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Problem-solving Pack
MathematicsYear 7 · 7.4 Decimals & Measurement
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
The weighing problem. You have a balance scale and four weights: 1 g, 3 g, 9 g, and 27 g. You can place weights on either side of the scale.
(a) Can you measure exactly 5 g? Show how.
(b) Can you measure exactly 11 g? Show how.
(c) What is the heaviest whole-number mass (in grams) you cannot measure using these four weights?
(d) How many different whole-number masses from 1 g to 40 g can you measure?
(a) Can you measure exactly 5 g? Show how.
(b) Can you measure exactly 11 g? Show how.
(c) What is the heaviest whole-number mass (in grams) you cannot measure using these four weights?
(d) How many different whole-number masses from 1 g to 40 g can you measure?
Working space
2Problem 2 of 12
Unit conversion chain. A snail moves at 0.048 km/h.
(a) Convert this speed to metres per minute.
(b) Convert to centimetres per second.
(c) How far (in mm) does the snail travel in 5 seconds?
(a) Convert this speed to metres per minute.
(b) Convert to centimetres per second.
(c) How far (in mm) does the snail travel in 5 seconds?
Working space
3Problem 3 of 12
Perimeter and algebra. Three rectangles are placed end to end along their lengths to form one large rectangle. Each small rectangle has the same width cm. Their lengths are 3.2 cm, 4.7 cm, and 5.1 cm.
The total perimeter of the large rectangle is 35.4 cm. Find .
The total perimeter of the large rectangle is 35.4 cm. Find .
Working space
4Problem 4 of 12
Repeating decimal mystery. Without a calculator, work out the decimal expansions of the following fractions by doing the long division. Then explain the pattern.
(a)
(b)
(c)
(d) Without calculating, write down the decimal expansions of , , and . Explain why they follow a cycle.
(a)
(b)
(c)
(d) Without calculating, write down the decimal expansions of , , and . Explain why they follow a cycle.
Working space
5Problem 5 of 12
Best buy with decimals. Three brands of olive oil:
| Brand | Size | Price |
|---|---|---|
| Alpha | 500 ml | £2.40 |
| Beta | 750 ml | £3.45 |
| Gamma | 1.2 litres | £5.16 |
Which brand offers the best value for money? Show your method clearly.
| Brand | Size | Price |
|---|---|---|
| Alpha | 500 ml | £2.40 |
| Beta | 750 ml | £3.45 |
| Gamma | 1.2 litres | £5.16 |
Which brand offers the best value for money? Show your method clearly.
Working space
6Problem 6 of 12
Missing digits. Each box represents a single decimal digit (0–9):
Find the missing digits. Show your working column by column.
Find the missing digits. Show your working column by column.
Working space
7Problem 7 of 12
Directed decimals and temperature. A freezer is at °C. The room is at °C.
(a) What is the difference in temperature between the room and the freezer?
(b) After a power cut, the freezer warms up at 2.4°C per hour. How long until the inside reaches 0°C? Give your answer in hours and minutes.
(c) How long until it reaches the room temperature?
(a) What is the difference in temperature between the room and the freezer?
(b) After a power cut, the freezer warms up at 2.4°C per hour. How long until the inside reaches 0°C? Give your answer in hours and minutes.
(c) How long until it reaches the room temperature?
Working space
8Problem 8 of 12
L-shape perimeter. A shape is made by removing a rectangular notch from the top-right corner of a rectangle.
Outer rectangle: 8.4 cm wide, 6.2 cm tall.
Notch removed: 3.1 cm wide, 2.5 cm tall (from the top-right corner).
Find the perimeter of the L-shaped figure.
Outer rectangle: 8.4 cm wide, 6.2 cm tall.
Notch removed: 3.1 cm wide, 2.5 cm tall (from the top-right corner).
Find the perimeter of the L-shaped figure.
Working space
9Problem 9 of 12
Painted cube investigation. A cube measuring is painted blue on all six faces, then cut into small cubes.
(a) How many small cubes are there in total?
(b) How many small cubes have exactly 3 painted faces?
(c) How many small cubes have exactly 2 painted faces?
(d) How many small cubes have exactly 1 painted face?
(e) How many small cubes have 0 painted faces?
(f) Verify that your answers to (a)–(e) add up to the total from (a).
(a) How many small cubes are there in total?
(b) How many small cubes have exactly 3 painted faces?
(c) How many small cubes have exactly 2 painted faces?
(d) How many small cubes have exactly 1 painted face?
(e) How many small cubes have 0 painted faces?
(f) Verify that your answers to (a)–(e) add up to the total from (a).
Working space
10Problem 10 of 12
Rounding and bounds. A plank of wood is measured as 2.4 m to the nearest 0.1 m.
(a) Write the lower and upper bounds of the true length.
(b) Planks are cut into shelves each exactly 0.8 m long. Using the upper bound, how many complete shelves can be cut?
(c) Using the lower bound, how many complete shelves can be cut?
(d) What does this tell you about the reliability of the answer to (b)?
(a) Write the lower and upper bounds of the true length.
(b) Planks are cut into shelves each exactly 0.8 m long. Using the upper bound, how many complete shelves can be cut?
(c) Using the lower bound, how many complete shelves can be cut?
(d) What does this tell you about the reliability of the answer to (b)?
Working space
11Problem 11 of 12
Tiling a hall. A school hall floor measures 24.5 m by 18.6 m. It is being tiled with square tiles each 0.5 m × 0.5 m, costing £3.80 each. A 10% wastage allowance must be added to the number of tiles ordered.
Calculate the total cost of the tiles. Give your answer to the nearest pound.
Calculate the total cost of the tiles. Give your answer to the nearest pound.
Working space
12Problem 12 of 12
Calendar arithmetic. 1 January 2024 is a Monday.
(a) What day of the week is 1 February 2024? (January has 31 days.)
(b) What day of the week is 1 March 2024? (2024 is a leap year, so February has 29 days.)
(c) What day of the week is 1 July 2024?
(d) Explain why the day of the week advances by 1 for each non-leap year, but by 2 for each leap year.
(a) What day of the week is 1 February 2024? (January has 31 days.)
(b) What day of the week is 1 March 2024? (2024 is a leap year, so February has 29 days.)
(c) What day of the week is 1 July 2024?
(d) Explain why the day of the week advances by 1 for each non-leap year, but by 2 for each leap year.
Working space