(a) Sarah bought some soup, apples and mushrooms from her local shop. The table shows some of the amounts and prices. Items Price (\$) --- --- $p$ cans of soup at 90 cents per can 6.30 1.5 kilograms of apples at \$ $q$ per kilogram 4.35 $r$ kilograms of mushrooms at \$6.40 per kilogram 1.60 (i) Find the values of $p$, $q$ and $r$. $p$ = \_\_\_\_\_\_ $q$ = \_\_\_\_\_\_ $r$ = \_\_\_\_\_\_ (ii) Sarah gives the shopkeeper \$20.00 to pay for all these items. How much change does…

(a)(i) p = 7, q = 2.90, r = 0.25 · (a)(ii) $7.75 · (b) $80 · (c) 3.5%

Mathematics (Syllabus D) 4024/22 — May/June 2010

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Q1NumberFree sample

(a) Sarah bought some soup, apples and mushrooms from her local shop.
The table shows some of the amounts and prices.

Items	Price ($)
$p$ cans of soup at 90 cents per can	6.30
1.5 kilograms of apples at $ $q$ per kilogram	4.35
$r$ kilograms of mushrooms at $6.40 per kilogram	1.60

(a)

(i)

Find the values of $p$ , $q$ and $r$ .

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

DifficultyMedium-Easy

Worked solution

Approach

Use the relationship $\text{Total Price} = \text{Quantity} \times \text{Unit Price}$ for each item, making sure units of money are consistent (convert cents to dollars where necessary).

Working

For soup:
Each can costs 90 cents = $0.90.
The total cost is $6.30.

p = \frac{6.30}{0.90} = 7

For apples:
$1.5\text{ kg}$ of apples costs $4.35.

q = \frac{4.35}{1.5} = 2.90

For mushrooms:
The unit cost is $6.40 per kg, and the total cost is $1.60.

r = \frac{1.60}{6.40} = 0.25

Answer

p = 7, \quad q = 2.90, \quad r = 0.25

Final answer

p = 7, q = 2.90, r = 0.25

Detailed explanation

Walkthrough

Finding $p$ (number of cans of soup):
The price is given as $90\text{ cents}$ per can, but the total price in the table is given in dollars ($6.30). First convert $90\text{ cents}$ into dollars by dividing by $100$ , giving $0.90. To find the number of cans $p$ , divide the total cost by the cost per can: $\frac{6.30}{0.90} = 7$ .
Finding $q$ (price per kg of apples):
The total price for $1.5\text{ kg}$ is $4.35. Divide the total price by the weight to find the price per kg: $\frac{4.35}{1.5} = 2.90$ .
Finding $r$ (mass of mushrooms in kg):
The price is $6.40 per kg, and the total spent is $1.60. Divide the total price by the rate per kg: $\frac{1.60}{6.40} = 0.25$ (or $\frac{1}{4}$ ).

Key Takeaways

Always ensure currency units match (e.g., converting cents to dollars before dividing).
Unit price multiplied by quantity gives the total cost: $\text{Quantity} = \frac{\text{Total}}{\text{Unit Price}}$ and $\text{Unit Price} = \frac{\text{Total}}{\text{Quantity}}$ .

Common Mistakes

Forgetting to convert $90\text{ cents}$ to dollars, leading to $p = \frac{6.30}{90} = 0.07$ .
Inverting the division for $r$ , e.g., $\frac{6.40}{1.60} = 4\text{ kg}$ instead of $\frac{1.60}{6.40} = 0.25\text{ kg}$ .

Things to Be Careful About

Ensure each value is placed next to its corresponding variable label ( $p$ , $q$ , or $r$ ).

Techniques used

convert units of currencydivide total cost by unit costdivide total cost by quantity

(ii)

Sarah gives the shopkeeper $20.00 to pay for all these items.

How much change does she receive?

$ ______

DifficultyEasy

Worked solution

Approach

Calculate the total cost of all three items from the table, then subtract this total from $20.00 to find the change.

Working

Total cost of items:

\text{Total cost} = 6.30 + 4.35 + 1.60 = 12.25

Change received from $20.00:

\text{Change} = 20.00 - 12.25 = 7.75

Answer

7.75

Final answer

$7.75

Detailed explanation

Walkthrough

Add up the total price column for all three purchases:
- Soup: $6.30
- Apples: $4.35
- Mushrooms: $1.60
$\text{Total} = 6.30 + 4.35 + 1.60 = 12.25$
Subtract this total from the amount handed to the shopkeeper ($20.00): $20.00 - 12.25 = 7.75$ Sarah receives $7.75 in change.

