Ratio and direct proportion
🎯 In this topic you will
- Use the relationship between ratio and direct proportion
🧠 Key Words
- comparison
- justify
- proportion
- shades
Show Definitions
- comparison: Checking how two or more numbers, objects, or ideas are alike or different.
- justify: To explain or give reasons to support an answer or method.
- proportion: A statement showing that two ratios are equal, or a part in relation to the whole.
- shades: Variations in color or tone, often used in diagrams to show differences in value.
You can see ratios in a variety of situations, such as mixing ingredients in a recipe or sharing an amount among several people.
Ratios can also be used to make comparisons.
For example, suppose you wanted to compare two mixes of paint.
Pink paint is made from red and white paint in a certain ratio (red : white).
If two shades of pink paint have been mixed from red and white paint, how do you decide which shade is darker?
The shade which is darker is the shade with the greater proportion of red paint. You can change the ratios into fractions, decimals or percentages to compare the proportions of red paint in each shade.
❓ EXERCISES
1. Copy and complete the workings to change each ratio into a fraction.
a. A bag of nuts contains cashew nuts and peanuts in the ratio $2:7$. What fraction of the nuts are
i cashew nuts ii peanuts?
b. A box of toys has plastic and paper toys in the ratio $3:5$. What fraction of the toys are
i plastic ii paper?
c. A basket of fruit has apples and bananas in the ratio $3:1$. What fraction of the fruit are
i apples ii bananas?
👀 Show answer
1a.
Total parts $=2+7=9$
i Cashew nuts $=\dfrac{2}{9}$
ii Peanuts $=\dfrac{7}{9}$
1b.
Total parts $=3+5=8$
i Plastic $=\dfrac{3}{8}$
ii Paper $=\dfrac{5}{8}$
1c.
Total parts $=3+1=4$
i Apples $=\dfrac{3}{4}$
ii Bananas $=\dfrac{1}{4}$
🧠 Think like a Mathematician
Task: Compare two different methods for solving the same ratio problem and decide which approach you prefer.
Scenario: A school choir is made up of girls and boys in the ratio 2:1. There are 36 students in the choir altogether. How many are girls?
Methods:
Total parts = 2 + 1 = 3
Value of 1 part = 36 ÷ 3 = 12
Girls = 2 × 12 = 24
Fraction of choir that are girls = $\dfrac{2}{2+1} = \dfrac{2}{3}$
Girls = $\dfrac{2}{3} \times 36 = 24$
Questions:
👀 show answer
- a) Both methods give the same answer (24 girls). - Tio uses ratio parts: total → value of 1 part → multiply. - Kai uses fractions: find fraction of total directly. They are mathematically equivalent, but presented differently.
- b) Preference may vary: - Some prefer Tio’s method because it follows the ratio step by step and feels systematic. - Others prefer Kai’s method because it is quicker if you are confident with fractions. Both are valid and efficient depending on the learner’s strengths.
❓ EXERCISES
3. A tin of biscuits contains coconut and ginger biscuits in the ratio $3:7$. The tin contains $50$ biscuits.
a. What fraction of the biscuits in the tin are coconut biscuits?
b. How many coconut biscuits are in the tin?
👀 Show answer
Total parts $=3+7=10$
3a. Fraction that are coconut $=\dfrac{3}{10}$
3b. Number of coconut biscuits $=\dfrac{3}{10} \times 50 = 15$
4. A school tennis club has $35$ members. The ratio of boys to girls is $4:3$.
a. What fraction of the club members are girls?
b. How many girls are in the club?
👀 Show answer
Total parts $=4+3=7$
4a. Fraction that are girls $=\dfrac{3}{7}$
4b. Number of girls $=\dfrac{3}{7} \times 35 = 15$
🧠 Think like a Mathematician
Task: Use ratio reasoning to decide if the total number of counters can be 62 or 72.
Scenario: The ratio of red to blue counters in a bag is $5:4$. Sofia says: “It is possible that there are 62 counters in the bag.” Zara says: “It is possible that there are 72 counters in the bag.”
Questions:
👀 show answer
- Total ratio parts = 5 + 4 = 9.
- The total number of counters must be a multiple of 9.
- Check Sofia’s number: 62 ÷ 9 = 6 remainder 8 → not a multiple of 9 → ❌ not possible.
- Check Zara’s number: 72 ÷ 9 = 8 exactly → ✔ possible.
- Therefore, Zara is correct: 72 counters works with a 5:4 ratio, Sofia’s 62 does not.
❓ EXERCISES
6. The ratio of boys to girls in class 8C is $5:7$.
Which of these cards shows the number of learners that could be in class 8C?
A:25 B:28 C:32 D:36 E:38
Justify your choice.
👀 Show answer
Total parts $= 5+7=12$
The total number of learners must be a multiple of $12$.
Among the options: $36$ is a multiple of $12$.
Answer: D (36)
7. The ratio of men to women in a book club is $3:5$.
The number of adults in the book club is greater than $20$ but fewer than $30$.
How many adults are in the book club?
👀 Show answer
Total parts $= 3+5=8$
The total number of adults must be a multiple of $8$.
Multiples of $8$ between $20$ and $30$ are $24$ only.
Answer: 24 adults
8. This is part of Jan’s classwork.
Question:
A bag contains blue and yellow cubes. $\tfrac{3}{11}$ of the cubes are blue.
What is the ratio of blue to yellow cubes?
Answer (from Jan):
$\tfrac{3}{11}$ are blue, so $1-\tfrac{3}{11}=\tfrac{8}{11}$ are yellow.
