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Fractions, decimals and percentages

It is Bea! Fractions, decimals and percentages are three ways to write parts of a whole. Learn how they connect and a whole big chunk of the 11+ becomes friendly.

1. Cheat sheet: Fractions, decimals and percentages

Everything you need on one page. Press the button to print it on one sheet of A4.

Fractions

  • Top = numerator, bottom = denominator.
  • Equivalent: × or ÷ top and bottom by the same number.
  • Simplify: 12/18 = 2/3 (÷ 6).
  • Compare: same denominator, then compare tops.

Mixed numbers

  • 17/5 = 3 2/5 (17 ÷ 5 = 3 r 2).
  • 2 3/4 = (2 × 4 + 3)/4 = 11/4.

Fraction of an amount

  • ÷ bottom, × top: 3/5 of 40 = 8 × 3 = 24.
  • 4 × 3/8 = 12/8 = 1 1/2.
  • Of the remainder: take the first part away, then find the second.

Add and subtract fractions

  • Same denominator: add the tops only.
  • 3/4 + 1/8 = 6/8 + 1/8 = 7/8.
  • 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Multiply and divide

  • Tops × tops, bottoms × bottoms.
  • 2/3 × 3/4 = 6/12 = 1/2.
  • 3/4 ÷ 3 = 1/4.

Decimals

  • Columns: ones . tenths hundredths thousandths.
  • 0.5 = 0.50 > 0.25. Compare digit by digit.
  • Round 3.746: 1 dp = 3.7, 2 dp = 3.75.
  • Line up the points to add or subtract.

Equivalents to know

FrDec%
1/20.550%
1/40.2525%
3/40.7575%
1/50.220%
1/100.110%
1/80.12512.5%

Percentages

  • 10% = ÷ 10, 5% = half of 10%, 1% = ÷ 100.
  • 15% of £80 = £8 + £4 = £12.
  • 20% off £45: 20% = £9, pay £36.
  • Increase: £40 up 15% = £40 + £6 = £46.
  • Reverse (stretch, beyond KS2): £36 is 80%, so 10% = £4.50 and 100% = £45.

2. Mini lessons

Equivalent fractions and simplifying

FDP-01FDP-03FDP-04

A fraction names equal parts of a whole or of a set. Different fractions can name the same amount. Multiply or divide the top and bottom by the same number and the amount stays the same. Dividing both by the biggest possible number is called simplifying.

Animation: the same amount, cut into more pieces

1/2= 2/4= 4/8

Three bars, all the same length. Cut into 2 equal parts, shade 1: that is 1/2. Cut into 4 parts, shade 2. The shaded part is exactly the same size: 2/4. Cut into 8 parts, shade 4: 4/8. So 1/2 = 2/4 = 4/8. Multiply top and bottom by the same number to make an equivalent fraction.

Worked example: Write 2/3 as a fraction with denominator 15. Then simplify 18/24.

  1. From 3 to 15 you multiply by 5. Do the same to the top: 2 × 5 = 10. So 2/3 = 10/15.
  2. For 18/24, find a number that divides both: the biggest is 6.
  3. 18 ÷ 6 = 3 and 24 ÷ 6 = 4. So 18/24 = 3/4.

Answer: 10/15 and 3/4

Comparing and ordering fractions

FDP-05

To compare fractions, give them the same denominator, so that the pieces are the same size. Then the bigger numerator is the bigger fraction.

5/8 = 25/403/5 = 24/4025/40 is more than 24/40, so 5/8 > 3/5
Change to the same denominator, then compare the tops.

Worked example: Which is bigger, 5/8 or 3/5?

  1. Find a number both 8 and 5 go into: 40.
  2. 5/8 = 25/40 (× 5 top and bottom). 3/5 = 24/40 (× 8 top and bottom).
  3. 25/40 is more than 24/40, so 5/8 is bigger.

Answer: 5/8

Mixed numbers and improper fractions

FDP-06

An improper fraction has a top bigger than its bottom, like 7/4. A mixed number shows the whole ones and the leftover fraction, like 1 3/4. They are two ways of writing the same amount.

7/4 = 1 3/4
Seven quarters: one whole bar and 3 quarters of the next.

Worked example: Write 23/6 as a mixed number and 4 3/8 as an improper fraction.

