Fractions, decimals and percentages: work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8)

The idea

This skill means moving fluently between a terminating decimal and its exact fraction without rounding. A terminating decimal stops after a finite number of decimal places. To convert 0.375 to a fraction, write it over a power of ten: 0.375 = 375/1000. Divide numerator and denominator by their highest common factor, 125, giving 3/8. To convert 7/2 to a decimal, divide 7 by 2 to get 3.5. The examples 3.5 and 7/2, and 0.375 and 3/8, show that fractions can be improper and decimals may have several places. You must recognise equivalence and choose the most convenient form. This foundation and higher tier skill is assessed in multiple-choice questions requiring exact equivalence.

What earns the marks

• A terminating decimal has a finite number of decimal places and can be written exactly as a fraction with denominator a power of 10 before simplifying.
• To convert a decimal to a fraction, write the decimal digits over the correct power of ten, then simplify by dividing numerator and denominator by their highest common factor.
• To convert a fraction to a decimal, divide the numerator by the denominator; for example 7 ÷ 2 = 3.5.
• Equivalent forms must be exact, not rounded; for example 0.375 = 3/8 exactly, whereas 0.38 is only an approximation.
• In a multiple-choice question, test each option by converting it to the other form or by cross-multiplying to check equality.

Common mistakes and how to put them right

• Writing 0.375 as 375/100 instead of 375/1000.

  Put it right: Count three decimal places and use denominator 10³ = 1000.

• Simplifying 375/1000 by dividing only the numerator or only the denominator.

  Put it right: Divide both by the same highest common factor, 125, to reach 3/8.

• Treating 3.5 as 35/10 and then simplifying incorrectly to 7/5.

  Put it right: 35/10 = 7/2, so 3.5 = 7/2.

• Assuming a fraction such as 7/2 cannot equal a decimal greater than 1.

  Put it right: Improper fractions convert to decimals greater than 1, as 7 ÷ 2 = 3.5.

Exam tips

• For each option, convert it to the form used in the question and compare exactly; do not choose an option that is merely close.
• When simplifying, divide numerator and denominator by their highest common factor in one step where possible, then check by multiplying back.
• Use a quick reverse check: multiply the fraction by the denominator of the decimal's place value to see whether you recover the decimal digits.