Fractions of an amount: worked examples and method

One part

The denominator says how many equal parts.

1/5 of 40 is 40 ÷ 5 = 8.

Several parts

3/5 is three of those parts.

If one part is 8, three parts are 24.

Of means multiply

3/5 of 40 is the same as 40 × 3 ÷ 5.

Divide first when the amount splits exactly.

The whole

5/5 of an amount is the whole amount.

The fraction of the amount cannot be bigger than the amount when the fraction is less than 1.

Check

Multiply the answer by the denominator and divide by the numerator.

You should get the original amount back.

Worded questions

“Three fifths of the class” is a fraction of an amount.

Find the class size, then take the fraction.

A unit fraction by dividing

The denominator is the number of equal parts. 1/4 of 28 asks for one of four equal parts, so divide 28 by 4. The part is 7. 1/5 of 35 is 35 ÷ 5 = 7. You divide by the bottom number. Multiplying by the bottom makes the amount larger, and a fraction less than 1 cannot do that.

Check by multiplying the part back up. Four lots of 7 are 28, so one lot is a quarter. Five lots of 7 are 35, so one lot is a fifth. If the check misses the original amount, the division was by the wrong number.

Use a bar if the story is easier that way. A bar of 28 split into 4 equal pieces has 7 in each piece. Shade one piece for a quarter. The shaded value is the answer for a unit fraction. Do not shade the denominator itself: the 4 is the number of pieces, not the value of a piece.

Method: 1. The denominator says how many equal parts to make. 2. Divide the amount by the denominator. 3. That one part is the unit fraction of the amount.

Then multiply by the numerator

When the numerator is more than 1, find one part and then take that many parts. 3/5 of 45: one fifth is 45 ÷ 5 = 9. Three fifths is 3 × 9 = 27. Stopping at 9 answers 1/5 of 45, which the question did not ask.

The same two moves can be written as one line: divide by the denominator, then multiply by the numerator. 45 ÷ 5 × 3 = 27. Dividing first is the safe order when the amount splits exactly, because you stay with whole numbers. Multiplying 45 × 3 first gives 135, and 135 ÷ 5 is also 27, but the numbers are larger than they need to be.

Five fifths of 45 is the whole 45. Three fifths should be smaller than 45, and 27 is. Two fifths would be 18, and one fifth is 9. The answers go up in steps of 9 as the numerator goes up by 1. If your step size is not the unit fraction, the first division was wrong.

Method: 1. Divide by the denominator to find one part. 2. Multiply that part by the numerator. 3. Check by seeing that the full denominator of parts returns the original amount.

The amount left

Nia has 80 stickers and uses 2/5 of them. One fifth of 80 is 16. Two fifths is 32. The question 'how many are left?' is not asking for 32. Subtract from the whole: 80 - 32 = 48 stickers left.

You can also think about the fraction that stays. She uses 2/5, so she keeps 3/5. Three fifths of 80 is 3 × 16 = 48. The two routes, subtract the used part or take the remaining fraction, must give the same number. Use the second route only when the two fractions add to 1.

A common slip is to subtract the denominator. 80 - 5 = 75 is not the amount left after using two fifths. The 5 is the number of equal parts, not a number of stickers. Another slip is to stop once the used part is found and box 32. Read the last sentence of the question before you stop.

Method: 1. Work out the fraction that was used. 2. Subtract that from the whole amount, or find the fraction that remains. 3. The amount left and the amount used must add back to the start.

Two fractions of one total

If both fractions are of the same original amount, work each one from that amount. A tin of 96 biscuits has 3/8 chocolate and 1/4 plain, both of the whole tin. One eighth of 96 is 12, so three eighths is 36 chocolate. One quarter of 96 is 24 plain. The ginger biscuits are the rest: 96 - 36 - 24 = 36.

Check the fractions add sensibly. 1/4 = 2/8, so chocolate and plain together are 3/8 + 2/8 = 5/8 of the tin. Ginger is the other 3/8, and three eighths of 96 is 36 again. If the two given fractions had added to more than 1, the story would be impossible. Here they do not.