So the ratio of blue to yellow is $\tfrac{3}{11}:\tfrac{8}{11}=3:8$.
Use Jan’s method to work out the following:
a. A bag contains green and red counters. $\tfrac{2}{3}$ of the counters are green.
What is the ratio of green to red counters?
b. A box of books contains history and science books. $\tfrac{3}{7}$ of the books are science books.
What is the ratio of history to science books?
c. A café sells sandwiches and cakes. $\tfrac{4}{9}$ of the items they sell are cakes.
What is the ratio of sandwiches to cakes that the café sells?
👀 Show answer
8a.
Green $=\tfrac{2}{3}$, so red $=1-\tfrac{2}{3}=\tfrac{1}{3}$.
Ratio green:red $=2:1$.
8b.
Science $=\tfrac{3}{7}$, so history $=1-\tfrac{3}{7}=\tfrac{4}{7}$.
Ratio history:science $=4:3$.
8c.
Cakes $=\tfrac{4}{9}$, so sandwiches $=1-\tfrac{4}{9}=\tfrac{5}{9}$.
Ratio sandwiches:cakes $=5:4$.
9. Shani mixes two shades of blue paint in the following ratios of blue:white:
| Sky blue $3:2$ | Sea blue $7:3$ |
a. What fraction of each shade of blue paint is white?
b. Which shade of blue paint is lighter?
Show all your working. Justify your choice.
🔎 Reasoning Tip
The paint which is lighter has a greater proportion of white.
👀 Show answer
Sky blue: total parts $=3+2=5$. White $=\tfrac{2}{5}$.
Sea blue: total parts $=7+3=10$. White $=\tfrac{3}{10}$.
Comparing: $\tfrac{2}{5}=0.4$ and $\tfrac{3}{10}=0.3$.
Sky blue has a higher fraction of white, so Sky blue is lighter.
10. Angelica mixes a fruit drink using mango juice and orange juice in the ratio $3:5$.
Sanjay mixes a fruit drink using mango juice and orange juice in the ratio $5:11$.
a. What fraction of each fruit drink is orange juice?
b. Whose fruit drink, Angelica’s or Sanjay’s, has the higher proportion of orange juice?
Show all your working. Justify your choice.
👀 Show answer
Angelica: total parts $=3+5=8$. Orange $=\tfrac{5}{8}=0.625$.
Sanjay: total parts $=5+11=16$. Orange $=\tfrac{11}{16}=0.6875$.
Sanjay’s drink has a higher proportion of orange juice.
11. In the Seals swimming club there are $13$ girls and $17$ boys.
a. What fraction of the children are boys?
In the Sharks swimming club there are $17$ girls and $23$ boys.
b. What fraction of the children are boys?
c. Which swimming club has the higher proportion of boys?
Show all your working. Justify your choice.
👀 Show answer
Seals: total $=13+17=30$. Boys $=\tfrac{17}{30}\approx0.567$.
Sharks: total $=17+23=40$. Boys $=\tfrac{23}{40}=0.575$.
Comparing: $0.567$ vs $0.575$. Sharks have the higher proportion of boys.
🧠 Think like a Mathematician
Task: Compare proportions in ratios to decide who has the greater share of black counters.
Scenario:
- Lin has black and white counters in the ratio 40:840.
- Ian has black and white counters in the ratio 25:535.
Questions:
👀 show answer
- a) - Lin: total = 40 + 840 = 880, proportion black = $\tfrac{40}{880} = \tfrac{1}{22} \approx 0.045$ (≈4.5%). - Ian: total = 25 + 535 = 560, proportion black = $\tfrac{25}{560} = \tfrac{5}{112} \approx 0.045$ (≈4.5%). Both proportions are equal – Lin and Ian have the same proportion of black counters.
- b) The best method is to convert each ratio into a fraction of the total. This makes comparison straightforward and avoids large numbers. Simplifying ratios to lowest terms is another quick check. Here, both simplify to almost the same fraction of about 1 part in 22.
❓ EXERCISES
13. Liam and Hannah collect coins and stamps.
Liam has $20$ coins and $320$ stamps.
Hannah has $15$ coins and $270$ stamps.
Use your favourite method from Question 12 to decide who has the greater proportion of stamps. Justify your choice.
👀 Show answer
Liam: total items $=20+320=340$. Stamps $=\tfrac{320}{340}=\tfrac{32}{34}=\tfrac{16}{17}\approx0.941$.
Hannah: total items $=15+270=285$. Stamps $=\tfrac{270}{285}=\tfrac{18}{19}\approx0.947$.
Comparing: $0.941$ vs $0.947$. Hannah has the greater proportion of stamps.
14. Two jewellery shops sell watches and rings.
Bright Jewellery has $12$ watches and $180$ rings for sale.
Mega-Jewellery has $30$ watches and $438$ rings for sale.
Which shop has the greater proportion of watches? Justify your choice.
👀 Show answer
Bright: total $=12+180=192$. Watches $=\tfrac{12}{192}=\tfrac{1}{16}=0.0625$.
Mega: total $=30+438=468$. Watches $=\tfrac{30}{468}=\tfrac{5}{78}\approx0.0641$.
Comparing: $0.0625$ vs $0.0641$. Mega-Jewellery has the greater proportion of watches.
⚠️ Be Careful
Always check that the ratio is simplified before you use it. If the ratio has units or decimals, convert them into the same units and whole numbers first. Using an unsimplified ratio can lead to wrong answers.