How to divide a quantity in a ratio
A ratio tells you how quantities compare. To turn that comparison into actual amounts, identify what the given number represents. A total, one share and the difference between shares each require a different calculation.
Label the shares before calculating
Write the names beside the ratio in the same order as the question. If blue and yellow counters are in the ratio 2:3, blue has two equal parts and yellow has three. This does not mean there are only two blue counters and three yellow counters. There could be 20 and 30, or another pair with the same relationship.
Each part has the same size. Blue is twice the value of one part, and yellow is three times that value. Together they account for five parts. Keep these part counts visible in your working so you do not divide by the wrong number.
Use comparable units when forming a ratio. A length of 1 metre compared with 50 centimetres is 100:50, which simplifies to 2:1. Writing 1:50 would compare numbers expressed in different units.
When you know the total, add the parts
Consider this invented practice question: share 60 counters between blue and yellow in the ratio 2:3. The total represents 2 + 3 = 5 equal parts. One part is 60 ÷ 5 = 12 counters. Blue gets 2 × 12 = 24, and yellow gets 3 × 12 = 36.
Check both conditions. The shares add to 24 + 36 = 60, and their ratio simplifies from 24:36 to 2:3. Checking only the total is insufficient: 20 and 40 also add to 60 but give a different ratio.
The same method works for three shares. For a practice total of 72 divided in the ratio 1:2:3, there are six parts. One part is 12, so the shares are 12, 24 and 36. Keep the labels in the original order when writing the answer.
When you know one share, use its own part count
Now suppose the ratio of blue to yellow is still 2:3, but the question says there are 18 yellow counters. That number represents three parts, rather than the five parts in the total. One part is 18 ÷ 3 = 6, so there are 2 × 6 = 12 blue counters. The total is 30.
Dividing 18 by five would treat a single share as the whole collection. Before dividing, complete the sentence: “The given amount of __ represents __ parts because __.” If you cannot fill in the reason, reread the question.
You can also scale directly from one share to the other: blue = yellow × 2/3. With 18 yellow counters, that gives 18 × 2/3 = 12 blue counters. The fraction comes from the labelled ratio; reversing it would make the smaller share larger.
Use the gap when the difference is given
For blue:yellow = 2:3, yellow exceeds blue by one part. If there are eight more yellow counters than blue counters, one part is eight. The shares are therefore 16 blue and 24 yellow, with a total of 40.
For a ratio of 2:5 and a difference of 21, the gap is 5 − 2 = 3 parts. One part is 21 ÷ 3 = 7, giving shares of 14 and 35. Their difference is 21 and their ratio simplifies to 2:5. The difference is not the total, so dividing 21 by seven would answer a different question.
Use this working template: “Share labels and ratio: __. Given amount: __. It represents a total / one share / a difference. Corresponding number of parts: __. Value of one part: __. Each labelled share: __. Check against the given amount: __. Check the ratio: __.”
Consider what can be divided in the problem. Lengths may allow fractional answers, while a count of individual counters must be whole. If sharing whole objects gives a fraction, check the wording and your calculation before rounding. Rounding individual shares can change the required ratio.
