Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Number
Practise Fractions and mixed numbers. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Simplifying, ordering and converting fractions, and adding, subtracting, multiplying and dividing fractions and mixed numbers exactly. These questions are common on the non-calculator paper, both as straight calculations and inside word problems.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
1
Simplify a fractionDivide the top and bottom by a common factor, e.g. \(\frac{18}{24} = \frac{3}{4}\).
2
Find a fraction of an amountDivide by the denominator, then multiply by the numerator: \(\frac{3}{5}\) of 40 = 40 ÷ 5 × 3 = 24.
Add and subtract fractionsUse a common denominator: \(\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}\).
4
Multiply and divide fractionsMultiply tops and bottoms; to divide, multiply by the reciprocal of the second fraction.
5
Calculate with mixed numbersChange mixed numbers to improper fractions first, e.g. \(2\frac{1}{3} = \frac{7}{3}\), then add, subtract, multiply or divide.
5
Solve multi-step fraction problemsE.g. find a fraction of what is left, or find the whole amount from a fraction of it.
Notes
Equivalent fractions and mixed numbers
Multiply or divide the numerator (top) and denominator (bottom) by the same number to get an equivalent fraction: \(\frac{3}{4} = \frac{9}{12}\).
Simplest form: divide the top and bottom by their HCF: \(\frac{18}{24} = \frac{3}{4}\) (HCF 6).
To order fractions, write them with a common denominator or change them to decimals.
Improper fraction to mixed number: \(\frac{17}{5} = 3\frac{2}{5}\) (5 goes into 17 three times, remainder 2).
Mixed number to improper fraction: \(2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}\).
Fractions, decimals and percentages
Fraction to decimal: divide the top by the bottom, e.g. with short division: \(\frac{3}{8}\) = 3 ÷ 8 = 0.375.
Decimal to fraction: use place value, then simplify: 0.35 = \(\frac{35}{100} = \frac{7}{20}\).
Percentage to fraction: write it over 100, then simplify: 45% = \(\frac{45}{100} = \frac{9}{20}\).
Multiply: tops together, bottoms together: \(\frac{2}{3} \times \frac{9}{10} = \frac{18}{30} = \frac{3}{5}\). Cancelling common factors first keeps the numbers small.
Divide: keep the first fraction, change ÷ to ×, and flip the second fraction (use its reciprocal): \(\frac{3}{4} \div \frac{9}{10} = \frac{3}{4} \times \frac{10}{9} = \frac{30}{36} = \frac{5}{6}\).
Always change mixed numbers to improper fractions before multiplying or dividing: \(1\frac{1}{2} \times 2\frac{2}{3} = \frac{3}{2} \times \frac{8}{3} = \frac{24}{6} = 4\).
'Of' means multiply: \(\frac{3}{5}\) of 40 = 24.
Whole from a part: if \(\frac{2}{7}\) of a number is 18, then \(\frac{1}{7}\) is 9 and the whole number is 63.
Cheatsheet
Add or subtract: common denominator first, then add or subtract the numerators
Find a common denominator and write the equivalent fractions.
Add or subtract the numerators; keep the denominator.
Simplify, and give a mixed number if the question asks for one.
Example. Work out \(2\frac{3}{5} + 1\frac{3}{4}\) Give your answer as a mixed number.
Show the model answer
\(\frac{13}{5} + \frac{7}{4}\) (M1 for both improper fractions) \(= \frac{52}{20} + \frac{35}{20}\) (M1 for a correct common denominator) \(= \frac{87}{20} = 4\frac{7}{20}\) (A1)
Show that: multiply or divide mixed numbers
3 marks5
Change each mixed number to an improper fraction.
To divide, keep the first fraction, change ÷ to × and flip the second.
Multiply, then simplify, writing every step to reach the given answer.
Example. Show that \(1\frac{7}{8} \div 2\frac{1}{4} = \frac{5}{6}\)
Show the model answer
\(\frac{15}{8} \div \frac{9}{4}\) (M1 for both improper fractions) \(= \frac{15}{8} \times \frac{4}{9}\) (M1 for multiplying by the reciprocal) \(= \frac{60}{72} = \frac{5}{6}\) (A1 for a complete, correct method)
Fraction problem in context
3 to 4 marks4
Find the first fraction of the amount.
Work out what is left before taking the next fraction, if the question says 'of the rest' or 'of what is left'.
Label each value so you don't lose track, and answer the question with units.
Example. Priya has £240. She spends \(\frac{3}{8}\) of it on a coat. She then spends \(\frac{2}{5}\) of the money she has left on shoes. How much money does Priya have left?
