Sharing in a ratio
🎯 In this topic you will
- Divide an amount into two or more parts in a given ratio
🧠 Key Words
- conjecture
- divide
- profit
- share
- twice as much
Show Definitions
- conjecture: A statement believed to be true based on reasoning or evidence, but not yet proven.
- divide: To split a number or quantity into equal parts.
- profit: The amount of money gained after subtracting costs from revenue.
- share: To split or distribute something among people or groups.
- twice as much: An amount that is double another amount.
You can use ratios to divide things up or to share them.
Example: Sally and Bob buy a car for $15000$.
Sally pays $10000$ and Bob pays $5000$.
You can write the amounts they paid as a ratio:
Sally : Bob
$10000 : 5000$
$\div 5000 \quad \ \ \div 5000$
$2 : 1$
🔎 Reasoning Tip
The highest common factor (HCF) of 10 000 and 5 000 is 5 000, so you divide by 5 000 to simplify the ratio.
So, Sally paid twice as much as Bob.
Five years later they sell the car for $9000$. They need to share the money fairly between them.
Sally paid twice as much as Bob, so she should get twice as much as him.
How do you work out how much each of them gets?
To share in a given ratio:
- Add the numbers in the ratio to find the total number of parts.
- Divide the amount to be shared by the total number of parts to find the value of one part.
- Use multiplication to work out the value of each share.
📘 Exercise
1a. Share $45 between Ethan and Julie in the ratio 1:4.
1b. Complete the table:
| Name | Parts | Amount |
|---|---|---|
| Ethan | 1 | |
| Julie | 4 | |
| Total | 5 | $45 |
👀 Show answer
Total parts = 1 + 4 = 5. Value of 1 part = 45 ÷ 5 = 9.
Ethan = 1 × 9 = $9. Julie = 4 × 9 = $36.
| Name | Parts | Amount |
|---|---|---|
| Ethan | 1 | $9 |
| Julie | 4 | $36 |
| Total | 5 | $45 |
2. Share the amounts in the given ratios.
👀 Show answer
a. $24 in the ratio 1:2 Total = 3 parts → 24 ÷ 3 = 8. Shares: $8 and $16.
b. $65 in the ratio 1:4 Total = 5 parts → 65 ÷ 5 = 13. Shares: $13 and $52.
c. $48 in the ratio 3:1 Total = 4 parts → 48 ÷ 4 = 12. Shares: $36 and $12.
d. $30 in the ratio 5:1 Total = 6 parts → 30 ÷ 6 = 5. Shares: $25 and $5.
e. $21 in the ratio 1:6 Total = 7 parts → 21 ÷ 7 = 3. Shares: $3 and $18.
f. $64 in the ratio 7:1 Total = 8 parts → 64 ÷ 8 = 8. Shares: $56 and $8.
3. Share these amounts between Lin and Kuan-yin in the given ratios.
👀 Show answer
a. $35 in the ratio 2:3 Total = 5 parts → 35 ÷ 5 = 7. Shares: $14 and $21.
b. $49 in the ratio 3:4 Total = 7 parts → 49 ÷ 7 = 7. Shares: $21 and $28.
c. $32 in the ratio 5:3 Total = 8 parts → 32 ÷ 8 = 4. Shares: $20 and $12.
d. $90 in the ratio 7:3 Total = 10 parts → 90 ÷ 10 = 9. Shares: $63 and $27.
4. Raine and Abella share an electricity bill in the ratio 5:4. The total bill is $72. How much does each of them pay?
👀 Show answer
Total number of parts = 5 + 4 = 9. Value of 1 part = 72 ÷ 9 = 8.
Raine pays = 5 × 8 = $40. Abella pays = 4 × 8 = $32.
Check: $40 + $32 = $72 ✔️
🧠 Think like a Mathematician
Task: Convert a ratio into fractions of the total and explain why the method works.
Scenario: A shop sells bags of Brazil nuts and walnuts in the ratio $4:5$.
Questions:
👀 show answer
- a) - Total parts = 4 + 5 = 9. i. Brazil nuts = $\tfrac{4}{9}$ of the total. ii. Walnuts = $\tfrac{5}{9}$ of the total.
