Fraction Problems
Theory
To solve a fraction problem, find the key words, choose the operation they describe, then calculate and answer in words with units. This Year 7 maths page has a table of key words, bar models for working backwards, and worked problems on the rest, fractions of fractions, sharing, rates and speed.
Step 1. Find the key words (see the table) and what you are asked to find. Write a number sentence.
Step 2. Calculate. To add or subtract, use a common denominator. To multiply or divide, change mixed numerals to improper fractions and cancel a top with a bottom.
Step 3. Answer in words, with units, and check the size. \(\dfrac{4}{9}\) of \(\$6.30\) must be a little less than half of \(\$6.30\), because \(\dfrac{4}{9} < \dfrac{1}{2}\).
| Key words | Operation | Example |
|---|---|---|
| altogether, total, in all | add | \(\dfrac{1}{3} + \dfrac{1}{4} = \dfrac{7}{12}\) |
| left, less, how many more | subtract | \(2 - \dfrac{3}{4} = 1\dfrac{1}{4}\) |
| the rest | take the fraction from \(1\) | \(1 - \dfrac{3}{8} = \dfrac{5}{8}\) |
| a fraction of an amount | multiply | \(\dfrac{2}{3}\) of \(24\) is \(16\) |
| per hour, each (a rate) | multiply by the rate | \(2\) h at \(3\dfrac{1}{2}\) km/h is \(7\) km |
| how many fit, how many pieces | divide | \(3 \div \dfrac{3}{4} = 4\) |
| a part is known: find the whole | find one part first | \(\dfrac{2}{5}\) is \(14\), so \(\dfrac{5}{5}\) is \(35\) |
Common number sentences:
How to solve a fraction problem
- Read the problem and underline the key words.
- Write a number sentence (add, subtract, multiply or divide).
- Calculate: common denominators to add or subtract; improper fractions to multiply or divide.
- Answer in words with units, and check the size of the answer.
Working backwards
- Find what fraction of the whole the known amount is.
- Divide to find one part (one fifth, one quarter, ...).
- Multiply to find the whole, then check with the story.
| \(1 - \dfrac{5}{8}\) | \(=\) | \(\dfrac{3}{8}\) |
| \(\dfrac{3}{8} \times 48\) | \(=\) | \(48 \div 8 \times 3\) |
| \(\) | \(=\) | \(18\) |
There are \(18\) seniors. Check: juniors \(= 6 \times 5 = 30\) and \(30 + 18 = 48\).
| \(\dfrac{3}{25} \times 2150\) | \(=\) | \(\dfrac{3 \times 2150}{25}\) |
| \(\) | \(=\) | \(\dfrac{6450}{25}\) |
| \(\) | \(=\) | \(258\) |
About \(258\) white cars.
| \(6\dfrac{1}{2} - \dfrac{2}{3}\) | \(=\) | \(6\dfrac{3}{6} - \dfrac{4}{6}\) |
| \(\) | \(=\) | \(5\dfrac{9}{6} - \dfrac{4}{6}\) |
| \(\) | \(=\) | \(5\dfrac{5}{6}\) |
\(\dfrac{3}{6}\) is smaller than \(\dfrac{4}{6}\), so borrow a whole: \(6\dfrac{3}{6} = 5\dfrac{9}{6}\). Mary works \(5\dfrac{5}{6}\) hours.
| \(\dfrac{1}{3} \times \dfrac{3}{4}\) | \(=\) | \(\dfrac{1}{4}\) |
| \(\dfrac{3}{4} - \dfrac{1}{4}\) | \(=\) | \(\dfrac{2}{4} = \dfrac{1}{2}\) |
He eats \(\dfrac{1}{4}\) of the whole cake, so \(\dfrac{1}{2}\) of the cake is left.
(ii) A bag holds \(7\dfrac{1}{2}\) kg of dog food. Each scoop is \(\dfrac{3}{8}\) kg. How many scoops are in the bag?
| \(5 \times 3 \times \dfrac{17}{2}\) | \(=\) | \(\dfrac{255}{2} = 127\dfrac{1}{2}\) |
| \(\dfrac{15}{2} \div \dfrac{3}{8}\) | \(=\) | \(\dfrac{15}{2} \times \dfrac{8}{3}\) |
| \(\) | \(=\) | \(20\) |
(i) Area \(= 15\) m², so \(127\dfrac{1}{2}\) stones (part of one stone is cut to fit).
(ii) \(20\) scoops. Check: \(20 \times \dfrac{3}{8} = 7\dfrac{1}{2}\).
| \(1 - \dfrac{1}{4}\) | \(=\) | \(\dfrac{3}{4}\) |
| \(\dfrac{2}{3} \times \dfrac{3}{4}\) | \(=\) | \(\dfrac{1}{2}\) |
| \(2 \times 24\) | \(=\) | \(48\) |
The snacks are \(\dfrac{1}{2}\) of his money, so Sam started with \(\$48\). Check: movie \(\$12\), rest \(\$36\), snacks \(\dfrac{2}{3} \times 36 = 24\).
(i) What fraction of the tank is \(150\) L?
(ii) How many litres does the full tank hold?
(iii) How many litres are in the tank now?
| \(\dfrac{7}{10} - \dfrac{4}{10}\) | \(=\) | \(\dfrac{3}{10}\) |
| \(150 \div 3 \times 10\) | \(=\) | \(500\) |
| \(\dfrac{7}{10} \times 500\) | \(=\) | \(350\) |
(i) \(\dfrac{3}{10}\)
(ii) \(500\) L
(iii) \(350\) L
(i) How far does he walk in \(1\dfrac{1}{3}\) hours?
(ii) How long does he take to walk \(10\dfrac{1}{2}\) km? Give the answer in minutes too.
| \(\dfrac{21}{4} \times \dfrac{4}{3}\) | \(=\) | \(7\) |
| \(\dfrac{21}{2} \div \dfrac{21}{4}\) | \(=\) | \(\dfrac{21}{2} \times \dfrac{4}{21}\) |
| \(\) | \(=\) | \(2\) |
(i) \(7\) km
(ii) \(2\) hours \(= 120\) minutes
Common pitfalls
Frequently asked questions
How do you know which operation to use in a fraction problem?
Look for key words: altogether means add, left means subtract, of means multiply, how many fit means divide.
What does the rest mean?
The part not yet used. Take the fraction from \(1\): if \(\dfrac{5}{8}\) are juniors, \(\dfrac{3}{8}\) are the rest.
How do you find the whole when you know a part?
Find one part first. If \(\dfrac{2}{5}\) is \(14\), then \(\dfrac{1}{5}\) is \(7\) and the whole is \(35\).
How do you find a fraction of a fraction?
Multiply. \(\dfrac{1}{3}\) of \(\dfrac{3}{4}\) is \(\dfrac{1}{3} \times \dfrac{3}{4} = \dfrac{1}{4}\).
How do you check an answer to a fraction problem?
Put the answer back into the story, and check the size: a fraction less than one half of an amount must be less than half of it.