(a) A shop buys some fruit. The table shows the bill for this fruit. Item Quantity (kg) Price per kg (\$) Cost price (\$) --- --- --- --- Bananas 50 0.51 25.50 Oranges 72 1.35 $p$ Avocados $r$ 1.95 $q$ Pears 45 $s$ 51.30 Total cost price 240.30 (i) Find the value of each of $p$, $q$, $r$ and $s$. $p$ = \_\_\_\_\_\_ $q$ = \_\_\_\_\_\_ $r$ = \_\_\_\_\_\_ $s$ = \_\_\_\_\_\_ (ii) The shop sells all this fruit for a total of \$325. Calculate the percentage profit. \_\_\_\_\_\_ %…

(a)(i) p = 97.20, q = 66.30, r = 34, s = 1.14 · (a)(ii) 35.2% · (b)(i)(a) 98° · (b)(i)(b) $4468.90 · (b)(ii) $33500

Mathematics (Syllabus D) 4024/22 — May/June 2023

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

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Topics Number · Algebra and Graphs · Statistics · Geometry · Trigonometry · Probability · +2 more

Q1NumberStatisticsFree sample

(a)

A shop buys some fruit.
The table shows the bill for this fruit.

Item	Quantity (kg)	Price per kg ($)	Cost price ($)
Bananas	50	0.51	25.50
Oranges	72	1.35	$p$
Avocados	$r$	1.95	$q$
Pears	45	$s$	51.30
		Total cost price	240.30

(i)

Find the value of each of $p$ , $q$ , $r$ and $s$ .

$p$ = ______
$q$ = ______
$r$ = ______
$s$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Use the relationship $\text{Cost price} = \text{Quantity} \times \text{Price per kg}$ and the total cost price to find each unknown value.

Working

To find $p$ (cost price of oranges):

p = 72 \times 1.35 = 97.20

To find $s$ (price per kg of pears):

s = \frac{51.30}{45} = 1.14

To find $q$ (cost price of avocados), subtract all other item costs from the total cost price of $240.30:

q = 240.30 - (25.50 + 97.20 + 51.30) = 240.30 - 174.00 = 66.30

To find $r$ (quantity of avocados in kg):

r = \frac{q}{1.95} = \frac{66.30}{1.95} = 34

Answer

p = 97.20, \quad q = 66.30, \quad r = 34, \quad s = 1.14

Final answer

p = 97.20, q = 66.30, r = 34, s = 1.14

Detailed explanation

Walkthrough

For oranges, both the quantity ( $72\,\text{kg}$ ) and the price per kilogram ($1.35) are given. Multiply them to get the total cost price: $72 \times 1.35 = 97.20$ , so $p = 97.20$ .
For pears, the total cost ($51.30) and quantity ( $45\,\text{kg}$ ) are given. Divide the total cost by the quantity to find the price per kilogram: $51.30 \div 45 = 1.14$ , so $s = 1.14$ .
The total cost price for all four types of fruit is $240.30. Subtract the known cost prices of bananas ($25.50), oranges ($97.20), and pears ($51.30) from $240.30 to determine the cost price of avocados: $240.30 - 174.00 = 66.30$ , so $q = 66.30$ .
Finally, divide the cost price of avocados by their price per kilogram ($1.95) to find the quantity: $66.30 \div 1.95 = 34$ , so $r = 34$ .

Key Takeaways

$\text{Total Cost} = \text{Quantity} \times \text{Unit Price}$ .
Rearranging gives $\text{Unit Price} = \frac{\text{Total Cost}}{\text{Quantity}}$ and $\text{Quantity} = \frac{\text{Total Cost}}{\text{Unit Price}}$ .
The sum of individual costs must equal the grand total.

Common Mistakes

Calculating $r$ before finding $q$ .
Arithmetic errors when summing the individual costs to find $q$ .

Things to Be Careful About

Write currency values clearly (e.g., $97.2$ or $97.20$ ).
Ensure follow-through consistency if a calculation error was made in an earlier variable.

Techniques used

multiply unit price by quantitysubtract known costs from total to find unknown costdivide cost by unit price to find quantitydivide cost by quantity to find unit price

(ii)

The shop sells all this fruit for a total of $325.

Calculate the percentage profit.

______ %

DifficultyMedium-Easy

Worked solution

Approach

Calculate the profit by subtracting the total cost price from the selling price, then divide by the total cost price and multiply by $100$ to obtain the percentage profit.

Working

Total cost price = $240.30
Total selling price = $325

\text{Profit} = 325 - 240.30 = 84.70

\text{Percentage profit} = \frac{84.70}{240.30} \times 100 = 35.2476\ldots\%

Rounding to $3$ significant figures gives $35.2\%$ .

Answer

35.2\%

Final answer

35.2%

Detailed explanation

Walkthrough

Determine the profit in dollars by subtracting the cost price ($240.30) from the total selling price ($325): $325 - 240.30 = 84.70$ .
Compute the percentage profit by dividing the profit by the original cost price ($240.30) and multiplying by $100$ : $\frac{84.70}{240.30} \times 100 \approx 35.248\%$ .
Standard Cambridge accuracy requires non-exact answers to be rounded to $3$ significant figures, giving $35.2\%$ (or values in the range $35.24\%$ to $35.25\%$ if kept to more decimal places).

