Practice Maths

Dividing a Quantity in a Given Ratio

Key Ideas

Key Terms

divide in ratio
share a quantity so that each portion's amount is proportional to its ratio number.
total parts
the sum of all numbers in the ratio; e.g. for ratio 3:2, total parts = 5.
value of one part
the total quantity divided by the total number of parts; each share equals its ratio number multiplied by one part.
share
each person's allocation after dividing; all shares must sum back to the original total.

Method: Share $120 in ratio 3:1

Step 1: Total parts = 3 + 1 = 4
Step 2: Value of 1 part = $120 ÷ 4 = $30
Step 3: First share = 3 × $30 = $90; Second share = 1 × $30 = $30
Check: $90 + $30 = $120 ✓

Hot Tip Always check that your shares add up to the original amount. If they don’t, you have made an error somewhere.

The Three-Step Method

Dividing a quantity in a given ratio follows a clear, reliable three-step method:

Step 1: Find the total number of parts by adding all the numbers in the ratio.
Step 2: Find the value of one part by dividing the total quantity by the total number of parts.
Step 3: Multiply the value of one part by each ratio number to find each share.

Example: Share $120 in the ratio 3:5.
Step 1: Total parts = 3 + 5 = 8
Step 2: Value of 1 part = $120 ÷ 8 = $15
Step 3: First share = 3 × $15 = $45.   Second share = 5 × $15 = $75
Check: $45 + $75 = $120. ✓

Worked Example: Three-Part Ratio

Mix paint in the ratio 2:1:3 (red:yellow:blue) to make 900 mL of a custom colour.

Step 1: Total parts = 2 + 1 + 3 = 6
Step 2: 1 part = 900 ÷ 6 = 150 mL
Step 3: Red = 2 × 150 = 300 mL,   Yellow = 1 × 150 = 150 mL,   Blue = 3 × 150 = 450 mL
Check: 300 + 150 + 450 = 900 mL ✓

The same method works for any number of parts — just add them all up in Step 1.

Finding a Quantity Given One Share

Sometimes you know one person's share and need to find the total or another share. Use the ratio to work backwards.

Example: Tom and Mia split their winnings in ratio 3:7. Tom receives $45. How much does Mia get, and what were the total winnings?
Tom has 3 parts = $45, so 1 part = $45 ÷ 3 = $15.
Mia has 7 parts = 7 × $15 = $105.
Total winnings = 10 × $15 = $150.

Multi-Step Ratio Problems

Some problems require ratios within ratios or involve finding a missing piece from given information.

Example: A school is allocated $4 200 for sport and arts in the ratio 5:2. The sport money is then split between football and swimming in the ratio 3:4. How much goes to swimming?
Step 1 — Split $4 200 in ratio 5:2: total parts = 7, 1 part = $600. Sport = 5 × $600 = $3 000.
Step 2 — Split $3 000 in ratio 3:4: total parts = 7, 1 part = $3 000 ÷ 7 ≈ $428.57. Swimming = 4 × $428.57 = $1 714.29.

Ratios in Context: Cooking, Sport and Design

Dividing in a ratio is used whenever something must be shared fairly but not equally. A recipe calls for butter, sugar and flour in ratio 1:2:3. If you want 360 g of mixture total, each ingredient's portion can be found exactly using the three-step method.

In sport, prize money might be split in ratio 5:3:2 among first, second, and third place. In design, a room might be divided into living and dining space in ratio 3:2. The method is always the same — total parts, value of one part, multiply.

Key tip: Always check your answer by adding all the shares together — they must equal the original total. If they don't, you've made an arithmetic error somewhere. This is the quickest and easiest way to verify your answer is correct.

Mastery Practice

  1. Divide each quantity in the given ratio. Show all steps. Fluency

    1. Divide 60 in ratio 2:3
    2. Divide $80 in ratio 3:1
    3. Divide 150 m in ratio 1:4
    4. Divide $200 in ratio 3:2
    5. Divide 90 kg in ratio 5:4
    6. Divide $720 in ratio 5:3
    7. Divide 240 mL in ratio 1:5
    8. Divide $1000 in ratio 7:3

  2. Divide each quantity in the given three-way ratio. Fluency

    1. Divide 240 in ratio 1:2:3
    2. Divide $360 in ratio 2:3:4
    3. Divide 180 cm in ratio 1:2:3
    4. Divide $500 in ratio 1:4:5
    5. Divide 120 kg in ratio 3:4:5
    6. Divide $900 in ratio 2:3:4
    7. Divide 1000 mL in ratio 1:3:6
    8. Divide $280 in ratio 1:2:4

  3. Find the original total given one share of the ratio. Fluency

    1. Sam’s share is $45 in ratio 3:2. What was the total amount?
    2. Alex gets $60 from a ratio of 3:1. What was the total?
    3. In ratio 2:5, the smaller share is $16. Find the total.
    4. In ratio 4:1, the larger share is $80. Find the total.
    5. The smaller share in ratio 3:7 is 30 g. Find the total.
    6. In ratio 5:3, one share is 40 km. Which part does this represent and what is the total?
    7. In ratio 2:3:5, the middle share is $120. Find the total.
    8. In ratio 1:4, the smaller share is $25. Find the larger share.

