7.4 Decimals & Measurement

**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?

**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?

**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 $w$ 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 $w$.

**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) $\dfrac{1}{7}$ (b) $\dfrac{2}{7}$ (c) $\dfrac{3}{7}$ (d) Without calculating, write down the decimal expansions of $\dfrac{4}{7}$, $\dfrac{5}{7}$, and $\dfrac{6}{7}$. Explain why they follow a cycle.

**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.

**Missing digits.** Each box represents a single decimal digit (0–9): $$3.\square\,7 + \square.4\,\square = 9.03$$ Find the missing digits. Show your working column by column.

**Directed decimals and temperature.** A freezer is at $-18.5$°C. The room is at $23.7$°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?

**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.

**Painted cube investigation.** A cube measuring $4 \times 4 \times 4$ is painted blue on all six faces, then cut into $1 \times 1 \times 1$ 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).

**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)?

**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.

**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.