Key Takeaways

Percentage profit is always calculated on the cost price, not the selling price: $\text{Percentage Profit} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100\%$ .

Common Mistakes

Dividing the profit by the selling price ($325) instead of the cost price ($240.30).
Forgetting to multiply by $100$ to convert the fraction into a percentage.

Things to Be Careful About

Give the answer to at least 3 significant figures unless stated otherwise.

Techniques used

calculate profitcalculate percentage profit relative to cost price

(b)

In 2022, the shop's total sales were $34 974.

(i)

A pie chart is drawn to show the item types that make up these total sales.

(a)

The sales for fruit were $9520.70.

Calculate the angle representing fruit on the pie chart.

______

DifficultyMedium-Easy

Worked solution

Approach

Find the proportion of total sales that comes from fruit, then multiply by $360^\circ$ to find the corresponding sector angle.

Working

\text{Sector angle} = \frac{9520.70}{34974} \times 360^\circ

\text{Sector angle} = 0.272222\ldots \times 360^\circ = 98^\circ

Answer

98^\circ

Final answer

98°

Detailed explanation

Walkthrough

A complete pie chart represents the total sales of $34,974 across $360^\circ$ .
The sales for fruit are $9520.70. The fraction of total sales represented by fruit is $\frac{9520.70}{34974}$ .
Multiply this fraction by $360^\circ$ to find the angle of the sector: $\frac{9520.70}{34974} \times 360^\circ = 98^\circ$ .

Key Takeaways

To find a sector angle in a pie chart: $\text{Angle} = \frac{\text{Category Value}}{\text{Total Value}} \times 360^\circ$ .

Common Mistakes

Calculating the percentage ( $27.2\%$ ) instead of multiplying by $360^\circ$ to find the sector angle.
Dividing $360$ by the fruit sales instead of the total sales.

Things to Be Careful About

Ensure the total sales figure used is $34,974 from the stem.

Techniques used

calculate pie chart sector angle from proportion

(b)

The angle representing frozen food is $46^\circ$ .

Calculate the sales for frozen food.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Divide the given sector angle by $360^\circ$ to find the fraction of the total pie chart it represents, then multiply by the total sales.

Working

\text{Sales for frozen food} = \frac{46^\circ}{360^\circ} \times 34974

\text{Sales for frozen food} = \frac{46 \times 34974}{360} = 4468.90

Answer

\$4468.90

Final answer

$4468.90

Detailed explanation

Walkthrough

The sector angle for frozen food is $46^\circ$ out of a full circle of $360^\circ$ .
The proportion of total sales is therefore $\frac{46}{360}$ .
Multiply this proportion by the total sales of $34,974 to find the dollar value: $\frac{46}{360} \times 34974 = 4468.90$ .

Key Takeaways

To find the value represented by a pie chart sector: $\text{Value} = \frac{\text{Sector Angle}}{360^\circ} \times \text{Total Value}$ .

Common Mistakes

Multiplying by $100$ instead of dividing by $360$ .
Prematurely rounding the intermediate fraction $\frac{46}{360}$ , leading to an inaccurate final dollar amount.

Things to Be Careful About

In financial contexts, answers should generally be given to 2 decimal places (cents) or exact if specified.

Techniques used

calculate category value from pie chart sector angle

(ii)

The shop's total sales of $34974 in 2022 were a $4.4\%$ increase on the total sales in 2021.

Calculate the total sales in 2021.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

A $4.4\%$ increase means the 2022 sales represent $100\% + 4.4\% = 104.4\%$ of the 2021 sales. Divide the 2022 sales by $1.044$ to find the 2021 sales.

Working

Let $x$ be the total sales in 2021.

1.044x = 34974

x = \frac{34974}{1.044} = 33500

Answer

\$33\,500

Final answer

$33500

Detailed explanation

Walkthrough

The sales in 2022 ($34,974) represent the sales after a $4.4\%$ increase on the 2021 total.
In percentage terms, the 2022 figure corresponds to $100\% + 4.4\% = 104.4\%$ of the 2021 figure.
Expressing $104.4\%$ as a decimal gives $1.044$ .
To find the original amount, divide the final amount by the multiplier: $\frac{34974}{1.044} = 33\,500$ .

Key Takeaways

To reverse a percentage increase of $r\%$ , divide the new quantity by $\left(1 + \frac{r}{100}\right)$ .
Never simply subtract $r\%$ of the new value from the new value.

Common Mistakes

Incorrectly calculating $4.4\%$ of $34,974 and subtracting it from $34,974 ( $34974 - 0.044 \times 34974 = 33435.14$ ), which incorrectly uses the new value as the base.

Things to Be Careful About

Ensure the decimal multiplier is correct: a $4.4\%$ increase is $1.044$ , not $1.44$ .

Techniques used

use a percentage multiplier to solve a reverse percentage problem

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