Ratio, proportion and percentage change – Year 10 GCSE maths worksheet answers to download as PDF

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1. Answer to Question 1

   1. Divide by 12: \(2:3\) 12 is the HCF of 24 and 36, so dividing both parts by 12 gives the simplest form \(2:3\).
   2. Multiply by 2 then divide by 3: \(3:9=\) \(1:3\) First multiply by 2 to remove the decimals, giving \(3:9\), then divide by 3. The simplest form is \(1:3\).
   3. Multiply by 4: \(2:3\), so \(2:3\) Multiplying both parts by 4, the common denominator, removes the fractions. The simplest form is \(2:3\).
   4. \(1.5\text{ kg}=1500\text{ g}\), so \(250:1500=\) \(1:6\) Both parts must be in the same units before simplifying, so change kilograms into grams, then divide by 250. The simplest form is \(1:6\).

2. Answer to Question 2

   1. \(3+5=8\) parts, \(240\div8=30\). Shares: \(3\times30=\) £90 and \(5\times30=\) £150. Add the parts to find the total number of parts, then find the value of one part. Check: \(90+150=240\).
   2. \(2+5+1=8\) parts, \(56\div8=7\). Shares: 14, 35 and 7. Check: \(14+35+7=56\).
   3. 4 parts, \(120\text{p}\div4=30\)p. Shares: 30p and 90p. Change £1.20 into 120p to make the division easier. Check: \(30+90=120\).
   4. The larger share is \(\dfrac58\) of the total. The larger share is 5 of the 8 equal parts.

3. Answer to Question 3

   1. 3 parts \(=21\), so 1 part \(=7\). Blue \(=7\times7=\) 49. Find the value of one part first, then multiply by 7. There are 49 blue counters.
   2. \(21+49=\) 70 (or \(10\times7\)). The total is \(3+7=10\) parts. There are 70 counters altogether.
   3. 1 part \(=45\div5=9\), girls \(=6\times9=\) 54. Five parts represent the 45 boys, so one part is 9. There are 54 girls.
   4. \(4+3=7\) parts, \(56\div7=8\), so pears \(=3\times8=\) 24. Add the parts to find the total number of parts, then find one part. Check: \(32+24=56\).

4. Answer to Question 4

   1. 1 apple \(=39\)p, so 9 apples \(=\) £3.51 Divide £2.34 by 6 to find the cost of one apple, then multiply by 9.
   2. 1 pencil \(=25\)p, so 20 pencils \(=\) £5.00 Divide 300p by 12 to find the cost of one pencil, then multiply by 20.
   3. 1 person \(=75\) g, so 5 people \(=\) 375 g Divide 600 g by 8 for one person, then multiply by 5.
   4. 1 litre covers 8 m², so 22 litres cover 176 m² Divide 120 m² by 15 to find what 1 litre covers, then multiply by 22.

5. Answer to Question 5

   1. \(180\div5=\) 36p per 100 g 500 g is 5 lots of 100 g, so we divide 180p by 5.
   2. \(255\div7.5=\) 34p per 100 g 750 g is 7.5 lots of 100 g, so we divide 255p by 7.5.
   3. Cereal B is better value (34p against 36p per 100 g). The lower price per 100 g gives more cereal for the same money.
   4. Juice A: \(1.35\div1.5=£0.90\) per litre. Juice B: \(1.90\div2=£0.95\) per litre. Juice A is better value. Divide each price by the number of litres. The lower price per litre is the better buy.

6. Answer to Question 6

   1. \(6.4\times25\,000=160\,000\) cm \(=1600\) m \(=\) 1.6 km Every centimetre on the map is 25 000 cm in real life. Then divide by 100 for metres and by 1000 for kilometres.
   2. 3 km \(=300\,000\) cm. \(300\,000\div25\,000=\) 12 cm Change kilometres into centimetres first, then divide by 25 000.
   3. \(25\,000\) cm \(=\) 250 m 25 000 cm divided by 100 is 250 m.
   4. 1 cm² on the map is \(250\times250=62\,500\) m². \(4\times62\,500=250\,000\) m² \(=\) 0.25 km². For areas, the scale is squared.