Show the model answer
Coat: 240 ÷ 8 × 3 = £90, so £150 is left (P1) Shoes: 150 ÷ 5 × 2 = £60 (P1) Left: 150 − 60 = £90 (A1)
Order fractions, decimals and percentages
2 marks3
Change every number to a decimal (or all to percentages).
Compare them, lining up the decimal points.
Write the numbers in their original form, in the order asked for (check whether it says smallest or largest first).
Example. Write these numbers in order of size. Start with the smallest number. 0.62, \(\frac{5}{8}\), 61%, \(\frac{3}{5}\)
Show the model answer
\(\frac{5}{8}\) = 0.625, 61% = 0.61, \(\frac{3}{5}\) = 0.6 (M1 for at least two converted to the same form) \(\frac{3}{5}\), 61%, 0.62, \(\frac{5}{8}\) (A1)
Shortcuts and memory tricks
Dividing fractions: Keep, Change, Flip (keep the first, change ÷ to ×, flip the second).
Cancel before you multiply: in \(\frac{4}{9} \times \frac{15}{8}\), cancel 4 with 8 and 15 with 9 to get \(\frac{1}{3} \times \frac{5}{2} = \frac{5}{6}\).
Sense check: multiplying by a proper fraction makes a number smaller; dividing by a proper fraction makes it bigger.
On the calculator paper, the fraction key gives exact answers, and the S⇔D key (on many calculators) switches between fraction and decimal.
Where marks are lost
Adding the denominators: \(\frac{1}{3} + \frac{1}{4}\) is \(\frac{7}{12}\), not \(\frac{2}{7}\).
Multiplying mixed numbers without converting: \(2\frac{1}{2} \times 3\frac{1}{2}\) is \(\frac{35}{4} = 8\frac{3}{4}\), not \(6\frac{1}{4}\).
Flipping the wrong fraction when dividing: only the second fraction is turned upside down.
Not simplifying when the question asks for the simplest form, or giving an improper fraction when it asks for a mixed number.
Taking a fraction of the original amount when the question says 'of the money left'.
Exam technique
On the non-calculator paper, show every stage: the improper fractions, the common denominator, then the answer. The method marks come from these lines.
In 'show that' questions, write every step; just writing the given answer earns nothing.
In multi-step problems, say what each number is, e.g. 'left after coat = £150'.
Check your final answer is in the form asked for: simplest form, mixed number or decimal.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
Ben works out \(\frac{2}{3} + \frac{1}{4}\). His answer is \(\frac{3}{7}\).
(a) Without working out the correct answer, explain how you can tell that Ben's answer must be wrong.[1]
(b) Work out the correct answer.[2]
Show the answer and mark scheme
(a)Answer: \(\frac{3}{7}\) is less than \(\frac{2}{3}\), but adding \(\frac{1}{4}\) to \(\frac{2}{3}\) must give more than \(\frac{2}{3}\).
C1 for a correct reason, e.g. \(\frac{3}{7}\) is smaller than \(\frac{2}{3}\) (or smaller than \(\frac{1}{2}\)), but the answer must be bigger than \(\frac{2}{3}\)
Worked solution: Adding a positive number makes a number bigger, so the answer must be more than \(\frac{2}{3}\). But \(\frac{3}{7}\) is less than \(\frac{1}{2}\).
(b)Answer: \(\frac{11}{12}\)
M1 for a correct common denominator, e.g. \(\frac{8}{12} + \frac{3}{12}\)
A water tank is \(\frac{3}{5}\) full. After 42 litres of water are used, the tank is \(\frac{1}{4}\) full.
(a) Work out the capacity of the tank.[3]
(b) The tank is then filled until it is \(\frac{5}{6}\) full. How many litres of water are added?[2]
Show the answer and mark scheme
(a)Answer: 120 litres
P1 for \(\frac{3}{5} - \frac{1}{4} = \frac{7}{20}\)
P1 for \(42 \div \frac{7}{20}\) or \(42 \div 7 \times 20\)
A1 for 120
Worked solution: \(\frac{3}{5} - \frac{1}{4} = \frac{12}{20} - \frac{5}{20} = \frac{7}{20}\) of the tank is 42 litres. \(\frac{1}{20}\) is 6 litres, so the capacity is \(20 \times 6 = 120\) litres.
(b)Answer: 70 litres
M1 for \(\frac{5}{6} \times 120\) (= 100) or \(\frac{5}{6} - \frac{1}{4} = \frac{7}{12}\) (ft their capacity)
A1 for 70 (ft)
Worked solution: Now the tank holds \(\frac{1}{4} \times 120 = 30\) litres. \(\frac{5}{6} \times 120 = 100\) litres, so \(100 - 30 = 70\) litres are added.