Sharing and Solving Ratio Problems
Ratio problems often ask you to split a total, work backwards from one share, or combine information from more than one ratio. The key is to keep track of what one equal part is worth.
Foundation & Higher. You should already be able to simplify and interpret ratios.
Sharing a total in a ratio
To share £84 in the ratio 3:4, first find the total number of parts: 3 + 4 = 7. One part is £84 ÷ 7 = £12. The shares are £36 and £48.
A bar model can help: draw 3 equal boxes for the first share and 4 for the second.
A part-to-whole ratio
Suppose A:total = 3:7 and the total is £84. Here 7 already represents the whole: one part is £12, so A gets £36 and the remainder gets £48. Do not add 3 and 7. The equivalent part-to-part ratio is A:remainder = 3:4.
Worked example — use a bar model
Share £126 in the ratio 2:5.
| Share | Equal parts | Value |
|---|---|---|
| A | 2 parts | 2 × £18 = £36 |
| B | 5 parts | 5 × £18 = £90 |
There are 7 equal parts altogether, so one part is £126 ÷ 7 = £18. The bar model keeps the equal-part structure visible and makes it easier to check that £36 + £90 = £126.
Three-part sharing
Share 180 in the ratio 2:3:4. There are 9 parts, so one part is 20. The shares are 40, 60 and 80.
When one share is known
A:B = 5:3 and A = 40. Five parts are 40, so one part is 8. Therefore B = 24.
When the difference is known
A:B = 7:4 and A is 18 greater than B. The difference is 3 parts, so one part is 18 ÷ 3 = 6. Therefore A = 42 and B = 24.
Combining compatible ratios
If A:B = 2:3 and B:C = 4:5, make the B parts match. A:B = 8:12 and B:C = 12:15, so A:B:C = 8:12:15.
Quantities added or removed
If a ratio changes after something is added or removed, write the original quantities using a common multiplier. Then form an equation from the new ratio.
Worked example — a changing ratio
Red:blue = 3:5. Then 8 red counters are added and the new ratio is 1:1. Find the original numbers.
Write the original quantities as 3k and 5k. After adding 8 red counters: 3k + 8 = 5k.
So 8 = 2k and k = 4. Originally there were 12 red and 20 blue counters.
Check: after adding 8 red counters there are 20 red and 20 blue, giving 1:1.
This is where Introduction to Algebra can help.
Worked example — removing a quantity
Red:blue = 5:3. After 8 red counters are removed, the numbers are equal. Initially there are 5k red and 3k blue, so 5k − 8 = 3k. Thus 2k = 8, k = 4: originally 20 red and 12 blue. Check that removing 8 leaves 12 of each colour.
Mixing and concentration
A drink mixed syrup:water = 1:4 contains 5 equal parts. The syrup concentration is 1/5 of the mixture. Always state clearly what each part represents.
Worked example — combine mixtures
Mix 2 L of drink with syrup:water = 1:4 and 3 L with syrup:water = 1:9. The syrup amounts are 2 × 1/5 = 0.4 L and 3 × 1/10 = 0.3 L. Together there are 0.7 L syrup in 5 L drink, so the syrup concentration is 0.7/5 = 14%. Water is 4.3 L, so syrup:water = 7:43. Assume the volumes add. Add the actual ingredient amounts, not the ratio entries or the two concentrations.
Think Like a Mathematician
The ratio gives a structure, not the actual amounts. Once you know the value of one part—or can work it out—the entire problem becomes a scaling problem.
Common Mistakes
Dividing by one ratio number: when sharing a total, divide by the sum of the parts.
Comparing incompatible middle terms: match the shared quantity before combining ratios.
Assuming all reverse-ratio questions are Higher: many are simply multi-step Foundation reasoning.
Practice — sharing and solving
1. Share £96 in the ratio 5:3.
8 parts, so one part = £12. Shares: £60 and £36.
2. Share 270 in the ratio 2:3:4.
9 parts, one part = 30. Shares: 60, 90, 120.
3. A:B = 7:2 and A = 49. Find B.
One part = 49 ÷ 7 = 7. B = 14.
4. P:Q = 9:5 and P exceeds Q by 28. Find P and Q.
Difference = 4 parts, so one part = 7. P = 63, Q = 35.
5. A:B = 3:5 and B:C = 2:7. Find A:B:C.
LCM of 5 and 2 is 10. A:B = 6:10 and B:C = 10:35. 6:10:35.
6. A:total = 2:5. A total of £150 is shared between A and B. Find both shares.
Five parts are £150, so one part is £30. A gets £60; B gets the remaining £90.
7. Mix 1 L of 20% syrup drink with 3 L of 10% syrup drink. Find the final syrup percentage, assuming volumes add.
Syrup = 0.20 × 1 + 0.10 × 3 = 0.5 L. Total drink = 4 L. Concentration = 0.5/4 × 100% = 12.5%.
8. A:B = 2:3. If 6 is added to A, the ratio becomes 4:3. Find the original A and B.
Let A=2k, B=3k. (2k+6)/(3k)=4/3. 6k+18=12k, so k=3. A=6, B=9.
9. Paint is mixed red:white = 3:2. How much white paint is needed with 750 ml red?
3 parts = 750, one part = 250. White = 2 parts = 500 ml.
10. A prize is shared A:B:C = 1:2:5. C receives £160. Find the total prize.
5 parts = £160, so one part = £32. Total 8 parts = £256.
Frequently Asked Questions
Should I add the ratio numbers before dividing?
For a part-to-part ratio, add them to find the whole. In a part-to-whole ratio, the whole is already one of the entries.
Can I average two mixture concentrations?
Only when the amounts of mixture are equal. Otherwise find each ingredient amount and divide their combined amount by the combined total.
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