Fractions, decimals and percentages: one idea, three ways

Fractions, decimals and percentages are one idea

A fraction, a decimal and a percentage can all describe the same part of a whole. The national curriculum asks teachers to show children throughout Year 5 that they are "different ways of expressing proportions", rather than three separate topics.

Picture a 10 by 10 grid of 100 squares, with 25 of them shaded. You could say a quarter is shaded, ¹⁄₄. You could say 25 hundredths are shaded, which is 0.25. So ¹⁄₄ = 0.25. Or you could say 25 out of 100 are shaded, which is 25%. The grid is the same each time.

The whole grid is 100%, and each square is 1%, or ¹⁄₁₀₀. Once a child sees that a percentage is just a number of hundredths, the conversions make sense rather than being rules to remember.

The key fractions, decimals and percentages to know

These are the fractions, decimals and percentages worth knowing by heart. Year 5 starts with halves, quarters, fifths and tenths, and in Year 6 children recall and use equivalences like these in all sorts of problems:

Fraction	Decimal	Percentage
1 whole	1	100%
1⁄2	0.5	50%
1⁄4	0.25	25%
3⁄4	0.75	75%
1⁄5	0.2	20%
2⁄5	0.4	40%
4⁄5	0.8	80%
1⁄10	0.1	10%
1⁄100	0.01	1%
1⁄8	0.125	12.5%
1⁄3	about 0.333	about 33.3%

Many of the rest can be built from these. ³⁄₁₀ is three lots of ¹⁄₁₀, so ³⁄₁₀ = 0.3, which is 30%. ³⁄₅ is three lots of ¹⁄₅, so ³⁄₅ = 0.6, which is 60%.

How to convert a fraction into a percentage

To convert a fraction to a percentage, find out how many hundredths it is. There are two ways.

Make the denominator 100. If the denominator goes into 100, multiply the top and bottom by the same number. ³⁄₅ = ⁶⁰⁄₁₀₀, so ³⁄₅ = 60%. ⁷⁄₂₀ = ³⁵⁄₁₀₀, so ⁷⁄₂₀ = 35%. ⁹⁄₂₅ = ³⁶⁄₁₀₀, so ⁹⁄₂₅ = 36%. Year 5 children work with fractions like these, with denominators of 10, 20, 25 and 50.

Change it to a decimal first. If the denominator doesn't go into 100, divide the numerator by the denominator to get a decimal, then multiply by 100. For ³⁄₈, 3 ÷ 8 = 0.375, so ³⁄₈ = 0.375. Then 0.375 × 100 = 37.5, so ³⁄₈ = 37.5%. Fractions to decimals shows that division step by step.

How to change a percentage to a decimal

To change a percentage to a decimal, divide it by 100. Every digit moves two places to the right, into smaller places:

• 45% is 45 hundredths: 45 ÷ 100 = 0.45.
• 7% is 7 hundredths: 7 ÷ 100 = 0.07. Watch the zero in the tenths place.
• 30% is 30 hundredths: 30 ÷ 100 = 0.3.

A place value chart shows where the digits land:

0.45 in a place value chart

Ones	Tenths	Hundredths
0	4	5

Going the other way, from a decimal to a percentage, multiply by 100. 0.6 × 100 = 60, so 0.6 is 60%. 0.08 × 100 = 8, so 0.08 is 8%.

How to change a percentage to a fraction

To change a percentage to a fraction, write it over 100, then simplify. The national curriculum puts it as writing percentages "as a fraction with denominator 100":

• 40% is ⁴⁰⁄₁₀₀. Divide the top and bottom by 20: ⁴⁰⁄₁₀₀ = ²⁄₅.
• 75% is ⁷⁵⁄₁₀₀. Divide the top and bottom by 25: ⁷⁵⁄₁₀₀ = ³⁄₄.
• 12% is ¹²⁄₁₀₀. Divide the top and bottom by 4: ¹²⁄₁₀₀ = ³⁄₂₅.