- b) In any ratio, add the parts together to get the total. Each part as a fraction is then: $\dfrac{\text{part}}{\text{total parts}}$. This ensures the fractions always add up to 1 (the whole).
📘 Exercise
6. A school choir is made up of girls and boys in the ratio 4:3. There are 35 students in the choir altogether.
a) How many of the students are boys?
b) What fraction of the students are boys?
👀 Show answer
Total parts = 4 + 3 = 7. Each part = 35 ÷ 7 = 5.
Number of boys = 3 × 5 = 15.
Fraction of boys = 15 ÷ 35 = 3/7.
🧠 Think like a Mathematician
Task: Rafina has solved a profit-sharing problem correctly, but her method involved difficult calculations. Find a simpler way to solve it and explain why it works.
Scenario: Brad and Lola buy a painting for $120. Brad pays $80 and Lola pays $40. They later sell the painting for $630. How much should each of them get?
Rafina’s solution:
- Ratio Brad:Lola = 80:40 = 2:1
- Total parts = 120
- Value of 1 part = $630 ÷ 120 = $5.25
- Brad gets $80 × 5.25 = $420
- Lola gets $40 × 5.25 = $210
- Check: $420 + $210 = $630
Questions:
👀 show answer
- a) Simplify the ratio first: Brad:Lola = 80:40 = 2:1. This avoids multiplying by $80 or $40 later.
- b) - Total ratio parts = 2 + 1 = 3 - Value of 1 part = $630 ÷ 3 = $210 - Brad gets 2 parts = $210 × 2 = $420 - Lola gets 1 part = $210
- c) Some learners may prefer to scale Brad’s and Lola’s shares directly by their fraction of the total: - Brad = $\tfrac{80}{120} \times 630 = 420$ - Lola = $\tfrac{40}{120} \times 630 = 210$ Both methods give the same answer, but simplifying the ratio first is quickest.
📘 Exercise
8. Arun (14 yrs, 58 kg) and Marcus (16 yrs, 62 kg) will share $240 either in the ratio of their ages or of their masses.
👀 Show answer
a) Which ratio is better for Arun?
Ages: \(14:16 = 7:8\) → Arun’s fraction \(= \frac{7}{15} \approx 0.4667\).
Masses: \(58:62 = 29:31\) → Arun’s fraction \(= \frac{29}{60} \approx 0.4833\).
Since \(0.4833 > 0.4667\), the mass ratio is better for Arun.
b) Check the amounts.
Age share: Arun \(= \frac{7}{15}\times 240 = 112\), Marcus \(= 128\).
Mass share: Arun \(= \frac{29}{60}\times 240 = 116\), Marcus \(= 124\).
✔️ Arun gets $116 by mass vs $112 by age, so mass is indeed better (by $4).
9. Every New Year, Auntie Bea gives $320 to be shared between her nieces in the ratio of their ages.
This year the nieces are aged 3 and 7.
Show that in 5 years’ time the younger niece will get $32 more than she got this year.
👀 Show Answer
This year: ratio = 3 : 7 → total parts = 10 → $320 ÷ 10 = $32 per part.
Younger gets 3 parts → 3 × 32 = $96.
In 5 years: ages = 8 and 12 → ratio 8 : 12 (total parts = 20) → $320 ÷ 20 = $16 per part.
Younger gets 8 parts → 8 × 16 = $128.
Increase: $128 − $96 = $32 ✅
10. Carys and Damien share $\(E\) in the ratio \(c:d\). Write algebraic expressions for their amounts.
🔎 Reasoning Tip
In question 10, start by writing an expression for:
- The total number of parts
- The value of one part
👀 Show answer
a) Carys will get: \( \displaystyle \frac{c}{c+d}\,E \).
b) Damien will get: \( \displaystyle \frac{d}{c+d}\,E \).
🔗 Learning Bridge
You’ve just learned the core method to divide an amount in a given ratio (find one part, then scale). Next, you’ll apply the same idea to three-way sharing and longer contexts. Keep these pointers in mind:
- Total parts first: For $a:b(:c)$, add the parts to get $P$. One part is $T\!/P$.
- Match names to parts: Label clearly (e.g., Alan $=2u$, Bob $=3u$, Chris $=1u$).
- Same ratio, different totals: Whether the total is $9000$ or $840$, the steps are identical—only $T$ changes.
