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4024/21

Mathematics (Syllabus D) 4024/21 — May/June 2022

Cambridge O-Level · Calculator · worked solutions for every part, with the mark scheme

11
questions
100
marks
150
minutes

Topics Algebra and Graphs · Number · Mensuration · Trigonometry · Geometry · Probability · +3 more

Q1NumberFree sample
(a)

In 2020, the running cost for Frederick's car was $5200.

28%28\% of the running cost was spent on insurance.
325\frac{3}{25} of the running cost was spent on maintenance.
$740 of the running cost was spent on tax.
The remainder of the running cost was spent on petrol.

(i)

Calculate the amount Frederick spent on petrol.

$ ______

3M
DifficultyMedium-Easy
Worked solution

Approach

The running cost is split into insurance, maintenance, tax and petrol. Calculate the insurance and maintenance amounts, add them to the tax, then subtract this total from $5200 to find the petrol spend.

Working

Insurance:

28100×5200=0.28×5200=1456\frac{28}{100} \times 5200 = 0.28 \times 5200 = 1456

Maintenance:

325×5200=3×208=624\frac{3}{25} \times 5200 = 3 \times 208 = 624

Total spent on insurance, maintenance and tax:

1456+624+740=28201456 + 624 + 740 = 2820

Therefore petrol:

5200−2820=23805200 - 2820 = 2380

Answer

Frederick spent $2380 on petrol.

Final answer

2380 dollars

Detailed explanation

Walkthrough

The total running cost is $5200. Insurance is 28% of this, so multiply $5200 by 28100\frac{28}{100} to get $1456. Maintenance is 325\frac{3}{25} of the total, so multiply $5200 by 325\frac{3}{25} to get $624. The tax is already given as $740. Adding these three amounts gives the total spent on everything except petrol: $1456 + $624 + $740 = $2820. Since the whole $5200 is made up of insurance, maintenance, tax and petrol, subtracting $2820 from $5200 leaves the petrol cost.

Key Takeaways

  • To find a percentage of an amount, write the percentage as a fraction (or decimal) and multiply.
  • To find a fraction of an amount, multiply by the numerator and divide by the denominator.
  • When a total is split into parts, the remaining part is found by subtracting the known parts from the total.

Common Mistakes

  • Using 28% as 28 instead of 0.28.
  • Taking 325\frac{3}{25} as 3.25 or 0.3 instead of 0.12.
  • Forgetting to include the $740 tax when finding the non-petrol total.
  • Subtracting only one of the costs from $5200.

Things to Be Careful About

Keep all amounts in dollars and do not round early. On this calculator paper, decimal multiplication is acceptable, but write the percentage and fraction expressions first so the method is clear. The final answer should be $2380, not just the sum of the other costs.

Techniques used
calculate a percentage of an amountcalculate a fraction of an amountsubtract component costs from a total
(ii)

In 2021, the tax increased by 1.5%1.5\%.

Calculate the tax in 2021.

$ ______

2M
DifficultyMedium-Easy
Worked solution

Approach

The tax increases by 1.5% of its 2020 value, so calculate the increase and add it to $740.

Working

Increase:

1.5100×740=11.1\frac{1.5}{100} \times 740 = 11.1

Tax in 2021:

740+11.1=751.1740 + 11.1 = 751.1

Answer

The tax in 2021 was $751.10.

Final answer

751.1 dollars

Detailed explanation

Walkthrough

The 2020 tax is $740. A 1.5% increase means we add 1.5% of $740 to the original amount. First find the increase: 1.5100×740=11.1\frac{1.5}{100} \times 740 = 11.1. Then add this to $740 to get $751.10. This is the new tax in 2021.

Key Takeaways

  • Percentage increase = original amount + percentage increase of the original.
  • 1.5% means 1.5 out of 100, or 0.015.

Common Mistakes

  • Adding 1.5 to 740 instead of 1.5% of 740.
  • Writing 1.5% as 15% or 0.15.
  • Forgetting to add the increase to the original tax.

Things to Be Careful About

The mark scheme allows $751.1 or $751.10. Show the increase calculation and the addition. Since this is a calculator paper, you may evaluate directly, but the expression 1.5100×740\frac{1.5}{100}\times 740 should be seen.

Techniques used
calculate a percentage increaseadd the increase to the original amount
(b)

In January, the cost of petrol is $2.20 per litre.

(i)

Find the cost of 38.7 litres of petrol.

$ ______

1M
DifficultyEasy
Worked solution

Approach

The total cost is the number of litres multiplied by the cost per litre.

