To solve an equation with fractions, multiply every term on both sides by the lowest common multiple of the denominators, then solve the ordinary equation that remains. It appears in Core and Extended style questions whenever an unknown sits inside a fraction.
This follows solving with brackets and uses the fraction skills from number sense.
What is the idea behind it?
Multiplying both sides of an equation by the same number keeps it balanced. If you choose the lowest common multiple of the denominators, every fraction cancels down to a whole-number multiplier.
A numerator with more than one term, like (2x + 1), behaves as if it were in brackets. After multiplying, keep it in brackets until you expand.
How to solve it, step by step
- Find the lowest common multiple of all the denominators.
- Multiply every term on both sides by it, including whole numbers.
- Cancel each denominator and write the result with brackets round any multi-term numerator.
- Expand and collect as in the previous lesson.
- Solve and check in the original equation.
Worked example
Solve (2x + 1)/3 + x/2 = 5
Step 1, common multiple: the denominators are 3 and 2, so use 6.
Step 2, multiply every term by 6: 6 × (2x + 1)/3 + 6 × x/2 = 6 × 5.
Step 3, cancel: 2(2x + 1) + 3x = 30.
Step 4, expand and collect: 4x + 2 + 3x = 30, so 7x + 2 = 30.
Step 5, solve: 7x = 28, so x = 4.
Check: (2 × 4 + 1)/3 = 9/3 = 3 and 4/2 = 2. Then 3 + 2 = 5. ✓
The mistake to watch for
The usual slip is multiplying the fraction terms but leaving the whole number alone.
Mistaken working: 2(2x + 1) + 3x = 5, so 7x = 3 and x = 3/7.
The student multiplied the fractions by 6 but did not multiply the 5.
Check x = 3/7 in the original: (2 × 3/7 + 1)/3 = 13/21 and x/2 = 3/14. Together they make 35/42 = 5/6, not 5.
The correction is to write ”× 6” in front of the whole equation and apply it to each term in turn, ticking them off as you go.
Check yourself
1. Solve x/4 + 3 = 7
Show answer
Subtract 3: x/4 = 4. Multiply by 4: x = 16.
Check: 16/4 + 3 = 4 + 3 = 7. ✓
2. Solve (x + 2)/3 = (x − 1)/2
Show answer
Multiply both sides by 6: 2(x + 2) = 3(x − 1). Expand: 2x + 4 = 3x − 3. Then 7 = x, so x = 7.
Check: (7 + 2)/3 = 3 and (7 − 1)/2 = 3. ✓
3. Solve (3x − 1)/4 − (x + 2)/6 = 1
Show answer
The common multiple of 4 and 6 is 12. Multiply every term: 3(3x − 1) − 2(x + 2) = 12. Expand, noting the minus changes both signs: 9x − 3 − 2x − 4 = 12. So 7x − 7 = 12, then 7x = 19 and x = 19/7.
Check: (57/7 − 1)/4 = (50/7)/4 = 25/14. And (19/7 + 2)/6 = (33/7)/6 = 11/14. The difference is 14/14 = 1. ✓
Where this leads next
Next, rearranging a formula with the variable on both sides uses the same balancing with letters. The equations and formulas practice set mixes all the types together.
If clearing fractions works on paper but slows you down in timed papers, a teacher in online one-to-one Mathematics tuition can watch your first two steps and tighten the routine.