  4. Solve each problem using ratio division. Understanding

    1. A prize of $500 is shared between two winners in ratio 3:2. How much does each winner receive?
    2. A 240 cm piece of ribbon is cut in ratio 5:3. Find the length of each piece.
    3. Two business partners share profits of $84 000 in ratio 5:2. How much does each receive?
    4. Three siblings inherit $90 000 in ratio 2:3:4. How much does each sibling receive?
    5. A recipe uses flour, sugar, and butter in ratio 4:2:1. There are 350 g of ingredients total. How much of each is used?
    6. A school council distributes a $2400 budget between sport, arts, and science in ratio 3:2:1. How much goes to each area?
    7. Two friends pool their money ($360 total) to buy a gift, contributing in ratio 5:4. How much does each person contribute?
    8. A 60 L mixture of cordial and water is in ratio 1:5. How much of each ingredient is there?

  5. Compare ratio sharing to percentage sharing. Understanding

    1. $200 is split 60% and 40%. Write this as a ratio in simplest form, then find each person’s share.
    2. $150 is divided in ratio 2:3. What percentage does each person receive?
    3. $600 is shared so that Alice gets 25% and the rest goes to Ben. Write the sharing as a ratio. How much does each person get?
    4. Is dividing $80 in ratio 1:3 the same as one person getting 25% and the other getting 75%? Explain.
    5. A class of 30 students is split into two groups in ratio 2:3. Express each group’s size as a percentage of the class.
    6. $480 is shared in ratio 5:7. What fraction and percentage of the total does the smaller share represent?

  6. Multi-step ratio division problems. Show all working. Problem Solving

    1. Three cousins inherit money in ratio 3:4:5. The eldest receives $3000 more than the youngest. Find the total inheritance and each person’s share.
    2. A sports club shares prize money between the first, second, and third-place teams in ratio 5:3:2. First place receives $1200 more than third place. Find the total prize money.
    3. Two solutions are mixed in ratio 2:5. The total volume needed is 350 mL. If solution A costs $0.80/mL and solution B costs $0.50/mL, find the total cost of the mixture.
    4. A business divides its annual profit between reinvestment and distribution to owners in ratio 3:2. The owners’ share is $48 000. How much is reinvested? What is the total profit?

  7. Share $240 in the ratio 3:5. Use the bar diagram and complete the working below. Understanding

    $? $? $? $? $? $? $? $? Share A (3 parts) Share B (5 parts)

    Total parts = ___ + ___ = ___
    Each part = $240 ÷ ___ = $___
    Share A = 3 × $___ = $___
    Share B = 5 × $___ = $___
    Check: $___ + $___ = $240 ✓

    1. Complete the working above and fill in the bar diagram.
    2. What fraction of $240 does Share A represent?
    3. What percentage of $240 does Share B represent?

  8. The bar diagram shows a total of 120 mL split in ratio 1:2:3. Understanding

    Part A Part Part Part B (2) Part Part Part Part C (3) Part A (1)
    1. How many total parts are there?
    2. What is the value of each part (in mL)?
    3. Find the volume of Part A, Part B, and Part C.
    4. Check: do all three parts add up to 120 mL?

  9. Three students — Alice, Ben, and Carlos — contribute to a project in ratio 2:3:4. The total materials budget is $180. Problem Solving

    1. How many total parts are there in the ratio 2:3:4?
    2. What is the value of each part?
    3. How much does each student contribute?
    4. Carlos says he is contributing twice as much as Alice. Is this correct? Show working to explain.
    5. If the budget increases to $270, how much would each person now contribute (keeping the same ratio)?

  10. Working backwards: find the total or the ratio. Show all steps. Problem Solving

    1. Two amounts are shared in ratio 3:7. The larger share is $105. What is the smaller share and what was the total?
    2. A sum of money is split between two people so that the first gets $180 and the second gets $120. Write this as a ratio in simplest form.
    3. Ali and Beth share $560 so that Ali gets $80 more than Beth. Find each person’s share and write the sharing as a ratio.
    4. A mixture of red and blue paint is made. For every 2 tins of red, there are 5 tins of blue. If there are 35 tins of blue paint, how many tins of red paint are there? What is the total number of tins?
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