Working

38.7×2.20=85.1438.7 \times 2.20 = 85.14

Answer

The cost is $85.14.

Final answer

85.14 dollars

Detailed explanation

Walkthrough

Petrol costs $2.20 per litre. To find the cost of 38.7 litres, multiply the number of litres by the cost per litre: 38.7×2.20=85.1438.7 \times 2.20 = 85.14. This gives $85.14.

Key Takeaways

  • Total cost = quantity multiplied by unit price.
  • Multiplying decimals can be done by treating 2.20 as 2.2.

Common Mistakes

  • Adding 38.7 and 2.20 instead of multiplying.
  • Misplacing the decimal point in 85.14.
  • Forgetting the dollar unit.

Things to Be Careful About

This is a one-mark question; the answer is $85.14. Ensure the decimal point is in the correct place.

Techniques used
multiply a quantity by a unit rate
(ii)

The average amount of petrol Frederick's car uses is 7 litres per 100 km100\,\text{km}.
In January, he spends $215.60 on petrol.

Calculate the number of kilometres he drives in January.

______ km\text{km}

3M
DifficultyMedium
Worked solution

Approach

First find how many litres were bought by dividing the total cost by the price per litre. Then use the consumption rate of 7 litres per 100 km to convert litres into kilometres.

Working

Litres bought:

215.602.20=98\frac{215.60}{2.20} = 98

Distance driven:

987×100=14×100=1400\frac{98}{7} \times 100 = 14 \times 100 = 1400

Answer

Frederick drives 1400 km in January.

Final answer

1400 km

Detailed explanation

Walkthrough

Frederick spends $215.60 on petrol at $2.20 per litre, so the number of litres bought is 215.602.20=98\frac{215.60}{2.20} = 98. The car uses 7 litres per 100 km, so 98 litres is 987=14\frac{98}{7} = 14 groups of 7 litres. Each group of 7 litres takes the car 100 km, so the distance is 14×100=140014 \times 100 = 1400 km. Equivalently, 987×100=1400\frac{98}{7}\times 100 = 1400.

Key Takeaways

  • To find a quantity from total cost, divide by the unit price.
  • A consumption rate of 7 litres per 100 km can be used as a proportion: litres7×100=km\frac{\text{litres}}{7} \times 100 = \text{km}.

Common Mistakes

  • Dividing 2.20 by 215.60 instead of 215.60 by 2.20.
  • Using 7100\frac{7}{100} instead of 1007\frac{100}{7} when converting litres to km.
  • Forgetting to multiply by 100 after dividing by 7.
  • Giving the answer in litres instead of km.

Things to Be Careful About

The mark scheme awards a method mark for 215.602.20\frac{215.60}{2.20} and another for their 987×100\frac{\text{their }98}{7}\times100. Show both steps. The final answer is 1400 km, with no decimal rounding needed.

Techniques used
divide total cost by unit price to find litresconvert litres to distance using a consumption ratescale proportionally to 100 km
(iii)

In February, the cost of petrol increases to $2.24 per litre.

Calculate the percentage increase in the cost of petrol from January to February.

______ %\%

2M
DifficultyMedium-Easy
Worked solution

Approach

The percentage increase is the increase divided by the original price, multiplied by 100.

Working

Increase:

2.24−2.20=0.042.24 - 2.20 = 0.04

Percentage increase:

0.042.20×100=1.818…\frac{0.04}{2.20} \times 100 = 1.818\ldots

Answer

The percentage increase is 1.82%.

Final answer

1.82%

Detailed explanation

Walkthrough

The price increased from $2.20 to $2.24. The increase is 2.24−2.20=0.042.24 - 2.20 = 0.04. Percentage increase is always the increase divided by the original value, multiplied by 100: 0.042.20×100=1.818…%\frac{0.04}{2.20}\times 100 = 1.818\ldots\%, which is 1.82% to 3 significant figures.

Key Takeaways

  • Percentage increase = new−originaloriginal×100%\frac{\text{new} - \text{original}}{\text{original}} \times 100\%.
  • The denominator is the original price, not the new price.

Common Mistakes

  • Dividing the increase by the new price ($2.24) instead of the original ($2.20).
  • Writing the increase as 0.04% instead of multiplying by 100.
  • Rounding 1.818... to 1.8 instead of 1.82.

Things to Be Careful About

The mark scheme accepts 1.82 or 1.818... . Give the percentage to a sensible accuracy, usually 3 significant figures. Show the subtraction and the division by the original price.

Techniques used
find the difference between two valuesexpress the difference as a percentage of the original

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