7. Answer to Question 7

   1. 5 parts \(=15\) litres, so 1 part \(=3\) litres. Blue \(=9\) litres, yellow \(=6\) litres.
   2. Blue \(9\times8=72\). Yellow \(6\times6.50=39\). Total \(=\) £111.
   3. \(111\div15=\) £7.40 per litre. Divide the total cost by the number of litres.
   4. Blue is 3 parts so 1 part \(=1.5\) litres and the total is \(5\times1.5=\) 7.5 litres. Blue is the limiting colour, so the most green paint he can make is 7.5 litres.

8. Answer to Question 8

   1. 9 parts: \(1260\div9=140\). Amir £280, Beth £420, Chen £560. Add the parts: \(2+3+4=9\). Divide the prize by 9 to find one part. Check: \(280+420+560=1260\).
   2. \(\dfrac49\) Chen has 4 of the 9 equal parts, so his fraction is \(\tfrac49\).
   3. \(420-390=\) £30 Subtract the price of the bike from Beth's share. Beth has £30 left.
   4. Yes. Chen has 4 parts and Amir has 2 parts, and \(560=2\times280\). So Amir is correct.

9. Answer to Question 9

   1. 2 parts \(=400\) g, so 1 part \(=200\) g. Flour \(=\) 1000 g, sugar \(=\) 600 g.
   2. \(10\) parts \(=\) 2000 g.
   3. \(\dfrac{3}{10}=\) 30% Sugar is 3 of the \(5+3+2=10\) parts.
   4. 1.5 kg \(=1500\) g, 1 part \(=150\) g: flour 750 g, sugar 450 g, butter 300 g.

10. Answer to Question 10

    1. \(4.5\times20\,000=90\,000\) cm \(=\) 900 m. Every centimetre on the map is 20 000 cm in real life, and there are 100 cm in a metre.
    2. \(0.9\text{ km}\div5=0.18\) h \(=0.18\times60=\) 10.8 minutes. Change the distance into kilometres, then use time \(=\) distance \(\div\) speed. Multiply by 60 to change hours into minutes.
    3. \(90\,000\div9=10\,000\), so the scale is \(1:10\,000\). The same real distance of 90 000 cm is shown by 9 cm, so 1 cm represents 10 000 cm.
    4. \(1\text{ cm}^2\) represents \(200\times200=40\,000\) m². \(0.5\text{ km}^2=500\,000\) m². \(500\,000\div40\,000=\) 12.5 cm². The area scale is the square of the length scale.

11. Answer to Question 11

    1. \(250\times1.16=\) €290
    2. Left \(=290-180=€110\). \(110\times0.84=\) £92.40
    3. She spent €180, which is worth \(180\div1.16=£155.17\) at the first rate. So she has £92.40 and spent about £155.17, a total of \(247.57\). Her loss from the exchange is £2.43 (to the nearest penny).
    4. \(\dfrac{2.43}{250}=\) 1.0% (1 d.p.)

12. Answer to Question 12

    1. \(40\times1.15=\) £46 A 15% increase gives the multiplier 1.15.
    2. \(46\times0.85=\) £39.10 A 15% decrease gives the multiplier 0.85, applied to the new price.
    3. £39.10 is less than £40. The 15% reduction is of the larger price £46. So the final price is 90p lower than the original price.
    4. \(1.15\times0.85=0.9775\), so a 2.3% decrease (1 d.p.). The overall multiplier is less than 1, and \(1-0.9775=0.0225\), so the price has fallen by 2.3%.

13. Answer to Question 13

    1. \(800\times0.035=\) £28 3.5% of £800 is \(0.035\times800\). Simple interest is the same every year because it is always worked out on the original £800.
    2. \(4\times28=\) £112, total \(=800+112=\) £912 Multiply the yearly interest by 4, then add it to the original amount. Ravi has £912 after 4 years.
    3. \(100I=PRT\), so \(T=\dfrac{100I}{PR}\) Multiply both sides by 100, then divide by \(PR\).
    4. \(T=\dfrac{100\times210}{800\times3.5}=\dfrac{21\,000}{2800}=\) 7.5 years Substitute \(I=210\), \(P=800\) and \(R=3.5\). Check: \(28\times7.5=210\). It takes 7.5 years.

14. Answer to Question 14

    1. 7 parts \(=1.4\), so 1 part \(=0.2\) m³: cement 0.2 m³, sand 0.4 m³, gravel 0.8 m³.
    2. Cement \(2\times9=18\), sand \(4\times4=16\), gravel \(8\times5=40\). Total \(=\) £74.
    3. \(\dfrac47=\) 57.1%
    4. Sand is 2 parts \(=0.5\), so 1 part \(=0.25\) and the total is \(7\times0.25=\) 1.75 m³.

