Mathematics (Syllabus D) 4024/22 — October/November 2021

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

questions

100

marks

150

minutes

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

Q1NumberFree sample

(a)

Jasmine buys a family holiday to India.
Here is some information about the cost.

Flights	$700
Hotel	$1550
Total cost	$2250

(i)

In October, Jasmine pays a deposit of 12% of the total cost.
She pays the rest of the total cost in December.

Calculate the amount she pays in December.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

The deposit is 12% of the total cost, so the December payment is the remaining part of the total. Calculate 12% of $2250 and subtract it from the total.

Working

The deposit is:

\frac{12}{100} \times 2250 = 270

So the amount paid in December is:

2250 - 270 = 1980

Answer

The amount Jasmine pays in December is $1980.

Final answer

1980 dollars

Detailed explanation

Walkthrough

We are told the total holiday cost is $2250 and Jasmine pays 12% as a deposit. The December payment is the rest, so first find 12% of the total. 12% means $\frac{12}{100}$ , so multiply 2250 by $\frac{12}{100}$ to get 270. Then subtract this deposit from the total to find the remaining amount: $2250 - 270 = 1980$ . This is the amount paid in December.

Key Takeaways

To find a percentage of an amount, write the percentage as a fraction or decimal and multiply.
"The rest" after a percentage payment means subtract the percentage amount from the whole.

Common Mistakes

Giving 270 as the final answer: this is only the deposit, not the December payment.
Forgetting to show the subtraction expression $2250 - \frac{12}{100} \times 2250$ , which is the method mark.
The mark scheme gives SC1 for 3960 after 0 scored, which is an incorrect doubling of the deposit.

Things to Be Careful About

The question asks for the amount paid in December, not the deposit.
Show the percentage calculation and the subtraction so both method marks are visible.
On this calculator paper, decimal evaluation is allowed, but the expression should still be written down.

Techniques used

calculate a percentage of an amountsubtract the deposit from the total cost

(ii)

Find the ratio cost of flights : cost of hotel.
Give your answer in its simplest form.

______ : ______

DifficultyEasy

Worked solution

Approach

Write the two costs as a ratio in the order flights : hotel, then divide both parts by their highest common factor.

Working

The ratio is:

700 : 1550

Divide both parts by 50:

700 \div 50 : 1550 \div 50 = 14 : 31

Answer

14 : 31

Final answer

14 : 31

Detailed explanation

Walkthrough

The ratio must be written in the order flights : hotel, so start with $700 : 1550$ . Both numbers are divisible by 50, so divide both parts by 50. This gives $700 \div 50 = 14$ and $1550 \div 50 = 31$ . There is no further common factor, so the simplest form is $14 : 31$ .

Key Takeaways

The order of a ratio matters: flights : hotel is not the same as hotel : flights.
To simplify a ratio, divide both parts by their highest common factor.

Common Mistakes

Reversing the ratio to $31 : 14$ . The mark scheme gives SC1 for this only after 0 scored, so it is not full credit.
Dividing by 10 only, leaving $70 : 155$ , which is not in simplest form.

Things to Be Careful About

The final answer must be in the order requested: flights : hotel.
Check that the two numbers have no common factor other than 1 after simplifying.

Techniques used

write a ratio in the given orderdivide both parts by the highest common factor

(b)

Jasmine changes $350 into rupees.
The exchange rate is $1 = 71.6 rupees.

On holiday, she spends 19 500 rupees.
She changes the rest back to dollars at the same exchange rate.

Calculate the amount of money she receives.
Give your answer correct to the nearest cent.

$ ______

DifficultyMedium

Worked solution

Approach

Convert the $350 into rupees, subtract the amount spent, then convert the remaining rupees back into dollars at the same exchange rate.

Working

Dollars to rupees:

350 \times 71.6 = 25060

Rupees left after spending:

25060 - 19500 = 5560

Rupees back to dollars:

\frac{5560}{71.6} = 77.6536\ldots

Rounded to the nearest cent:

77.65

Answer

Jasmine receives $77.65.

Final answer

77.65 dollars

Detailed explanation

Walkthrough

The exchange rate is $1 = 71.6 rupees, so to change dollars into rupees we multiply by 71.6. Jasmine starts with $350, so she gets $350 \times 71.6 = 25060$ rupees. She spends 19500 rupees, leaving $25060 - 19500 = 5560$ rupees. To change rupees back into dollars, divide by 71.6, because 71.6 rupees are worth 1 dollar. $5560 \div 71.6 = 77.6536\ldots$ , which is 77.65 dollars to the nearest cent.

Key Takeaways

To convert from dollars to rupees, multiply by the exchange rate.
To convert from rupees to dollars, divide by the exchange rate.
"Nearest cent" means rounding to two decimal places.

Common Mistakes

Forgetting to subtract the 19500 rupees before converting back.
Dividing instead of multiplying when converting dollars to rupees, or multiplying instead of dividing when converting back.
Rounding 77.6536 to 77.7 instead of 77.65.
The mark scheme allows M2 for the combined expression $350 - \frac{19500}{71.6}$ , so an unsupported final answer may not get full credit.

Things to Be Careful About

The final answer must be correct to the nearest cent, so keep enough decimal places until the final rounding.
The mark scheme uses follow-through: if a candidate calculates $350 \times 71.6$ correctly, they can still earn the second method mark for dividing their remaining rupees by 71.6.

Techniques used

convert dollars to rupees using an exchange ratesubtract the amount spentconvert rupees back to dollarsround to the nearest cent

(c)

The table shows the number of tourists and the total tourist spending for some countries in 2016.

