Speed, density and compound units – Year 8 maths worksheet answers to download as PDF

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1. Answer to Question 1

   1. Speed = distance ÷ time: \(150\div2.5=\)60 km/h.
   2. Distance = speed × time: \(18\times3=\)54 km.
   3. Time = distance ÷ speed: \(90\div45=\)2 hours.
   4. Metres and seconds give m/s: \(400\div50=\)8 m/s.

   In each case the units of the answer come from the units used: km and hours give km/h, metres and seconds give m/s.

2. Answer to Question 2

   There are 60 minutes in an hour, so divide minutes by 60 to get a decimal of an hour and multiply the decimal part of an hour by 60 to get minutes.

   1. \(45\div60=0.75\), so 2.75 hours
   2. \(0.4\times60=24\), so 3 h 24 min
   3. 1 h 40 min \(=1\frac23\) h. \(85\div\frac53=\) 51 mph
   4. \(27\div18=1.5\) h \(=\) 1 h 30 min

3. Answer to Question 3

   1. 1 km = 1 000 m and 1 hour = 3 600 s: \(72\times1000\div3600=\)20 m/s.
   2. Multiply by 3.6: \(15\times3.6=\)54 km/h.
   3. Divide by 3.6: \(90\div3.6=\)25 m/s.
   4. \(8\times3.6=\)28.8 km/h.

   Dividing by 3.6 is the same as multiplying by 1 000 and dividing by 3 600; m/s values are always smaller numbers than the matching km/h values.

4. Answer to Question 4

   1. Total \(150\) km in \(2.5\) h: 60 km/h
   2. Total \(32\) km in \(1.5\) h: \(32\div1.5=\) 21.3 km/h
   3. Distances: \(18+6=24\) km in 3 h: 8 km/h
   4. The runner spends more time at 9 km/h, so the average speed is closer to 9 than to 6; the simple mean 7.5 would ignore the times.

5. Answer to Question 5

   1. Density = mass ÷ volume: \(540\div200=\)2.7 g/cm³ (the density of aluminium).
   2. Mass = density × volume: \(8\times35=\)280 g.
   3. Volume = mass ÷ density: \(750\div1.5=\)500 cm³.
   4. Volume of the cube \(5^{3}=125\) cm³, so density \(1000\div125=\)8 g/cm³.

6. Answer to Question 6

   1. Rate = amount ÷ time: \(120\div8=\)15 litres per minute.
   2. People ÷ area: \(36000\div40=\)900 people per km².
   3. Miles ÷ litres: \(60\div5=\)12 miles per litre.
   4. Each mile uses \(\dfrac{1}{12}\) litre, so costs \(140\div12=\)11.7p per mile.

   The word “per” tells you which quantity to divide by.

7. Answer to Question 7

   1. 11:25 minus 09:10 \(=\) 2 h 15 min
   2. \(2+\dfrac{15}{60}=\) 2.25 hours
   3. \(200\div2.25=\) 88.9 mph
   4. \(1\text{ h }55\text{ min}=1.9167\) h. \(200\div1.9167=104.3\) mph. It is about 15.4 mph faster.

   Writing 2 h 15 min as 2.25 hours, not 2.15, is the key step: 15 minutes is a quarter of an hour.

8. Answer to Question 8

   1. 27.5 min \(=27.5\div60=\) 0.4583 h
   2. Ella: \(5\div0.4583=\) 10.9 km/h. Raj: \(5\div0.4=\) 12.5 km/h.
   3. Ella's speed \(=10.909\) km/h \(=3.03\) m/s. In 24 min \(=1440\) s she runs \(4364\) m, so she is 636 m behind.
   4. \(5\div\frac{25}{60}=\) 12 km/h

9. Answer to Question 9

   1. \(4.8\div16=0.3\) h \(=\) 18 min
   2. \(4.8\div12=0.4\) h \(=\) 24 min
   3. Total distance \(9.6\) km in \(0.7\) h: \(9.6\div0.7=\) 13.7 km/h
   4. No. The round trip takes longer at the slower speed, so the average is 13.7 km/h, not 14.

   The slower journey takes longer, so it counts for more in the average; this is why the round-trip average is below 14 km/h.

10. Answer to Question 10

    1. \(1260\div630=\) 2 hours
    2. 06:50 + 2 h \(=\) 08:50
    3. \(1260\div540=2.33\) h \(=\) 2 h 20 min
    4. Total \(2520\) km in \(4.33\) h: 582 km/h (to the nearest whole number).

