Higher tier ratio questions combine ratio with algebra. To combine two ratios that share a term, scale them so that the shared term matches. When a ratio changes because items are added or removed, work out the actual amounts first, then form the new ratio. Writing a ratio in the form 1 : n makes comparisons easy, and an unknown in a ratio is found by forming and solving an equation.
Ratio and algebra
- Combine a : b and b : c by making b the same in both (use the LCM)
- 1 : n form: divide both parts by the first part
- a : b = p : q means qa = pb
- Changing ratio: find the actual amounts, adjust them, then write the new ratio
Combining two ratios
If x : y = 3 : 4 and y : z = 6 : 5, the shared term y is 4 in one ratio and 6 in the other. Scale both so y is the lowest common multiple, 12: x : y = 9 : 12 and y : z = 12 : 10. Then x : y : z = 9 : 12 : 10.
Changing ratios
When items are added or removed, the ratio changes, but the actual numbers are what change. Work out the current amounts from the ratio and the total, adjust them, then write the new ratio. With 40 counters in the ratio red : blue = 5 : 3, there are 25 red and 15 blue; adding 6 blue gives 25 : 21.
The form 1 : n
To write a ratio in the form 1 : n, divide both parts by the first part. 6 : 15 becomes 1 : 2.5. The second number can be a decimal. This form is useful for comparing ratios and for map scales.
Forming equations from ratios
If a : b = 4 : 7, then a is 4 parts and b is 7 parts. If you know the difference b − a = 18, that difference is 3 parts, so one part is 6. More generally, a : b = p : q means qa = pb, which turns a ratio statement into an equation.
Ratios with algebraic expressions
If a ratio contains expressions, such as (x + 3) : (2x − 1) = 2 : 3, cross-multiply: 3(x + 3) = 2(2x − 1). Expanding and solving gives x = 11. Check by substituting: 14 : 21 simplifies to 2 : 3.
Worked examples
Example 1. Given x : y = 3 : 4 and y : z = 6 : 5, find x : y : z.
- The LCM of 4 and 6 is 12.
- x : y = 9 : 12 and y : z = 12 : 10.
Answer: 9 : 12 : 10.
Example 2. A bag has red and blue counters in the ratio 5 : 3. There are 40 counters. Six more blue counters are added. Find the new ratio of red to blue.
- Red: 5/8 × 40 = 25. Blue: 15.
- After adding 6 blue: 25 red, 21 blue.
Answer: 25 : 21.
Example 3. Write 6 : 15 in the form 1 : n.
- Divide both parts by 6.
Answer: 1 : 2.5.
Example 4. a : b = 4 : 7 and b − a = 18. Find a and b.
- The difference is 3 parts = 18, so one part is 6.
- a = 4 × 6 and b = 7 × 6.
Answer: a = 24, b = 42.
Example 5. (x + 3) : (2x − 1) = 2 : 3. Find x.
- Cross-multiply: 3(x + 3) = 2(2x − 1).
- 3x + 9 = 4x − 2.
Answer: x = 11.
Common mistakes
- Combining ratios without matching the shared term. Scale both ratios so the shared quantity is equal.
- Changing the ratio parts instead of the amounts. Work out the actual numbers first, then adjust them.
- Insisting on whole numbers in 1 : n. n can be a decimal, such as 1 : 2.5.
- Using the total when the difference is given. Match the difference in parts to the difference in amounts.
- Cross-multiplying the wrong way. For a : b = p : q, write qa = pb.
How ratio with algebra is examined
These ratio problems are Higher tier content for Edexcel GCSE (1MA1) and the International GCSE (4MA1), and they often carry 3 to 5 marks. Questions ask you to:
- combine two ratios into a three-part ratio
- find a new ratio after items are added or removed
- write a ratio in the form 1 : n
- form and solve an equation from a ratio statement
Practise with the sharing in a ratio worksheet, which includes answers, and the GCSE proportion engine.
This builds on ratio and uses linear equations.
Frequently asked questions
How do you combine two ratios?
Scale them so the term they share has the same value, then write all three parts together.
How do you write a ratio in the form 1 : n?
Divide both parts of the ratio by the first part.
How do you solve a ratio with an unknown?
Cross-multiply: a : b = p : q means qa = pb, then solve the equation.