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.
Bea’s tip: Make the parts equal first. Then use what you know about the whole.
Find 2/3 + 1/6.
Sixths are equal-sized parts: 2/3 = 4/6.
4/6 + 1/6 = 5/6. Only the numerators are added.
A bag costs £60 with 15% off. What do you pay?
10% = £6; 5% = £3. So 15% = £9.
£60 − £9 = £51. Check: a discount makes the price smaller.
1/2 + 1/3 is not 2/5. The pieces are different sizes! Change to sixths: 3/6 + 2/6 = 5/6.
Find 3/5 of 35.
21: 35 ÷ 5 × 3 = 21.
Write 7/8 as a decimal and a percentage.
0.875 and 87.5%: 7 ÷ 8 = 0.875, then × 100.
Find 3/4 − 1/8.
5/8: 3/4 = 6/8, then 6/8 − 1/8 = 5/8.
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
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.
Answer: 10/15 and 3/4
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.
Worked example: Which is bigger, 5/8 or 3/5?
Answer: 5/8
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.
Worked example: Write 23/6 as a mixed number and 4 3/8 as an improper fraction.
Answer: 3 5/6 and 35/8
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.
Worked example: (a) Find 3/5 of 40. (b) Work out 4 × 3/8.
Answer: 24, and 1 1/2
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.
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?
Answer: £30
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
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.
Answer: 7/8 and 5/6
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.
Worked example: Work out 2/3 × 3/4 and 3/4 ÷ 3.
Answer: 1/2 and 1/4
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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.
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.
Answer: 0.25, 0.405, 0.45, 0.5; then 3.7 and 3.75
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.
Worked example: Work out 4.75 + 3.6 and 2.4 × 3.
Answer: 8.35 and 7.2
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
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.
Answer: 0.35 and 35%
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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.
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.)
Answer: £36, then £46, then £45
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.
What is 1/3 of 27?
Answer: 9
Aliyah has 36 stickers. She gives 3/4 of them to her friends. How many does she give away?
Answer: 27
Work out 2/5 + 3/10.
Answer: 7/10
A coat costs £80. In a sale it has 25% off. Mina then uses a £5 voucher. How much does she pay?
Answer: £55
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?
Answer: £9
Lots of children make these slips. Spot them early and you will win marks.
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).
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.
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.
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.
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.
Five questions, easy to hard. Have a go on paper, then open the answer.
Find 3/8 of 56.
21
56 ÷ 8 = 7, then 7 × 3 = 21.
Write 0.45 as a fraction in its simplest form.
9/20
0.45 = 45/100. Divide top and bottom by 5: 9/20.
Put these in order, smallest first: 3/4, 0.7, 72%.
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).
A £60 jacket has 35% off. What is the sale price?
£39
10% = £6, so 30% = £18. 5% = £3. 35% = £21 off. £60 − £21 = £39.
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?
42
1/10 of 70 = 7, leaving 63. 1/3 of 63 = 21. Ben keeps 63 − 21 = 42.