GCSE Mathematics
› M1.1 Operations with integers, decimals, fractions
Operations with integers, decimals, fractions
Free GCSE Mathematics practice questions on Operations with integers, decimals, fractions.
Aligned with the UK Department for Education GCSE subject content — works for any UK GCSE exam board.
Sample questions below with detailed mark schemes. Sign up to practise the full set with spaced repetition.
Show — 4 marks
A student is investigating the energy efficiency of different household appliances. They measure the power consumption of three devices: a kettle uses 2.4 kW, a microwave uses 0.8 kW, and a toaster uses 1.6 kW. The student needs to calculate total energy usage and compare efficiencies using fractional and decimal operations.
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(a)
[1 mark]
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(b)
[2 marks]
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(c)
[1 mark]
Show mark scheme
- (a) Correct addition: 2.4 + 0.8 + 1.6 = 4.8 kW (or clear working shown)
- (b) Mark 1: Correct calculation for kettle: 2.4 × 1/3 = 0.8 kWh
- (b) Mark 2: Correct calculation for microwave: 0.8 × 1/4 = 0.2 kWh, and addition 0.8 + 0.2 = 1.0 kWh (or equivalent working shown)
- (c) Correct simplified fraction: 1.6 ÷ 4.8 = 1/3 (or equivalent such as 16/48 simplified to 1/3)
Describe — 2 marks
A student is conducting an experiment to measure the density of different materials. They measure the mass of a copper block as 178.5 g and its volume as 20 cm³. To find the density, they need to divide the mass by the volume.
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Describe the steps you would follow to calculate the density of the copper block using the values 178.5 g and 20 cm³.
[2 marks]
Show mark scheme
- {'mark': 1, 'description': 'Divide 178.5 by 20 (or equivalent statement showing correct operation)'}
- {'mark': 1, 'description': 'Obtain the answer of 8.925 g/cm³ (or 8.9 g/cm³ to 1 d.p.) with correct units'}
Explain — 3 marks
A student is investigating the efficiency of different renewable energy systems. A solar panel array produces 2.5 kW of power on a sunny day. However, due to system losses, only 3/4 of this power is actually delivered to the home. On a cloudy day, the output is reduced by 0.4 of the sunny day's delivered power.
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(a) Calculate the power actually delivered to the home on a sunny day. Show your working.
[1 mark]
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(b) Calculate the power delivered on a cloudy day. Show your working.
[1 mark]
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(c) Explain why the difference between sunny and cloudy day outputs (as a fraction of the sunny day delivered power) is important when designing a battery storage system for the home.
[1 mark]
Show mark scheme
- (a) Award 1 mark for correctly calculating 2.5 × 3/4 = 1.875 kW (or 1.88 kW to 2 d.p.)
- (b) Award 1 mark for correctly calculating 1.875 - (0.4 × 1.875) = 1.125 kW (or 1.13 kW to 2 d.p.)
- (c) Award 1 mark for explaining that the significant variation/difference in power output between weather conditions means the battery must be sized large enough to store sufficient energy to meet demand during low-output periods (cloudy days)
Describe — 4 marks
A physics student is investigating the density of different materials. They measure the mass of a copper block as 178.5 g and calculate its volume using water displacement. The volume reading changes from 15.2 cm³ to 35.7 cm³ when the block is submerged.
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(a) Calculate the volume of the copper block in cm³. Show your working.
[1 mark]
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(b) The student needs to convert the mass to kilograms. Describe the process of converting 178.5 g to kilograms as a decimal.
[1 mark]
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(c) Using your answers from parts (a) and (b), describe the steps you would use to calculate the density of copper in kg/cm³. You do not need to give a final numerical answer.
[2 marks]
Show mark scheme
- (a) Subtract 15.2 from 35.7 to get 20.5 cm³ (or equivalent correct subtraction of decimals)
- (b) Divide 178.5 by 1000 (or move the decimal point three places to the left) to convert grams to kilograms, giving 0.1785 kg
- (c) Divide the mass in kilograms (0.1785) by the volume in cm³ (20.5)
- (c) Use the formula density = mass ÷ volume (or equivalent statement showing the correct operation with the values from parts a and b)
Calculate — 2 marks
A bakery uses standard recipes for its cakes. One recipe requires \frac{3}{4} kg of flour and 1.25 kg of sugar. The baker needs to prepare multiple batches for upcoming orders.
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(a) Calculate the total amount of flour needed for 2 batches.
[1 mark]
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(b) Calculate the total amount of sugar needed for 3 batches.
[1 mark]
Show mark scheme
- (a) 1.5 kg or equivalent (e.g., \frac{3}{2} kg, 1\frac{1}{2} kg)
- (b) 3.75 kg or equivalent (e.g., \frac{15}{4} kg, 3\frac{3}{4} kg)
GCSE Perfect