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Mathematics

Converting Fractions, Decimals and Percentages: Practice Questions

Twelve conversion questions worked in all three directions, including 1/3 and 1/6, where the decimal recurs and the rounding is stated rather than hidden.

The short answer

Twelve questions converting between fractions, decimals and percentages in all three directions. Every fraction becomes a decimal by dividing the numerator by the denominator, then a percentage by multiplying by 100. Two of them, 1/3 and 1/6, recur, and the answers say so instead of rounding quietly.

The questions

  1. Question 1Foundation2 marks

    Write 3/4 as a decimal and as a percentage.

    Show the working

    0.75 and 75%

    1. 3 ÷ 4 = 0.75
    2. 0.75 × 100 = 75
    3. So 3/4 = 0.75 = 75%
  2. Question 2Foundation2 marks

    Write 0.6 as a fraction in its lowest terms and as a percentage.

    Show the working

    3/5 and 60%

    1. The 6 sits in the tenths column, so 0.6 = 6/10.
    2. Divide numerator and denominator by 2: 6/10 = 3/5.
    3. 0.6 × 100 = 60, so 0.6 = 3/5 = 60%.
  3. Question 3Foundation2 marks

    Write 45% as a decimal and as a fraction in its lowest terms.

    Show the working

    0.45 and 9/20

    1. Per cent means out of 100, so 45% = 45/100.
    2. 45 ÷ 100 = 0.45
    3. Divide numerator and denominator by 5: 45/100 = 9/20.
  4. Question 4Core2 marks

    Write 7/8 as a decimal and as a percentage.

    Show the working

    0.875 and 87.5%

    1. 7 ÷ 8: 8 into 70 goes 8 remainder 6; 8 into 60 goes 7 remainder 4; 8 into 40 goes 5.
    2. So 7 ÷ 8 = 0.875
    3. 0.875 × 100 = 87.5, so 7/8 = 87.5%.
  5. Question 5Core2 marks

    Write 0.125 as a fraction in its lowest terms and as a percentage.

    Show the working

    1/8 and 12.5%

    1. The last digit is in the thousandths column, so 0.125 = 125/1000.
    2. Divide numerator and denominator by 125: 125/1000 = 1/8.
    3. 0.125 × 100 = 12.5, so 0.125 = 1/8 = 12.5%.
  6. Question 6Core2 marks

    Write 1/3 as a decimal and as a percentage. Give exact answers.

    Show the working

    0.333... (0.3 recurring) and 33 1/3 %

    1. 1 ÷ 3 = 0.333..., and the 3 repeats without ending.
    2. Exactly, that is 0.3 recurring.
    3. 0.333... × 100 = 33.333...%, which as an exact figure is 33 1/3 %.
    4. If a rounded answer is wanted: 0.333 to 3 d.p., or 33.3% to 1 d.p. - and say so.
  7. Question 7Core2 marks

    Write 1/6 as a decimal and as a percentage. Give exact answers.

    Show the working

    0.1666... (0.16 with the 6 recurring) and 16 2/3 %

    1. 1 ÷ 6 = 0.1666..., and the 6 repeats without ending.
    2. × 100 gives 16.666...%.
    3. Exactly, that is 16 2/3 %, since two thirds of one per cent is 0.666...%.
    4. Rounded: 0.167 to 3 d.p., or 16.7% to 1 d.p.
  8. Question 8Core3 marks

    Put these in ascending order: 0.62, 3/5, 63%, 0.605

    Show the working

    3/5, 0.605, 0.62, 63%

    1. Convert everything to a decimal. 3 ÷ 5 = 0.6 and 63 ÷ 100 = 0.63.
    2. The four values are now 0.62, 0.6, 0.63, 0.605.
    3. Write them to the same number of decimal places: 0.620, 0.600, 0.630, 0.605.
    4. In order: 0.600, 0.605, 0.620, 0.630 - that is 3/5, 0.605, 0.62, 63%.
  9. Question 9Core3 marks

    A shirt costs PKR 2,400 and is reduced by 15% in a sale. What is the sale price?

    Show the working

    PKR 2,040

    1. 10% of 2,400 = 240, so 5% of 2,400 = 120.
    2. 15% = 240 + 120 = 360
    3. The question asks for the price, not the discount: 2,400 - 360 = 2,040.
    4. Check with a multiplier: 100% - 15% = 85%, and 0.85 × 2,400 = 2,040.
  10. Question 10Stretch3 marks

    Write 5/16 as a decimal and as a percentage.

