Mathematics (Syllabus D) 4024/22 — May/June 2018

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

questions

100

marks

150

minutes

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

Q1NumberFree sample

(a)

Each week Leah works 5 days and is paid a total of $682.
Each day she works from 0845 until 1215 and then from 1315 until 1730.

Calculate Leah’s hourly rate of pay.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Calculate the total time Leah works each day by finding the duration of each shift. Convert this time into decimal hours, multiply by the number of working days per week to find the total weekly hours, and divide the total weekly earnings by the total hours to find the hourly rate of pay.

Working

Calculate the hours worked in each shift:

Morning shift: from $08:45$ to $12:15$

12:15 - 08:45 = 3\text{ hours } 30\text{ minutes}

Afternoon shift: from $13:15$ to $17:30$

17:30 - 13:15 = 4\text{ hours } 15\text{ minutes}

Total time worked per day:

3\text{ h } 30\text{ min} + 4\text{ h } 15\text{ min} = 7\text{ hours } 45\text{ minutes} = 7.75\text{ hours}

Total hours worked in $5$ days:

5 \times 7.75 = 38.75\text{ hours}

Hourly rate of pay:

\frac{682}{38.75} = 17.6

Answer

17.60

Final answer

17.60

Detailed explanation

Walkthrough

First, find the time worked in the morning: from $08:45$ to $12:15$ is $3$ hours and $30$ minutes.
Next, find the time worked in the afternoon: from $13:15$ to $17:30$ is $4$ hours and $15$ minutes.
Add both shifts together to find the daily working hours: $3\text{ h } 30\text{ min} + 4\text{ h } 15\text{ min} = 7\text{ hours } 45\text{ minutes}$ .
Convert the time to hours in decimal form: $45\text{ minutes} = \frac{45}{60}\text{ hours} = 0.75\text{ hours}$ , so Leah works $7.75$ hours each day.
Multiply by $5$ to find total hours per week: $5 \times 7.75 = 38.75\text{ hours}$ .
Divide her total pay of $682 by the total hours worked: $\frac{682}{38.75} = 17.60$ , i.e. $17.60 per hour.

Key Takeaways

Time must be converted to decimal hours (dividing minutes by $60$ ) before performing division with money.
Rates are calculated using $\text{Rate} = \frac{\text{Total Pay}}{\text{Total Hours}}$ .

Common Mistakes

Incorrectly converting $7\text{ hours } 45\text{ minutes}$ to $7.45$ instead of $7.75$ hours.
Forgetting to multiply the daily hours by $5$ before dividing into the weekly total.

Things to Be Careful About

Ensure the final money answer is formatted correctly, preferably to 2 decimal places ($17.60).

Techniques used

calculate time durationconvert minutes to hours as a decimalcalculate hourly rate of pay

(b)

Carlos buys a new bicycle.
After one year he sells it for $231.
He makes a loss of 16% on the price he paid.

Calculate the price Carlos paid for the bicycle.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

A loss of $16\%$ means the selling price of $231 corresponds to $100\% - 16\% = 84\%$ of the original cost price. Let $x$ be the original price paid, set up a linear equation, and solve for $x$ .

Working

(100 - 16)\% \text{ of } x = 231

0.84x = 231

x = \frac{231}{0.84} = 275

Answer

275

Final answer

275

Detailed explanation

Walkthrough

The bicycle was sold at a $16\%$ loss, meaning the selling price represents $100\% - 16\% = 84\%$ of the original price.
Let the original price be $x$ . We write the relationship as $0.84x = 231$ .
To find $x$ , divide $231$ by $0.84$ :

x = \frac{231}{0.84} = 275

Thus, Carlos paid $275 for the bicycle.

Key Takeaways

A percentage loss reduces the original amount to $(100 - r)\%$ .
To find the original amount before a loss, divide the reduced price by the multiplier $1 - \frac{r}{100}$ .

Common Mistakes

Finding $16\%$ of $231 and adding it to $231, which gives $231 + 36.96 = 267.96$ (this incorrectly uses the selling price as the base instead of the original price).

Things to Be Careful About

Always check if the question asks for the original value (reverse percentage) or the result of a percentage decrease.