Country	Number of tourists	Total spending in dollars
China	$5.93 \times 10^7$	$4.44 \times 10^{10}$
India	$1.46 \times 10^7$	$2.31 \times 10^{10}$
Kenya	$1.27 \times 10^6$	$1.62 \times 10^9$
Madagascar	$2.93 \times 10^5$	$9.13 \times 10^5$

(i)

Calculate how many more tourists visited India than Kenya in 2016.
Give your answer in standard form.

______

DifficultyEasy

Worked solution

Approach

Rewrite both numbers with the same power of 10, subtract the coefficients, then convert the result back into standard form.

Working

India: $1.46 \times 10^7 = 14.6 \times 10^6$ .

Kenya: $1.27 \times 10^6$ .

Subtract:

14.6 \times 10^6 - 1.27 \times 10^6 = 13.33 \times 10^6

Convert to standard form:

13.33 \times 10^6 = 1.333 \times 10^7

Answer

1.333 \times 10^7

Final answer

1.333 × 10^7

Detailed explanation

Walkthrough

To subtract numbers in standard form, they must have the same power of 10. Write India's $1.46 \times 10^7$ as $14.6 \times 10^6$ . Now both are in millions ( $10^6$ ). Subtract the coefficients: $14.6 - 1.27 = 13.33$ , so the difference is $13.33 \times 10^6$ . Standard form requires one non-zero digit before the decimal point, so rewrite $13.33 \times 10^6$ as $1.333 \times 10^7$ .

Key Takeaways

Standard form is $a \times 10^n$ where $1 \le a < 10$ .
To add or subtract in standard form, align the powers of 10 first.

Common Mistakes

Subtracting $1.46 - 1.27 = 0.19$ and leaving the power as $10^7$ .
Giving $13.33 \times 10^6$ as the final answer, which is not standard form.
Forgetting that $1.46 \times 10^7 = 14.6 \times 10^6$ .

Things to Be Careful About

The final answer must be in standard form, so the coefficient must be between 1 and 10.
This is a 1-mark question; the final answer must be exactly $1.333 \times 10^7$ .

Techniques used

rewrite numbers in standard form with the same power of tensubtract the coefficientsconvert the result back to standard form

(ii)

Calculate the average amount spent per tourist in China in 2016.
Give your answer correct to the nearest dollar.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

The average amount spent per tourist is the total spending divided by the number of tourists. Substitute the China values and divide the standard-form numbers.

Working

\frac{4.44 \times 10^{10}}{5.93 \times 10^7} = \frac{4.44}{5.93} \times 10^{10-7}

= 0.748735\ldots \times 10^3 = 748.735\ldots

Rounded to the nearest dollar:

749

Answer

The average amount spent per tourist in China is $749.

Final answer

749 dollars

Detailed explanation

Walkthrough

The average is found by dividing the total spending by the number of tourists. For China, total spending is $4.44 \times 10^{10}$ and the number of tourists is $5.93 \times 10^7$ . Dividing gives $\frac{4.44}{5.93} \times 10^{10-7} = 0.748735\ldots \times 10^3 = 748.735\ldots$ dollars. To the nearest dollar, this is 749.

Key Takeaways

Average = total ÷ number.
When dividing numbers in standard form, divide the coefficients and subtract the powers of 10.

Common Mistakes

Dividing the wrong way round, e.g. number of tourists ÷ total spending.
Rounding 748.735 to 748 instead of 749.
Giving $7.49 \times 10^2$ without recognising it is 749.

Things to Be Careful About

The answer must be correct to the nearest dollar.
The mark scheme gives M1 for the division expression $\frac{4.44 \times 10^{10}}{5.93 \times 10^7}$ ; show this expression before evaluating.

Techniques used

divide total spending by the number of touristsdivide numbers in standard formround to the nearest dollar

(iii)

From 2014 to 2016, the total amount spent by tourists in Madagascar increased by 23.5%.

Calculate the amount spent by tourists in Madagascar in 2014.

$ ______

DifficultyMedium

Worked solution

Approach

The 2016 amount is 23.5% more than the 2014 amount, so 2016 = 2014 × 1.235. Rearrange to find the 2014 amount.

Working

Let the 2014 amount be $x$ .

x \times 1.235 = 9.13 \times 10^5

x = \frac{9.13 \times 10^5}{1.235} = 739271.25\ldots

Rounded to 3 significant figures:

739000 \quad \text{or} \quad 7.39 \times 10^5

Answer

The amount spent by tourists in Madagascar in 2014 was $739000.

Final answer

739000 dollars

Detailed explanation

Walkthrough

An increase of 23.5% means the 2016 amount is 123.5% of the 2014 amount. As a multiplier, $123.5\% = 1.235$ . If the 2014 amount is $x$ , then $x \times 1.235 = 9.13 \times 10^5$ . To undo the increase, divide by 1.235: $x = \frac{9.13 \times 10^5}{1.235} = 739271.25\ldots$ . Since the given value is in standard form to 3 significant figures, round to 739000, which is $7.39 \times 10^5$ .

Key Takeaways

A percentage increase of 23.5% corresponds to multiplying by 1.235.
To reverse a percentage change, divide by the multiplier.

Common Mistakes

Multiplying $9.13 \times 10^5$ by 1.235 instead of dividing.
Using 0.235 as the multiplier instead of 1.235.
Giving 739271.25 as the final answer when the mark scheme expects 739000 or $7.39 \times 10^5$ .

Things to Be Careful About

The mark scheme gives M1 for the equation $\frac{100 + 23.5}{100}x = 9.13 \times 10^5$ seen or implied; show the equation.
The final answer is rounded to 3 significant figures because the given value is in standard form to 3 significant figures.
The answer can be written as 739000 or $7.39 \times 10^5$ .

Techniques used

form an equation for a percentage increasedivide by the multiplier to reverse the percentageround to 3 significant figures

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