  1. 23 ÷ 6 = 3 remainder 5, so 23/6 = 3 5/6.
  2. For 4 3/8: 4 wholes is 4 × 8 = 32 eighths. Add the 3 eighths: 32 + 3 = 35.
  3. So 4 3/8 = 35/8.

Answer: 3 5/6 and 35/8

Fractions of amounts and multiplying by a whole number

FDP-02FDP-09

To find a fraction of an amount, divide by the bottom number to find one part, then multiply by the top number. To multiply a fraction by a whole number, multiply the top only.

888883/5 of 40 = 3 × 8 = 24
Divide by the bottom number, multiply by the top number.

Worked example: (a) Find 3/5 of 40. (b) Work out 4 × 3/8.

  1. (a) 40 ÷ 5 = 8 is one fifth. 3 fifths is 8 × 3 = 24.
  2. (b) Multiply the top: 4 × 3 = 12, so the answer is 12/8.
  3. 12/8 = 1 4/8 = 1 1/2.

Answer: 24, and 1 1/2

Fraction of the remainder

FDP-18

In a multi-step problem, the second fraction usually refers to what is LEFT after the first part has gone. Work out the remainder first, then take the fraction of that.

book £20£10left after the book: £40, in four parts of £10£40 − £10 = £30 left
The second fraction is of the REMAINDER, not of the original amount.

Worked example: Zara has £60. She spends 1/3 on a book, then 1/4 of what is left on a snack. How much does she have left?

  1. 1/3 of £60 = £20 on the book.
  2. Left: £60 − £20 = £40.
  3. 1/4 of £40 = £10 on the snack.
  4. Left: £40 − £10 = £30.

Answer: £30

Adding and subtracting fractions

FDP-07FDP-08

You can only add or subtract fractions when the pieces are the same size. Change to a common denominator first. Then add or subtract the tops, and keep the bottom the same.

Animation: 1/2 + 1/3

1/21/31/2 = 3/6+1/3 = 2/6=5/6

Halves and thirds are different-sized pieces, so we cannot add them yet. Cut into sixths. 1/2 is 3 sixths. 1/3 is 2 sixths. Now the pieces are the same size. Add the sixths: 3 + 2 = 5, so 1/2 + 1/3 = 5/6. Only the top numbers are added; the bottom number names the size of the piece.

Worked example: Work out 3/4 + 1/8 and 1/2 + 1/3.

  1. 3/4 + 1/8: 4 goes into 8, so change 3/4 to 6/8. Then 6/8 + 1/8 = 7/8.
  2. 1/2 + 1/3: 6 is a multiple of both. 3/6 + 2/6 = 5/6.

Answer: 7/8 and 5/6

Multiplying fractions and dividing by a whole number

FDP-10

To multiply two fractions, multiply the tops and multiply the bottoms. To divide a fraction by a whole number, multiply the bottom by it. A picture of a rectangle cut into rows and columns shows why.

1/2 × 1/31/2 of 1/3 = 1/6
The shaded cell is 1 of 6, so 1/2 × 1/3 = 1/6 (multiply the tops and multiply the bottoms).

Worked example: Work out 2/3 × 3/4 and 3/4 ÷ 3.

  1. Tops: 2 × 3 = 6. Bottoms: 3 × 4 = 12. So 6/12.
  2. Simplify: 6/12 = 1/2.
  3. 3/4 ÷ 3: share 3 quarters into 3 equal parts, 1 quarter each. So 1/4.

Answer: 1/2 and 1/4

Decimal place value, ordering and rounding

FDP-11FDP-12

After the decimal point come tenths, hundredths and thousandths. To compare decimals, line up the places and compare from the left. To round, look at the digit just after the place you are rounding to: 5 or more rounds up.

3.73.753.83.746
3.746 is between 3.74 and 3.75. It is nearer 3.75.

Worked example: Put 0.5, 0.25, 0.45 and 0.405 in order, smallest first. Then round 3.746 to 1 and to 2 decimal places.

  1. Give each three decimal places: 0.500, 0.250, 0.450, 0.405.
  2. Smallest first: 0.25, 0.405, 0.45, 0.5.
  3. 3.746 to 1 dp: the next digit is 4, so stay at 3.7.
  4. 3.746 to 2 dp: the next digit is 6, so round up to 3.75.