- Fractions of the whole: In $2:3:1$, shares are $\tfrac{2}{6},\tfrac{3}{6},\tfrac{1}{6}$ of the total. This is a quick accuracy check.
- Simplify first if needed: If you’re given amounts (e.g., who paid what), simplify to a clean ratio before sharing sale money.
- Common slip: Don’t divide by the largest number; divide by the sum of the parts.
Shares: first $=a\,u$, second $=b\,u$, third $=c\,u$. Check: they add to $T$.
Sometimes you need to share an amount in a given ratio.
For example, Zara, Sofia and Marcus buy a painting for $600$.
Zara pays $200, Sofia pays $300 and Marcus pays $100.
Zara : Sofia : Marcus
$200 : 300 : 100$
$\div 100 \qquad \div 100 \qquad \div 100$
$2 : 3 : 1$
You can see that Zara paid twice as much as Marcus, and Sofia paid three times as much as Marcus. There are now 6 equal parts in total ($2+3+1=6$). When they sell the painting, they need to share the six parts of the money fairly between them. They can do this by using the same ratio of $2:3:1$.
Follow these steps to share an amount in a given ratio.
- 1 Add all the numbers in the ratio to find the total number of parts.
- 2 Divide the amount to be shared by the total number of parts to find the value of one part.
- 3 Use multiplication to work out the value of each share.
📘 Exercise
1. Share $80 between So, Luana and Kyra in the ratio 3:2:5.
👀 Show answer
Total parts = 3 + 2 + 5 = 10
Value of 1 part = 80 ÷ 10 = $8
- So: 3 × 8 = $24
- Luana: 2 × 8 = $16
- Kyra: 5 × 8 = $40
✅ Check: 24 + 16 + 40 = 80
2. Share each amount in the given ratios.
👀 Show answer
a) $90 in ratio 1:2:3 → parts=6, one part=$15 → $15, $30, $45
b) $225 in ratio 2:3:4 → parts=9, one part=$25 → $50, $75, $100
c) $432 in ratio 3:5:1 → parts=9, one part=$48 → $144, $240, $48
d) $396 in ratio 4:2:5 → parts=11, one part=$36 → $144, $72, $180
🧠 Think like a Mathematician
Task: Reflect on how to check ratio-sharing answers and compare different checking strategies.
Questions:
👀 show answer
- a) Method: Add the shared amounts together and check that they equal the original total. Example: If $120 is shared in the ratio 2:1 as $80 and $40, check that $80 + $40 = $120.
- b) Some learners may check by simplifying the amounts back to the original ratio (e.g. $80:$40 simplifies to 2:1). Both methods work, but adding back to the total is the quickest check, while simplifying to the ratio confirms accuracy in another way. The best method depends on whether you want to check totals or ratios.
📘 Exercise
4. Dave, Ella and Jia share in the ratio 3:4:5.
👀 Show answer
5. Choir ratio Men:Women:Children = 5:7:3, total 285.
👀 Show answer
6. Fruit ratio Oranges:Apples:Peaches = 4:2:3, total 72.
👀 Show answer
7. Project earnings $450. Hours: Aden 6, Eli 4, Lily 3, Ziva 5.
👀 Show answer
Total hours = 6+4+3+5 = 18 → $ per hour = 450 ÷ 18 = $25.
- Aden: 6×25 = $150
- Eli: 4×25 = $100
- Lily: 3×25 = $75
- Ziva: 5×25 = $125
8. Here is a set of ratio cards. Sort the cards into their correct groups. Each group must have one pink, one yellow and three blue cards.
👀 Show answer
Group A: Share $150, ratio 2:3:1 → $50, $75, $25
Group B: Share $120, ratio 3:1:4 → $45, $15, $60
Group C: Share $132, ratio 1:5:6 → $11, $55, $66
Group D: Share $126, ratio 2:6:1 → $28, $84, $14
9. The angles in a triangle are in the ratio 2:3:5. Work out the size of the angles.
🔎 Reasoning Tip
The angles in a triangle add up to 180°.
👀 Show answer
Total ratio parts = 2 + 3 + 5 = 10.
Triangle total = 180° → 1 part = 180 ÷ 10 = 18°.