The equivalent fractions show up clearly in bars of the same length. Two fifths and four tenths cover the same amount, and so do 40 hundredths, which is 40%:

Simplifying fractions explains how to find the number to divide by.

How to convert a number into a percentage

To convert a score or a number "out of" something into a percentage, write it as a fraction first, then change the fraction. This is how children use percentages to compare things in Year 6, which the national curriculum includes.

Say your child scores 18 out of 25 on one quiz and 34 out of 50 on another. Which is better?

• 18 out of 25 is ¹⁸⁄₂₅. Multiply the top and bottom by 4: ¹⁸⁄₂₅ = ⁷²⁄₁₀₀, so it's 72%.
• 34 out of 50 is ³⁴⁄₅₀. Multiply the top and bottom by 2: ³⁴⁄₅₀ = ⁶⁸⁄₁₀₀, so it's 68%.

The first score is better, even though 34 is a bigger number than 18. Percentages put both scores out of 100, so they can be compared fairly.

How to convert a ratio to a percentage

A ratio compares parts with each other, so add the parts to find the whole first. In Year 6, children may write ratios as a:b. If a bag has 3 red sweets for every 1 green sweet, the ratio is 3:1, and there are 4 sweets in each group. Red is 3 out of 4, which is ³⁄₄, so 75% of the sweets are red. Green is ¹⁄₄, which is 25%.

A common mistake is to turn 3:1 into ³⁄₁ or ¹⁄₃. The denominator is the whole group, so it's 4.

Common mistakes to watch for

• Using the digits as they are. 0.3 is 30%, not 3%. 0.05 is 5%, not 50%.
• Forgetting the whole is 100%. If 35% of a class walk to school, 65% don't. The parts add up to 100%.
• Dividing the wrong way. ³⁄₈ is 3 ÷ 8, not 8 ÷ 3.
• Thinking a percentage can't be a decimal. ¹⁄₈ is 12.5%, which is fine.
• Ratio parts as the denominator. For a ratio of 3:1, the fraction is ³⁄₄, not ³⁄₁.
• Comparing raw scores. 34 out of 50 sounds better than 18 out of 25, until both become percentages.

How to help at home

• Spot percentages out and about. Battery icons, sale signs and food labels all use them. Ask what 50% or 25% would be as a fraction.
• Use money. A pound is 100 pence, so 1% of a pound is 1p, and 0.25 of a pound is 25p, which is 25% or a quarter.
• Play matching pairs. Write ¹⁄₄, 0.25 and 25% on separate cards, do the same for the other key equivalents, and match them up.
• Ask for all three. When your child gives an answer as a fraction, ask for the decimal and the percentage too.

When do children learn fractions, decimals and percentages?

Percentages start in Year 5 and come together in Year 6. In Year 5 maths children meet the per cent sign, learn that per cent means parts per hundred, and write percentages as fractions and decimals. In Year 6 maths they recall the common equivalents, use them to compare, and work out decimals and percentages for fractions like ³⁄₈. It builds on work with fractions, equivalent fractions and place value.

Fractions, decimals and percentages questions to try

1. Write ³⁄₄ as a percentage.
2. Write 0.4 as a percentage.
3. Write 65% as a decimal.
4. Write 20% as a fraction in its simplest form.
5. Write ¹¹⁄₂₀ as a percentage.
6. Ana scores 21 out of 25 and Ben scores 42 out of 50. What percentage does each of them get?

Answers

1. 75%
2. 40% (0.4 × 100 = 40)
3. 0.65 (65 ÷ 100 = 0.65)
4. ¹⁄₅ (²⁰⁄₁₀₀, then divide the top and bottom by 20)
5. 55% (¹¹⁄₂₀ = ⁵⁵⁄₁₀₀)
6. Both get 84%. ²¹⁄₂₅ = ⁸⁴⁄₁₀₀ and ⁴²⁄₅₀ = ⁸⁴⁄₁₀₀.

For the rest of what children learn in these years, see the KS2 maths guide. The bar model is a useful way to draw percentage problems.