $48 \div 24 = 2$ ✓ $72 \div 24 = 3$ ✓ $120 \div 24 = 5$ ✓. HCF = 24
- $48 \div 24 : 72 \div 24 : 120 \div 24$
$2 : 3 : 5$
(b) 360 students, girls : boys = 7 : 5. How many boys?
- Total parts: $7 + 5 = 12$ → place 360 in the Total column
- Multiplier: $360 \div 12 = 30$
- Multiply all parts by 30:
| Girls | Boys | Total | |
|---|---|---|---|
| Ratio | 7 | 5 | 12 |
| × 30 | 210 | 150 | 360 |
Boys = 150
(c) 30 more girls join. New ratio girls : boys.
- New number of girls: $210 + 30 = 240$. Boys unchanged: 150
- New ratio: $240 : 150$
- HCF of 240 and 150: test 30 → $240 \div 30 = 8$ ✓ $150 \div 30 = 5$ ✓. HCF = 30
New ratio girls : boys = 8 : 5
22.
(a) Tom and Sam share money in ratio $5 : 2$. Tom receives £42 more than Sam. How much does each receive?
(b) Find the total amount shared.
Answer
(a) Tom : Sam = 5 : 2, Tom gets £42 more.
The £42 is the difference. Use a Difference column.
- Difference in ratio parts: $5 – 2 = 3$ → place 42 in the Difference column
- Multiplier: $42 \div 3 = 14$
- Multiply all parts by 14:
| Tom | Sam | Difference | |
|---|---|---|---|
| Ratio | 5 | 2 | 3 |
| × 14 | 70 | 28 | 42 |
Tom = £70 Sam = £28
Check: $70 – 28 = 42$ ✓ Ratio $70 : 28 = 5 : 2$ ✓
(b) Total amount shared.
Total $= 70 + 28 = $ £98
23.
Map scale $1 : 40\,000$.
(a) Two villages are $7.5\text{ cm}$ apart on the map. Real distance in km?
(b) Real distance town to lake = $18\text{ km}$. Map distance in cm?
(c) A different map shows the same town and lake $6\text{ cm}$ apart. Scale in form $1 : n$?
Answer
(a) Map 7.5 cm → real distance
- Multiply by scale: $7.5 \times 40\,000 = 300\,000\text{ cm}$
- Convert: $300\,000\text{ cm} \div 100 = 3\,000\text{ m}$
- Convert: $3\,000\text{ m} \div 1\,000 = 3\text{ km}$
Real distance = 3 km
(b) Real 18 km → map distance
- Convert to cm: $18\text{ km} = 18 \times 1\,000 = 18\,000\text{ m} = 18\,000 \times 100 = 1\,800\,000\text{ cm}$
- Divide by scale: $1\,800\,000 \div 40\,000$
Simplify: $1\,800\,000 \div 40\,000 = 1\,800 \div 40 = 45$
Map distance = 45 cm
(c) Same 18 km shown as 6 cm — find the scale.
- Real distance in cm: $1\,800\,000\text{ cm}$ (from part b)
- Scale $n = 1\,800\,000 \div 6 = 300\,000$
Scale = $1 : 300\,000$
Note: A larger $n$ means a smaller scale map — more area shown but less detail. This map shows the same distance in less space than the 1:40,000 map.
24.
(a) £720 shared in ratio $5 : 4 : 3$. How much does each friend receive?
(b) The friend with the smallest share spends half. What fraction of £720 is this?
Answer
(a) £720 in ratio 5 : 4 : 3
- Total parts: $5 + 4 + 3 = 12$ → place 720 in the Total column
- Multiplier: $720 \div 12 = 60$
- Multiply all parts by 60:
| Friend A | Friend B | Friend C | Total | |
|---|---|---|---|---|
| Ratio | 5 | 4 | 3 | 12 |
| × 60 | 300 | 240 | 180 | 720 |
Friend A = £300 Friend B = £240 Friend C = £180
(b) Smallest share spends half — fraction of £720.
- Smallest share = £180
- Half of £180: $180 \div 2 = \text{\pounds}90$
- Fraction of total: $\dfrac{90}{720} = \dfrac{1}{8}$
$\dfrac{1}{8}$ of £720
25.
Peach paint: red : yellow : white = $5 : 3 : 2$. Sam has 200 ml red, 90 ml yellow, 80 ml white.
(a) Maximum peach paint Sam can make?
(b) How much of each colour is left over?
(c) Sam buys 60 ml more yellow. Can Sam make more paint? How much additional paint?
Answer
(a) Maximum paint
- Divide each available amount by its ratio number:
Red: $200 \div 5 = 40$
Yellow: $90 \div 3 = 30$
White: $80 \div 2 = 40$ - Lowest value = 30 (yellow is the limiting colour) → multiplier = 30
- Amounts used: Red $= 5 \times 30 = 150\text{ ml}$ Yellow $= 3 \times 30 = 90\text{ ml}$ White $= 2 \times 30 = 60\text{ ml}$
- Total paint: $150 + 90 + 60 = 300\text{ ml}$
Maximum peach paint = 300 ml
(b) Amounts left over
- Red left: $200 – 150 = 50\text{ ml}$
- Yellow left: $90 – 90 = 0\text{ ml}$
- White left: $80 – 60 = 20\text{ ml}$
Red: 50 ml left Yellow: 0 ml left White: 20 ml left
(c) Sam buys 60 ml more yellow — can Sam make more?
After buying: Red = 50 ml, Yellow = $0 + 60 = 60\text{ ml}$, White = 20 ml
- Divide each amount by its ratio number:
Red: $50 \div 5 = 10$
Yellow: $60 \div 3 = 20$
White: $20 \div 2 = 10$ - Lowest value = 10 → multiplier = 10
- Amounts used: Red $= 5 \times 10 = 50\text{ ml}$ Yellow $= 3 \times 10 = 30\text{ ml}$ White $= 2 \times 10 = 20\text{ ml}$
- Additional paint: $50 + 30 + 20 = 100\text{ ml}$
Yes — Sam can make an additional 100 ml of peach paint. Red and white are now both used up completely.
Note: Sam has $60 – 30 = 30\text{ ml}$ of yellow left over after this second batch.