Angles: 2×18 = 36°, 3×18 = 54°, 5×18 = 90°.
🧠 Think like a Mathematician
Task: Zara has solved a ratio-sharing problem correctly, but her working was long and calculator-heavy. Explore simpler strategies.
Scenario: A grandmother leaves $2520 in her will, to be shared among her grandchildren in the ratio of their ages: 6, 9, and 15. How much does each child receive?
Zara’s method:
- Ratio = 6:9:15
- Total parts = 6 + 9 + 15 = 30
- Value of one part = $2520 ÷ 30 = 84
- 6-year-old: $84 × 6 = $504
- 9-year-old: $84 × 9 = $756
- 15-year-old: $84 × 15 = $1260
- Check: $504 + $756 + $1260 = $2520 ✔
Questions:
👀 show answer
- a) She could simplify the ratio first. 6:9:15 simplifies to 2:3:5 (divide everything by 3). This avoids multiplying by large numbers later.
- b) - Ratio 2:3:5 → total = 10 parts - $2520 ÷ 10 = 252 per part - 2 parts = $504, 3 parts = $756, 5 parts = $1260 - Same results, but easier arithmetic.
- c) Some might still work with 6:9:15, but simplifying first is more efficient.
- d) Extra step: always simplify the ratio before calculating. This reduces the working and avoids unnecessary calculator use.
📘 Exercise
11. Every year David shares $300 among his children in the ratio of their ages. This year the children are aged 4, 9, 11. Show that, in two years’ time, the oldest child will receive $7.50 less than he receives this year.
👀 Show answer
This year: parts = 4+9+11 = 24 → 1 part = $300 ÷ 24 = $12.50.
Oldest gets 11 parts → 11×12.50 = $137.50.
In two years: ages = 6, 11, 13 → parts = 30 → 1 part = $300 ÷ 30 = $10.
Oldest then gets 13 parts → 13×10 = $130.
Difference: 137.50 − 130 = $7.50 less ✅
12. Zhi, Zhen and Lin buy a house for $180 000. Zhi pays $60 000, Zhen pays $90 000 and Lin pays the rest. Five years later they sell for $228 000 and share the money in the same ratio as they bought it. Lin thinks he will make $9000 profit. Is Lin correct? Show your working.
👀 Show answer
Contributions: 60 000 : 90 000 : 30 000 = 2 : 3 : 1 (Lin paid $30 000).
Proceeds $228 000 → parts = 2+3+1 = 6 → 1 part = 228 000 ÷ 6 = $38 000.
Zhi: 2 parts = $76 000 (profit $16 000)
Zhen: 3 parts = $114 000 (profit $24 000)
Lin: 1 part = $38 000 (profit $8 000).
Lin is not correct: his profit is $8 000, not $9 000. ❌
13. Akello, Bishara and Cora are going to share $960, either in the ratio of their ages or their heights.
Ages: Akello 22, Bishara 25, Cora 33. Heights (cm): Akello 168, Bishara 152, Cora 160.
a) Without working out the answer, which ratio do you think is better for Bishara, age or height? Explain your decision.
b) Work out whether your decision was correct. If not, explain why.
👀 Show answer
Reasoned guess (part a): One might think age is better since Bishara is the shortest in height (152 cm), so height might seem worse.
Actual shares (part b):
By ages 22:25:33 (sum 80) → $960 ÷ 80 = $12 per part.
Akello $264, Bishara $300, Cora $396.
By heights 168:152:160 (sum 480) → $960 ÷ 480 = $2 per part.
Akello $336, Bishara $304, Cora $320.
Bishara gets $304 by height vs $300 by age, so height is actually better for Bishara. If the initial guess was “age”, it was wrong because Bishara’s height is a slightly larger fraction of the total heights (152/480 = 31.67%) than his fraction of the total ages (25/80 = 31.25%).
⚠️ Be careful!
- Don’t divide by the biggest ratio number — always divide by the sum of the parts.
- Keep the order given. “Sally : Bob = 2:1” means Sally’s share comes first.
- Simplify ratios first if possible (e.g., 80:40 → 2:1) to make arithmetic easier.
- Convert units before using them in a ratio (kg ↔ g, hours ↔ minutes).
- Check your total. Add the shares to ensure they equal the original amount.