Answer: 0.25, 0.405, 0.45, 0.5; then 3.7 and 3.75

Adding, subtracting and multiplying decimals

FDP-13

Line up the decimal points so tenths sit under tenths. Fill any gaps with zeros. To multiply a decimal by a whole number, ignore the point, multiply, then put the point back so the answer has the same number of decimal places.

4.75+3.608.35← add a 0
Line up the decimal points. Fill gaps with 0.

Worked example: Work out 4.75 + 3.6 and 2.4 × 3.

  1. 4.75 + 3.60: 5 + 0 = 5 hundredths; 7 + 6 = 13 tenths, write 3 carry 1; 4 + 3 + 1 = 8 ones. So 8.35.
  2. 2.4 × 3: think 24 × 3 = 72.
  3. 2.4 has one decimal place, so the answer has one: 7.2.

Answer: 8.35 and 7.2

Fractions, decimals and percentages together

FDP-14

"Per cent" means out of 100. So 25% is 25/100, which simplifies to 1/4, and as a decimal is 0.25. Learn the common ones by heart and you can swap between all three.

Animation: shading a hundred square

10%20%25%= 25/100= 1/4= 0.25

A hundred square has 100 equal squares. "Per cent" means "out of 100". Shade a row of 10 squares: 10 out of 100 is 10%. Two rows: 20%. Shade 25 squares. 25 out of 100 is 25% = 25/100 = 1/4 = 0.25. One amount, three names: a percentage, a fraction and a decimal.

Worked example: Write 7/20 as a decimal and as a percentage.

  1. Make the denominator 100: 20 × 5 = 100, so 7/20 = 35/100.
  2. 35/100 = 0.35.
  3. 35 out of 100 is 35%.

Answer: 0.35 and 35%

Percentages of amounts, discounts, increases and reverse percentages

FDP-15FDP-16FDP-17

Find 10% by dividing by 10, then build the percentage you need (5% is half of 10%, 1% is a hundredth). A discount is a percentage you take off, and an increase is a percentage you add on. A reverse percentage starts with the sale price and works back to the original. Reverse percentages go beyond KS2, so treat them as a challenge.

£8£410% = £8, 5% = £415% of £80 = £8 + £4 = £12
Find 10% first, then build up.

Worked example: A coat costs £45. It is reduced by 20%. What is the sale price? A game costs £40 and its price goes up by 15%. What is the new price? Another coat costs £36 after a 20% reduction. What was its original price? (This last part is stretch, beyond KS2.)

  1. 10% of £45 = £4.50, so 20% = £9.
  2. Sale price: £45 − £9 = £36.
  3. Increase: 10% of £40 = £4 and 5% = £2, so 15% = £6. New price: £40 + £6 = £46.
  4. Reverse: after a 20% reduction you pay 80% of the original. 80% = £36.
  5. 10% = £36 ÷ 8 = £4.50, so 100% = £4.50 × 10 = £45.

Answer: £36, then £46, then £45

3. Worked examples from easy to hard

Level 1 is Year 3, one step. Level 5 is a challenge with 3 to 5 steps. Try each one yourself before you read the steps.

Level 1One step

What is 1/3 of 27?

  1. Divide by the bottom number: 27 ÷ 3 = 9.

Answer: 9

Level 2One or two steps

Aliyah has 36 stickers. She gives 3/4 of them to her friends. How many does she give away?

  1. One quarter: 36 ÷ 4 = 9.
  2. Three quarters: 9 × 3 = 27.

Answer: 27

Level 3Two steps

Work out 2/5 + 3/10.

  1. Change 2/5 to tenths: 2/5 = 4/10.
  2. Add: 4/10 + 3/10 = 7/10.

Answer: 7/10

Level 4Exam standard

A coat costs £80. In a sale it has 25% off. Mina then uses a £5 voucher. How much does she pay?

  1. 25% is a quarter: £80 ÷ 4 = £20 off.
  2. Sale price: £80 − £20 = £60.
  3. Take off the voucher: £60 − £5 = £55.

Answer: £55

Level 5Challenge

Omar has £90. He spends 2/5 on a bike light, then 1/3 of what is left on a book, and puts 25% of the rest into his savings box. How much goes into the savings box?

  1. Bike light: 2/5 of £90 = £36. Left: £90 − £36 = £54.
  2. Book: 1/3 of £54 = £18. Left: £54 − £18 = £36.
  3. Savings: 25% is a quarter. £36 ÷ 4 = £9.
  4. The question asks about the savings box, so the answer is £9, not what is left.

