Algebraic ratio problems give two expressions in terms of a variable — such as (2x + 1) and (x + 3) — and state their ratio. Set each expression equal to the matching multiple of the ratio, form an equation, and solve for x. The method fuses ratio and algebra into a single structured problem.
What does an algebraic ratio problem look like?
Instead of being given two plain numbers in a ratio, you are given algebraic expressions. The ratio fixes the relationship between the two expressions, which gives you enough information to find the unknown.
Typical question: Two lengths are in the ratio 3 : 2. The first length is (2x + 1) cm and the second is (x + 3) cm. Find x and both lengths.
This type of question appears on GCSE Higher and sometimes Foundation papers. It tests whether you can translate a ratio statement into a usable equation.
How do you form an equation from a ratio?
If two quantities A and B are in the ratio p : q, then:
A/B = p/q, or equivalently, q × A = p × B (cross-multiplying).
Use whichever form suits the question. Cross-multiplying avoids fractions and is often quicker.
Worked example 1: (2x + 1) : (x + 3) = 3 : 2
| Step | Working |
|---|---|
| Write as fraction equation | (2x + 1)/(x + 3) = 3/2 |
| Cross-multiply | 2(2x + 1) = 3(x + 3) |
| Expand | 4x + 2 = 3x + 9 |
| Solve | x = 7 |
| Find each quantity | First: 2(7) + 1 = 15 cm |
| Second: 7 + 3 = 10 cm | |
| Check the ratio | 15 : 10 = 3 : 2 ✓ |
Worked example 2: angles in a triangle in ratio 2 : 3 : 5
The three angles of a triangle sum to 180°. Express each angle as a multiple of a variable, form an equation.
- Write the angles as 2k, 3k, and 5k (where k is the scale factor).
- Sum to 180°: 2k + 3k + 5k = 180.
- Simplify: 10k = 180.
- Solve: k = 18.
- Angles are 2 × 18 = 36°, 3 × 18 = 54°, 5 × 18 = 90°.
- Check: 36 + 54 + 90 = 180° ✓
This shows that the scale-factor method (write each part as a multiple of k) is often cleaner than cross-multiplication when the ratio has more than two parts.
Which method works best for different problem types?
| Problem type | Recommended method |
|---|---|
| Two expressions in a given ratio | Cross-multiply to eliminate the fraction |
| Three or more quantities in ratio | Use scale factor k; write each part as nk |
| Ratio given with a total | Scale factor: find total number of parts, divide total by parts to find k |
| Ratio involving angles or sides | Scale factor k is usually simplest |
What mistakes do students commonly make?
- Cross-multiplying incorrectly. (2x + 1)/(x + 3) = 3/2 must give 2(2x + 1) = 3(x + 3), not 2(x + 3) = 3(2x + 1). The denominator on each side multiplies the numerator on the other.
- Forgetting to check. Always substitute x back into the original expressions and verify the ratio. A sign error in expansion gives a wrong value of x that still satisfies an incorrect equation.
- Not expanding brackets fully. 2(2x + 1) = 4x + 2, not 4x + 1. Write every bracket expansion out carefully.
- Finding x but not the quantities. Most mark schemes award marks for the quantities (lengths, angles, etc.), not just x itself.
Frequently asked questions
What if the ratio involves three quantities?
Use the scale-factor method: call the three parts nk, mk, and pk where n : m : p is the given ratio. Write an equation using any additional information (such as a total or a difference), solve for k, then multiply back. This is exactly the method used for angles in a triangle (example 2 above).
What if the ratio involves subtraction, e.g. (5x − 2) : (3x + 4)?
Cross-multiply exactly as before: if the ratio is a : b, then (5x − 2)/( 3x + 4) = a/b → b(5x − 2) = a(3x + 4). Expand both sides carefully, paying attention to signs with negative terms.
How do I know which expression is "first" in the ratio?
The order of the ratio corresponds to the order in which the quantities are listed in the question. If "length A to length B = 5 : 3", then A is compared to 5 and B is compared to 3. Reading carefully and labelling the quantities before writing the equation prevents this common mix-up.
Can the answer for x be negative?
Algebraically, yes — but you must check whether it makes physical sense. A negative length or a negative angle is meaningless in most geometry contexts. If x turns out negative, re-read the question to see whether you have set up the ratio the wrong way round.
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