    The average speed for the round trip uses the total distance and the total time, not the average of 630 and 540.

11. Answer to Question 11

    1. Aluminium: \(405\div150=\) 2.7. Iron: \(630\div80=\) 7.875. Gold: \(965\div50=\) 19.3.
    2. Aluminium and iron are less dense than mercury, so they float; gold (19.3) is more dense, so it sinks.
    3. \(19.3\times1000=19300\) g \(=\) 19.3 kg
    4. \(19.3\div2.7=7.15\), so yes, about 7 times.

12. Answer to Question 12

    1. Volume = length × width × depth: \(25\times10\times1.8=450\) m³, which is 450 000 litres.
    2. \(450000\div5000=90\) minutes, which is 1 h 30 min.
    3. \(450\times1.80=\)£810.
    4. A day has \(24\times60\times60=86400\) s: \(4\times86400=345600\) litres, which is 345.6 m³.

13. Answer to Question 13

    1. A: \(240\div5=\) 48 mpg. B: \(240\div6.4=\) 37.5 mpg.
    2. A: \(5\times6.30=\) £31.50. B: \(6.4\times6.30=\) £40.32.
    3. Car A is cheaper by \(40.32-31.50=\) £8.82.
    4. \(12\times37.5=\) 450 miles

    Miles per gallon is distance ÷ fuel used; the higher the mpg, the cheaper the car is to run.

14. Answer to Question 14

    1. 4 km in 1 h: 4 km/h
    2. Flat part from 1 h to 3 h: 2 hours
    3. 4 km in 2 h (from 3 h to 5 h): 2 km/h
    4. Total distance \(8\) km in \(5\) h: 1.6 km/h

15. Answer to Question 15

    1. \(120\div60=2\) h, so 10:00
    2. Stop for 30 min: 10:30
    3. \(150\div75=2\) h, so 12:30
    4. Total distance \(270\) km in \(4.5\) h: 60 km/h
    5. Moving time \(=4\) h: \(270\div4=\) 67.5 km/h

    Including the 30-minute stop lowers the average speed, because the coach covers no distance during that time.

16. Answer to Question 16

    1. Moving towards each other, their speeds add: \(90+50=\)140 km/h.
    2. The gap of 280 km closes at 140 km/h: \(280\div140=\)2 hours.
    3. The first train travels \(90\times2=\)180 km from A.
    4. Two hours after 10:00: 12:00.
    5. \(50\times2=\)100 km, and \(180+100=280\) km, the full distance.

17. Answer to Question 17

    1. Mass = density × volume. Copper \(9\times60=\)540 g, zinc \(7\times40=\)280 g.
    2. Total mass \(540+280=\)820 g; total volume \(60+40=\)100 cm³.
    3. Density = total mass ÷ total volume: \(820\div100=\)8.2 g/cm³, between the two metals as expected.
    4. The mould has volume \(5\times4\times5=100\) cm³, so the mass is \(8.2\times100=\)820 g.

18. Answer to Question 18

    1. Multiply by 3.6: \(2.5\times3.6=\)9 km/h.
    2. Going with the current, the speeds add: \(6+9=\)15 km/h.
    3. Time = distance ÷ speed: \(15\div15=\)1 hour.
    4. Against the current the speeds subtract: \(6-9=-3\) km/h. The boat is pushed backwards, so it cannot travel upstream.

19. Answer to Question 19

    1. Density = population ÷ area. Leeds \(800000\div552\approx\)1 449, Bristol \(470000\div110\approx\)4 273, London \(8900000\div1570\approx\)5 669 people per km².
    2. The highest density is London, although Bristol is close behind.
    3. \(4273\div1449\approx\)2.9 times as crowded.
    4. \(470000\div150\approx\)3 133 people per km²; a larger area with the same population lowers the density.

20. Answer to Question 20

    1. \(\pi\times1.2^2\times0.3=\) 1.357 cm³
    2. \(38.6\div1.357=\) 28.4 g/cm³
    3. 28.4 is greater than 19.3, so it is not pure gold; it must contain a metal denser than gold (or the measurements are wrong).
    4. \(19.3\times1.357=\) 26.2 g
    5. The measured mass is greater than the pure gold mass of 26.2 g, so it must contain something denser than gold; the density test shows that the coin is not simply gold.