Answer: £9

4. Common mistakes

Lots of children make these slips. Spot them early and you will win marks.

Thinking a longer decimal is bigger

DEC_LONGER_IS_LARGER

The mistake: Reading the digits after the point as a whole number, so 0.25 looks bigger than 0.5 because 25 is bigger than 5.

Why it is wrong: 0.5 is 5 tenths. 0.25 is only 2 tenths and 5 hundredths. 0.5 = 0.50, which is more than 0.25.

How to avoid it: Write both with the same number of decimal places, or use a number line from 0 to 1.

✗ Wrong: 0.25 > 0.5. And counting up in tenths, the next number after 4.9 is 4.10.
✓ Right: 0.50 > 0.25. Counting up in tenths, the next number after 4.9 is 5.0 (not 4.10).

Thinking tenths always beat hundredths

DEC_SHORTER_IS_LARGER

The mistake: Believing a shorter decimal is bigger because tenths are bigger than hundredths.

Why it is wrong: You must compare the whole number of tenths and hundredths, not just the name of the column. 0.83 has 8 tenths, more than 0.4.

How to avoid it: Add zeros to match the length: 0.40 and 0.83.

✗ Wrong: 0.4 > 0.83.
✓ Right: 0.40 < 0.83.

Adding tops and bottoms

FRAC_ADD_ACROSS

The mistake: Adding the numerators and also the denominators.

Why it is wrong: The bottom number tells the size of the piece, and that does not change when you add. Also 2/5 is smaller than 1/2, so it cannot be the sum of 1/2 and something.

How to avoid it: Make the denominators the same, then add the tops only.

✗ Wrong: 1/2 + 1/3 = 2/5.
✓ Right: 3/6 + 2/6 = 5/6.

Big numbers mean a big fraction

FRAC_BIGGER_NUMBERS_BIGGER

The mistake: Thinking a fraction with bigger numbers is bigger.

Why it is wrong: A bigger denominator means the whole is cut into more, smaller pieces.

How to avoid it: Picture the same pizza shared between 4 people and then 8 people.

✗ Wrong: 1/8 > 1/4 because 8 > 4.
✓ Right: 1/4 > 1/8, because quarters are bigger pieces than eighths.

Assuming a number line is in tenths

FRAC_NUMBERLINE_ASSUME_TENTHS

The mistake: Labelling the marks on every number line as tenths.

Why it is wrong: The marks show how many equal parts the line from 0 to 1 is cut into. That could be eighths, fifths or anything.

How to avoid it: Count the jumps from 0 to 1 first. That is your denominator.

✗ Wrong: The third mark after 0 on a line cut into 8 parts is 3/10.
✓ Right: The third mark after 0 is 3/8.

5. Quick check

Five questions, easy to hard. Have a go on paper, then open the answer.

  1. Find 3/8 of 56.

    Show answer

    21

    56 ÷ 8 = 7, then 7 × 3 = 21.

  2. Write 0.45 as a fraction in its simplest form.

    Show answer

    9/20

    0.45 = 45/100. Divide top and bottom by 5: 9/20.

  3. Put these in order, smallest first: 3/4, 0.7, 72%.

    Show answer

    0.7, 72%, 3/4

    As decimals they are 0.75, 0.70 and 0.72. So the order is 0.70 (0.7), 0.72 (72%), 0.75 (3/4).

  4. A £60 jacket has 35% off. What is the sale price?

    Show answer

    £39

    10% = £6, so 30% = £18. 5% = £3. 35% = £21 off. £60 − £21 = £39.

  5. Ben has 70 stickers. He gives 1/10 to his sister, then 1/3 of the stickers that are left to a friend. How many does he keep?

    Show answer

    42

    1/10 of 70 = 7, leaving 63. 1/3 of 63 = 21. Ben keeps 63 − 21 = 42.

6. For grown-ups

  • Use real objects: cut a pizza or chocolate bar. Ask "If we cut each piece in half, what is the fraction now?" to show equivalent fractions.
  • Percentages are everywhere. In shops, ask "What is 10% off this? So what is 20% off?" and let your child do it in their head.
  • Decimals go wrong when children see them as two separate whole numbers. Money helps: £0.50 is more than £0.25, even though 25 looks bigger.