15. Answer to Question 15

    1. \(\dfrac{3k+6}{5k+6}=\dfrac57\), so \(7(3k+6)=5(5k+6)\). In 6 years both ages increase by 6, so Maya will be \(3k+6\) and her father \(5k+6\). Cross-multiply to remove the fractions.
    2. \(21k+42=25k+30\), so \(12=4k\) and \(k=\) 3. Expand both sides, then subtract \(21k\) and 30 from both sides.
    3. Maya 9, her father 15. Substitute \(k=3\): Maya is \(3\times3=9\) and her father is \(5\times3=15\).
    4. In 6 years: \(15:21=5:7\) ✓. In 6 years Maya will be 15 and her father 21, and \(15:21\) simplifies to \(5:7\) by dividing by 3.

16. Answer to Question 16

    1. \(A:B=2:3=8:12\) and \(B:C=4:5=12:15\). 12 is the LCM of 3 and 4, so we multiply the first ratio by 4 and the second ratio by 3.
    2. \(A:B:C=\) \(8:12:15\). Now B is 12 in both ratios, so the ratios can be joined.
    3. 35 parts \(=175\), so 1 part \(=5\): A £40, B £60, C £75. Add the parts: \(8+12+15=35\). Check: \(40+60+75=175\).
    4. \(\dfrac{15}{35}=\dfrac37\) C collected 15 of the 35 parts, which simplifies to \(\tfrac37\).

17. Answer to Question 17

    1. \(1.1\times1.2=1.32\), so 32% increase. Multiply the two multipliers. The result is not 30%, because the 20% is worked out on the larger amount.
    2. \(1200\times1.04\times1.02=\) £1272.96. Multiplier \(1.0608\), so 6.08% increase. Use the multipliers 1.04 and 1.02 one after the other.
    3. \(\dfrac1{1.25}=0.8\), so a 20% decrease. To reverse a multiplier we divide by it, so the new multiplier is \(1\div1.25=0.8\).
    4. \(0.92^3=0.7787\) and \(0.92^4=0.7164\). After 4 years it is below 75%. Keep multiplying by 0.92 until the result first drops below 0.75. It takes 4 years.

18. Answer to Question 18

    1. A: \(0.2\times8=1.6\) kg. B: \(0.45\times12=5.4\) kg. Multiply each mass by its percentage of copper.
    2. Copper \(=1.6+5.4=7\) kg. Total \(=8+12=20\) kg. Add the copper from both alloys and the total masses.
    3. \(\dfrac7{20}=\) 35% Divide the mass of copper by the total mass and multiply by 100.
    4. Let \(b\) kg be added: \(1.6+0.45b=0.4(8+b)=3.2+0.4b\), so \(0.05b=1.6\) and \(b=\) 32 kg. Check: \(1.6+0.45\times32=16\) kg of copper in \(40\) kg, which is 40%.

19. Answer to Question 19

    1. \(3\times8+2\times13=24+26=\) £50 Multiply each mass by its price per kilogram, then add. The mixture costs £50.
    2. \(50\div5=\) £10 per kg Divide the total cost by the total mass of \(3+2=5\) kg. The blend costs £10 per kg.
    3. \(10\times1.3=\) £13 per kg A 30% profit gives the multiplier 1.3. The blend should be sold at £13 per kg.
    4. \(0.45\times13=5.85\), so £5.85. Change 450 g into 0.45 kg before multiplying. She pays £5.85.

20. Answer to Question 20

    1. Juice \(=2k\) and water \(=3k\). The ratio \(2:3\) means there are 2 equal parts of juice and 3 equal parts of water, each of size \(k\) litres.
    2. \(\dfrac{2k}{3k+10}=\dfrac12\), so \(4k=3k+10\) and \(k=10\). Adding water changes only the water, so the juice stays \(2k\). Cross-multiply, then subtract \(3k\) from both sides.
    3. Juice \(=2\times10=\) 20 litres (water \(30\) litres). Substitute \(k=10\): there were 20 litres of juice.
    4. \(20+30+10=\) 60 litres, in the ratio \(20:40=1:2\) ✓. The final drink has 20 litres of juice and 40 litres of water.