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To solve an equation with fractions, clear the fractions first: multiply every term on both sides by the lowest common denominator (LCD), then solve the resulting equation as normal. For example, to solve x/6 = 1 - x/4, multiply every term by 12 to get 2x = 12 - 3x, so 5x = 12 and x = 12/5.

Once the fractions are gone, the usual rules of algebra apply: collect the terms containing the variable on one side, move the numbers to the other side, then divide by the coefficient of the variable.

Fraction equations come up regularly in numerical reasoning tests, where you need to work quickly under time pressure. The five steps below walk through the method using the example above, and the worked examples further down apply it to harder questions.

Step 1: Find the least common denominator

First, find the least common denominator (LCD) of the fractions in the equation: the smallest number that every denominator divides into. For the example equation x/6 = 1 - x/4, the denominators are 6 and 4, so the LCD is 12, the lowest common multiple of 4 and 6.

Step 2: Multiply by the least common denominator

Multiply both sides of the equation by the LCD, making sure it multiplies every term on each side.

Step 3: Simplify the equation

Cancel each denominator into the LCD and simplify both sides so you are only working with whole numbers. Here 12(x/6) becomes 2x and 12(1 - x/4) becomes 12 - 3x, leaving 2x = 12 - 3x.

Step 4: Simplify until there's one term on both sides

Move all terms containing the variable to one side and any numbers to the other, then simplify. Adding 3x to both sides gives 2x + 3x = 12, which simplifies to 5x = 12.

Step 5: Divide by the coefficient on both sides

Once the variable term is isolated, divide both sides by its coefficient to solve for the unknown. Dividing both sides by 5 gives x = 12/5, or 2.4 as a decimal.

Worked examples of equations with fractions

Q1) Solve for x: x/5 = 3

There is only one fraction, so the lowest common denominator is simply 5. Multiply both sides by 5:

5(x/5) = 5 × 3

x = 15

Check the answer by substituting it back in: 15/5 = 3, which is correct.

Q2) Solve for x: x/4 + x/6 = 5

The denominators are 4 and 6, so the LCD is 12. Multiply every term by 12:

12(x/4) + 12(x/6) = 12 × 5

3x + 2x = 60

5x = 60

Divide both sides by 5:

x = 12

Check: 12/4 + 12/6 = 3 + 2 = 5, which is correct.

Q3) Find the value of x in:

First, let’s find the least common denominator (LCD) of the fractions:

6 = 2 × 3

15 = 3 × 5

LCD: 2 × 3 × 5 = 30

Multiply every term by 30 and simplify after distributing:

Move 6x to the left-hand side of the equation to collect the terms containing the variable:

20x - 6x = 60

14x = 60

Divide both sides by 14 and simplify the fraction to solve for x:

x = 60/14 = 30/7

Q4) An unknown fraction is added to 1 and we divide the sum by 3. The result is equal to 3/4. What is the value of the unknown fraction?

Let n be the unknown number and set up the equation:

First, look for the least common denominator (LCD). Since 3 and 4 don’t share any common factors, the LCD is their product:

LCD: 3 × 4 = 12

Multiply both sides by 12:

Simplifying gives 4(n + 1) = 9, which expands to:

4n + 4 = 9

Subtract 4 from both sides:

4n = 9 - 4

4n = 5

Divide both sides by 4 to isolate n:

The unknown fraction is equal to 5/4.

Tips for solving equations with fractions

  1. Cross-multiply when both sides are single fractions. If the equation is one fraction equal to another, such as x/6 = 5/3, multiply diagonally: 3x = 30, so x = 10. This is the LCD method in disguise, but quicker to apply.
  2. Multiply every term, not just the fractions. The most common mistake is forgetting to multiply the whole numbers by the LCD. In x/6 = 1 - x/4, the 1 must also be multiplied by 12.
  3. Check your answer by substitution. Substitute your value back into the original equation. Fractional answers such as 30/7 are common, so do not assume a non-whole answer is wrong.

Practice equations with fractions

The quickest way to make this method second nature is to use it against the clock. Try our numerical reasoning tests to work with fractions under timed conditions, brush up the underlying skills with our basic numeracy tests, or sample a wider range of question types with our free aptitude tests.