Key Takeaways

$\text{Change} = \text{Amount paid} - \text{Total cost}$ .

Common Mistakes

Arithmetic errors in adding decimals or subtracting from $20.00$ .

Things to Be Careful About

Make sure all three items are included in the sum.

Techniques used

sum total costscalculate change by subtraction

(b)

Fig. 1

Lavin decides to buy this washing machine.

How much more would it cost Lavin if he paid for the washing machine using the finance offer instead of paying the $980 immediately?

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Calculate the total cost under the finance offer by adding the $20\%$ deposit to the sum of the $24$ monthly payments, then subtract the cash price ($980) from this total.

Working

Calculate the deposit ( $20\%$ of $980):

\text{Deposit} = 0.20 \times 980 = 196

Calculate the total of the monthly payments:

\text{Instalments} = 24 \times 36 = 864

Calculate the total cost of the finance offer:

\text{Total finance cost} = 196 + 864 = 1060

Calculate the extra cost compared to paying immediately:

\text{Extra cost} = 1060 - 980 = 80

Answer

80

Final answer

$80

Detailed explanation

Walkthrough

Find the deposit: The finance offer requires a $20\%$ deposit on the $980 price. Multiply $980$ by $0.20$ to find the deposit amount: $0.20 \times 980 = 196$
Find the total instalments: There are $24$ monthly payments of $36 each. Multiply the number of months by the monthly payment: $24 \times 36 = 864$
Find the total finance cost: Add the deposit to the total instalments: $196 + 864 = 1060$
Find how much more it costs: Subtract the immediate price ($980) from the total finance cost: $1060 - 980 = 80$

Key Takeaways

Hire purchase / finance deals typically cost more than paying immediately; total cost = deposit + (number of instalments $\times$ instalment amount).
Read carefully to determine whether the question asks for the total finance cost or the difference between the two options.

Common Mistakes

Forgetting to include the deposit and only calculating $24 \times 36 = 864$ .
Providing the total finance cost ($1060) rather than the difference ($80).

Things to Be Careful About

Ensure the difference is taken with respect to $980.

Techniques used

calculate a percentage of a quantitycalculate total cost from instalment paymentsfind the difference between two payment methods

(c)

Asif deposits $650 into a bank paying simple interest.
He leaves the money there for 5 years.
At the end of the 5 years, the amount in the bank is $763.75.

Calculate the percentage rate of interest the bank paid per year.

______ %

DifficultyMedium-Easy

Worked solution

Approach

Find the total interest earned over the $5$ years by subtracting the principal from the final amount, then use the simple interest formula $I = \frac{PRT}{100}$ to solve for the annual rate $R$ .

Working

Find the interest earned ( $I$ ):

I = 763.75 - 650 = 113.75

Substitute $I = 113.75$ , $P = 650$ , and $T = 5$ into $I = \frac{PRT}{100}$ :

113.75 = \frac{650 \times R \times 5}{100}

113.75 = 32.5 \times R

Solve for $R$ :

R = \frac{113.75}{32.5} = 3.5

Answer

3.5

Final answer

3.5%

Detailed explanation

Walkthrough

Identify the values given:
- Principal ( $P$ ) = $650
- Time ( $T$ ) = $5\text{ years}$
- Total amount after 5 years ( $A$ ) = $763.75
Find the simple interest earned ( $I$ ):
The interest earned is the increase in the account balance: $I = A - P = 763.75 - 650 = 113.75$
Use the simple interest formula:
The simple interest formula is: $I = \frac{PRT}{100}$ where $R$ is the annual percentage rate. Rearranging for $R$ gives: $R = \frac{100 \times I}{P \times T}$
Calculate $R$ : $R = \frac{100 \times 113.75}{650 \times 5} = \frac{11375}{3250} = 3.5$ So the annual interest rate is $3.5\%$ .

Key Takeaways

Total amount $A = P + I$ , where $P$ is the principal and $I$ is the interest.
The simple interest formula is $I = \frac{PRT}{100}$ .

Common Mistakes

Using the total amount $763.75 in place of interest $I$ in the formula, leading to an incorrect rate of $23.5\%$ .
Forgetting to divide by time $T = 5$ , giving the total percentage increase over 5 years ( $17.5\%$ ) rather than the annual rate ( $3.5\%$ ).

Things to Be Careful About

Distinguish between simple interest and compound interest (the problem explicitly specifies simple interest).

Techniques used

calculate total simple interest earnedrearrange simple interest formula to find the rate

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