Dividing Quantities by Ratio
Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole
What a learner can do afterwards
- Share £60 between two people in the ratio 2:3 by finding the value of one part
- Express the result of sharing 24 sweets as 10 and 14 in ratio form as 5:7
- Solve problems involving three-part ratios (e.g. divide 180 in the ratio 1:2:3)
1 · Read
You already know how to write a ratio and picture it with a bar model. Now you'll use that picture to split a real amount. Add the numbers in the ratio to find the total number of parts. Divide the whole amount by that total to find what one part is worth. Then multiply each ratio number by that value to get every share.
Ben and Amy share £60 in the ratio 2:3. Total parts: 2 + 3 = 5. One part: £60 ÷ 5 = £12. Ben's share: 2 × £12 = £24. Amy's share: 3 × £12 = £36. As a fraction of the whole, Ben gets 2/5 of the £60 and Amy gets 3/5.
The same idea stretches to three parts. To split 180 in the ratio 1:2:3, add the parts: 1 + 2 + 3 = 6. One part is 180 ÷ 6 = 30. The shares are 1 × 30 = 30, 2 × 30 = 60, and 3 × 30 = 90. Check by adding them back: 30 + 60 + 90 = 180.
You can also work backwards. If a split comes out as 10 and 14, write it as a ratio, 10:14, then simplify by dividing both numbers by their highest common factor, 2. That gives you 5:7.
Add the parts, divide the whole to find one part, then multiply by each ratio number to get every share.
More to read
2: Introducing Ratios
A ratio compares two amounts while keeping their link. Saying squash uses 2 parts concentrate to 3 parts water tells you the taste of every glass, however big. The numbers 2 and 3 are not the amounts, they are the recipe, and scaling the recipe keeps the taste.
Sharing by a ratio means splitting the total into recipe-sized parts first. For £60 in the ratio 2:3, the whole splits into 5 equal parts of £12. The shares follow the recipe: 2 parts (£24) and 3 parts (£36).
9.1: Introducing Ratios and Ratio Language
Ratio language says for each with care. A ratio of 10 sweets to 14 sweets can be written 10:14, but the simplified form 5:7 says the same thing more cleanly. Simplifying works exactly like simplifying fractions: divide both sides by their highest common factor.
Order matters in a ratio. The ratio 5:7 is not the same as 7:5, because the first number belongs to the first person or item. When a question gives the shares, write them in the stated order, then simplify without swapping the sides.
Section 12: Solving Ratio and Rate Problems
Three-part ratios use the same one-part method with more adding. Dividing 180 in the ratio 1:2:3 means 6 parts, each worth 30, giving shares of 30, 60 and 90. The method never changes: total the parts, value one part, multiply for each share.
Working backwards starts from a known share instead of the total. If the smaller share of a 5:7 split is £35, one part is £35 / 5 = £7, and the total is 12 parts at £7 each. Check at the end by adding the shares: they must rebuild the total.
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