IGCSE Maths / Algebraic equations
Equations with fractions
Edexcel International GCSE Maths (4MA1), chapter 5: Algebraic equations, specification point 2.4a. Free lesson.
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Clear fractions by multiplying every term in the equation by the smallest number that all the denominators divide into. Then solve the resulting ordinary equation, taking care to keep each fraction's whole numerator together in brackets.
How to clear fractions
The denominators are the numbers at the bottom of fractions. Find the smallest common multiple, the smallest number that each denominator divides into exactly. Multiplying every term by this number removes the fractions, so you can solve the equation as usual. Multiply every term, including any term on the other side that has no fraction.
Solving with x on top
For x/2 + x/5 = 7, the denominators are 2 and 5, so use 10. The right-hand 7 must also be multiplied by 10. The resulting equation has no fractions and can be simplified normally. 5x and 2x are like terms because both are multiples of x, so they combine to make 7x. Check the answer by substituting it into the original equation.
Why the numerator needs brackets
When a fraction has more than one term on top, the whole numerator is multiplied after the denominator cancels. Brackets show that every term in the numerator is included. For example, 12 × (2x + 1)/3 = 4(2x + 1). Writing 4 × 2x + 1 would multiply only 2x and leave the 1 unchanged, giving the wrong expression.
Solving when there is algebra on top
For (2x + 1)/3 − (x − 3)/4 = 4, use 12 because it is the smallest number that both 3 and 4 divide into. Keep brackets round both numerators. The minus sign before the second fraction applies to every term inside its bracket when you multiply out. The first numerator is multiplied by 4 because 12 ÷ 3 = 4. The second is multiplied by 3 because 12 ÷ 4 = 3. A negative number multiplied by a negative number gives a positive number.
The same method works for any denominators
For x/4 + x/6 = 5, use 12, the smallest number that both 4 and 6 divide into. Multiplying every term by 12 clears both fractions at once. The check is 12/4 + 12/6 = 3 + 2 = 5.