Fractions, decimals and percentages: change recurring decimals into their corresponding fractions and vice versa (Higher tier only)

The idea

This higher tier skill means converting a recurring decimal to its exact fraction and converting a fraction to a recurring decimal. A recurring decimal has a digit or block of digits that repeats forever, shown by dots or a bar. For example, 0.7̇ means 0.777…, and 0.4̇5̇ means 0.454545…. To convert 0.7̇ to a fraction, let x = 0.777…, multiply by 10 to get 10x = 7.777…, subtract x to obtain 9x = 7, so x = 7/9. To convert 5/11 to a decimal, divide 5 by 11 to get 0.454545… = 0.4̇5̇. The method works because multiplying by a power of ten shifts the repeating block so subtraction eliminates the infinite tail. In multiple-choice questions, you must identify the exact fraction or recurring decimal, not a rounded value.

What earns the marks

• A recurring decimal has a repeating digit or block of digits that continues indefinitely, indicated by dots or a bar over the repeating part.
• To convert a recurring decimal to a fraction, set x equal to the decimal, multiply by a power of ten that shifts one full repeating block, subtract the original equation, and solve for x.
• For a single repeating digit such as 0.7̇, multiplying by 10 gives 10x = 7.777…, and subtracting x = 0.777… gives 9x = 7, so x = 7/9.
• For a repeating block of two digits such as 0.4̇5̇, multiplying by 100 gives 100x = 45.4545…, and subtracting x = 0.4545… gives 99x = 45, so x = 45/99 = 5/11.
• To convert a fraction to a recurring decimal, divide the numerator by the denominator and identify the repeating block; for example 5 ÷ 11 = 0.454545… = 0.4̇5̇.
• In a multiple-choice question, check exact equivalence by converting the option to the other form or by substituting into the original equation.

Common mistakes and how to put them right

• Using a rounded decimal such as 0.78 for 0.7̇ instead of the exact recurring form.

  Put it right: Keep the recurring notation or the exact fraction 7/9.

• Multiplying by the wrong power of ten when the repeating block has more than one digit.

  Put it right: Multiply by 10ⁿ where n is the number of digits in the repeating block.

• Subtracting the equations in the wrong order and losing the sign.

  Put it right: Subtract the smaller expression from the larger so the coefficient of x is positive.

• Forgetting to simplify the resulting fraction.

  Put it right: Divide numerator and denominator by their highest common factor, as 45/99 simplifies to 5/11.

Exam tips

• Write the recurring decimal with a clear dot or bar notation before starting, so you can see exactly which digits repeat.
• After solving for x, substitute back into the original recurring decimal to check that the fraction is exact.
• In multiple-choice questions, eliminate options that are rounded or have the wrong repeating block before doing a full conversion.