    Show the working

    0.3125 and 31.25%

    1. 5 ÷ 16: 16 into 50 goes 3 remainder 2; into 20 goes 1 remainder 4; into 40 goes 2 remainder 8; into 80 goes 5.
    2. So 5 ÷ 16 = 0.3125
    3. 0.3125 × 100 = 31.25, so 5/16 = 31.25%.
    4. Sense check: 5/16 is a little under 1/3, and 31.25% is a little under 33 1/3 %.
  11. Question 11Stretch2 marks

    17 students out of a class of 40 walk to school. What percentage is that?

    Show the working

    42.5%

    1. 17 out of 40 is the fraction 17/40.
    2. 17 ÷ 40 = 0.425
    3. 0.425 × 100 = 42.5%
    4. Check: 40 × 0.425 = 17
  12. Question 12Stretch3 marks

    Which is larger, 5/8 or 62%? By how much?

    Show the working

    5/8 is larger, by 0.5 of a percentage point

    1. 5 ÷ 8 = 0.625, so 5/8 = 62.5%.
    2. 62% = 0.62
    3. 0.625 > 0.620, so 5/8 is the larger.
    4. 62.5% - 62% = 0.5 of a percentage point.

Where these go wrong

  • Writing 1/3 as 0.33 and then using it as though it were exact, so that three thirds come to 0.99 rather than 1.
  • Moving the decimal point the wrong way between decimals and percentages, so 0.45 is given as 4.5% instead of 45%.
  • Reading 0.6 as six hundredths and writing 6/100, which cancels to 3/50 and is ten times too small.
  • Ordering 3/5, 0.62 and 63% by comparing 3, 0.62 and 63 rather than converting all three into one form first.
  • Finding 15% of PKR 2,400 correctly and then giving PKR 360 as the sale price instead of subtracting it from 2,400.

One route handles every conversion here

Fraction to decimal is a division: the numerator divided by the denominator, in that order. Decimal to percentage is a multiplication by 100. Run those two in sequence and any fraction becomes a percentage without a separate rule to remember.

Going the other way, reverse them. A percentage divided by 100 gives the decimal; the decimal read by its last column gives the fraction, because the last digit names the denominator - tenths, hundredths, thousandths - and then it is cancelled down. Every answer below uses this and nothing else.

1/3 is not 0.33

Two of these twelve do not convert to a decimal that stops. 1 ÷ 3 gives 0.333... and 1 ÷ 6 gives 0.1666..., and both go on forever. Writing 0.33 and moving on looks harmless until three of them are added and the total comes to 0.99 instead of 1.

There are two honest answers and both are acceptable. Give the exact percentage as a mixed number - 33 1/3 % and 16 2/3 % - or give a rounded decimal and say what it has been rounded to. What is not acceptable is a rounded figure written as though it were exact, because everything built on it inherits an error nobody can trace.

The conversions worth knowing without working them out

About a dozen conversions come up often enough that dividing them out each time is wasted effort. They are the eighths, the fifths, the quarters and the thirds, and between them they cover most of what an exam asks.

The eighths are the ones most often missed, and they are the easiest to rebuild: 1/8 = 0.125, and every other eighth is a multiple of it.

  • 1/2 = 0.5 = 50%
  • 1/4 = 0.25 = 25% and 3/4 = 0.75 = 75%
  • 1/5 = 0.2 = 20%, so 2/5 = 40%, 3/5 = 60%, 4/5 = 80%
  • 1/8 = 0.125 = 12.5%, so 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%
  • 1/10 = 0.1 = 10% and 1/20 = 0.05 = 5%
  • 1/3 = 33 1/3 % and 2/3 = 66 2/3 % - exact only as fractions

Common questions

Which way round do I divide to turn a fraction into a decimal?

Numerator divided by denominator - the top number by the bottom one. For 3/4 that is 3 ÷ 4 = 0.75. Doing it the other way gives 1.333..., which is bigger than 1 and therefore visibly wrong for a fraction that is less than a whole.

Why do 1/3 and 1/6 not give a decimal that stops?

A fraction terminates as a decimal only when its denominator, in lowest terms, is built from 2s and 5s. 4, 8, 16, 20 and 40 all are, which is why 3/4, 7/8, 5/16 and 17/40 stop. 3 and 6 are not, so those two recur forever.

What is the difference between 0.5 of a percentage point and 0.5 per cent?

A percentage point is the gap between two percentages. 5/8 is 62.5% and 62% is 62%, so the gap is 0.5 of a percentage point. Saying 5/8 is 0.5% larger would mean something different - about a hundredth of a point - and in a question about interest rates that difference matters.

When are children expected to know these conversions?

By the end of Year 6 in England. The national curriculum expects pupils to recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. The eighths and the thirds are the pair most often still missing when pupils reach us in Year 7.

Sources

  1. National curriculum in England: mathematics programmes of study — Department for Education

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