A second cut of what remains is a different instruction. A rope of 150 cm has 2/5 cut off, which is 60 cm, leaving 90 cm. Then 1/3 of the remaining rope is 90 ÷ 3 = 30 cm, and 60 cm is left. Taking 1/3 of the original 150 cm would be 50 cm, and that is not what the second sentence says. The word remaining tells you to change the total before the second fraction.

Method: 1. Find each fraction of the original total, not of what is left. 2. Add those parts if you need both groups together. 3. Subtract them from the total if you need what remains.

Reading worded fraction questions

In a real exam the fraction work is almost always hidden inside a story: spends, saves, gives away, uses, sells. Read the story one sentence at a time and draw a line under every fraction and every amount. Each fraction needs its own calculation, and each one is a fraction of a particular amount. Write which amount it belongs to, for example "3/8 of the £264", so you do not mix them up later.

The phrase "of the rest" or "of what is left" is the usual trap. The second fraction does not apply to the starting amount. It applies to what remains after the first piece has gone. Find the first piece, subtract it, and only then take the second fraction of the new, smaller amount. Questions often finish with "how much is left?", which means one more subtraction at the end.

Method: 1. Underline each fraction and the amount it belongs to. 2. Find the first piece: divide by the denominator, then multiply by the numerator. 3. Subtract the first piece to get the rest, before using any "of the rest" fraction. 4. Finish with what the question asks for: the piece, the amount left or the total spent.

Tip: Write "rest = " on its own line. That line is where most method marks are won in multi-step fraction questions.

Working back to the whole amount

Sometimes the question gives you the answer to a fraction of an amount and asks for the whole. For example, 3/8 of the members walk to school and 21 members walk. The 21 is three of the eight equal parts. So divide by the numerator to get one part, then multiply by the denominator to get all the parts.

This is the reverse of the usual order, and students often multiply when they should divide. A quick sense check protects you: the whole must be bigger than the piece. If your total comes out smaller than the number you were given, you used the wrong order. Check by finding the fraction of your total again. It must give back the number in the question.

Method: 1. Write what the given number is: for example, three parts out of eight. 2. Divide the given number by the numerator to find one part. 3. Multiply one part by the denominator to find the whole. 4. Check by taking the fraction of your whole. It should match the given number.

Comparing, "show that" and "is she correct"

Many exam questions ask you to decide something rather than just calculate. "Which is bigger, 5/6 of 48 or 7/10 of 60?" needs both amounts worked out and a clear statement of which is larger. The command phrase "show how you decide" means the numbers must be visible. A bare answer such as "the second one" scores nothing.

A "show that" question gives you the answer, so every step has to be written down. "Is she correct?" needs a calculation and then a clear conclusion that uses it, such as "No, because 3/5 of 85 is 51, not 52". A one-word Yes or No on its own does not earn the mark. Finish with a sentence that points back at your numbers.

Method: 1. Work out each fraction of its amount separately. 2. Write both results next to each other so they can be compared. 3. State the conclusion in a short sentence that uses the numbers. 4. For "show that", end on the exact value the question gave you.

Money, measures and the calculator

Fractions of money and measures follow exactly the same method, but the units matter. A fraction of £84 is in pounds, a fraction of 56 litres is in litres, and a fraction of 2 hours is easiest in minutes. Convert first when the amount has mixed units, and put the unit back into your answer.

On a calculator paper you can type the fraction in directly, but writing the divide-then-multiply line is still what earns the method mark. If the amount does not divide exactly, the question is probably asking for a fraction of a decimal amount in pounds, so keep two decimal places. Always read the result back against the story: a sale price must be lower than the original price.

Method: 1. Check the units, and convert to a single unit if the amount has two. 2. Divide by the denominator, then multiply by the numerator. 3. If the question is about a sale or a share, decide whether to subtract the piece. 4. Write the unit and check the answer is sensible for the story.

Tip: A sale price must be less than the original price. If it is not, you found the amount taken off but forgot to subtract it.