Techniques used

form a reverse percentage equationcalculate original value from percentage loss

(c)

The exchange rate between dollars ($) and euros (€) is $1 = €0.44.
Henry changes $850 to euros for his holiday.
He spends €260 when he is on holiday.
He changes the rest of the money back to dollars at the same exchange rate.

Calculate how much money in dollars he receives.
Give your answer correct to the nearest dollar.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Convert the initial dollar amount to euros using the exchange rate $1 = €0.44. Subtract the euros spent to find the remaining euros, then convert back to dollars by dividing by $0.44$ . Finally, round the result to the nearest dollar.

Working

Convert $850 to euros:

850 \times 0.44 = 374\text{ euros}

Calculate the remaining euros after spending €260:

374 - 260 = 114\text{ euros}

Convert €114 back to dollars:

\frac{114}{0.44} = 259.0909\dots

Rounding to the nearest dollar gives $259$ .

Answer

259

Final answer

259

Detailed explanation

Walkthrough

Multiply the $850 Henry takes with him by $0.44$ to find how many euros he receives:

850 \times 0.44 = €374

Subtract the €260 he spends to find how many euros are left over:

374 - 260 = €114

Convert €114 back to dollars by dividing by the exchange rate $0.44$ :

\frac{114}{0.44} \approx \$259.09

Rounding to the nearest dollar gives $259.

Key Takeaways

To convert from the base currency ( $\\$ ) to another currency (€), multiply by the rate.
To convert back, divide by the rate.
Always read the required rounding accuracy carefully.

Common Mistakes

Multiplying by $0.44$ when converting back from euros to dollars instead of dividing.
Forgetting to round the final answer to the nearest integer as requested.

Things to Be Careful About

Ensure exact values are maintained until the final step to avoid compounding rounding errors.

Techniques used

convert currency using exchange ratescalculate remaining balanceround to the nearest whole unit

(d)

Anya has $3000 to invest in a savings account for 3 years.
She can choose from these two accounts.

Account A
Year 1: 1.1% interest
Year 2: 1.2% interest added to end of Year 1 total
Year 3: 1.4% interest added to end of Year 2 total

Account B
Fixed rate of compound interest 1.3% per year

She chooses the account that will give her more money at the end of the 3 years.

Decide which account she chooses and find the amount she will have in her account at the end of 3 years.

Account ______
$ ______

DifficultyMedium

Worked solution

Approach

Calculate the total balance at the end of 3 years for both accounts. For Account A, apply the different annual multipliers sequentially. For Account B, apply the fixed compound interest formula. Compare the final amounts to determine which account yields more money.

Working

Account A:

\begin{aligned} \text{Amount} &= 3000 \times (1 + 0.011) \times (1 + 0.012) \times (1 + 0.014) \\ &= 3000 \times 1.011 \times 1.012 \times 1.014 \\ &= 3000 \times 1.0374567 \\ &= 3112.370\dots \\ &\approx \$3112.37 \end{aligned}

Account B:

\begin{aligned} \text{Amount} &= 3000 \times (1 + 0.013)^3 \\ &= 3000 \times 1.013^3 \\ &= 3000 \times 1.0395119 \\ &= 3118.535\dots \\ &\approx \$3118.54 \text{ (or } 3118.53\text{)} \end{aligned}

Comparing the two final amounts:

3118.54 > 3112.37

Hence, Anya chooses Account B.

Answer

Account B

3118.53

Final answer

Account B, 3118.53

Detailed explanation

Walkthrough

For Account A, interest changes each year and compounds on the previous year's total:
- Year 1 multiplier: $1 + \frac{1.1}{100} = 1.011$
- Year 2 multiplier: $1 + \frac{1.2}{100} = 1.012$
- Year 3 multiplier: $1 + \frac{1.4}{100} = 1.014$
- Total in Account A $= 3000 \times 1.011 \times 1.012 \times 1.014 = 3112.37$ , i.e. $3112.37.
For Account B, the interest rate is a fixed $1.3\%$ per year compounded:
- Multiplier per year: $1 + \frac{1.3}{100} = 1.013$
- Total in Account B $= 3000 \times 1.013^3 = 3000 \times 1.0395119 = 3118.53$ , i.e. $3118.53 (or $3118.54).
Compare the totals: Account B gives $3118.53, which is greater than Account A's $3112.37. Therefore, Anya chooses Account B.

Key Takeaways

When interest rates vary year by year, multiply by each individual annual multiplier: $\text{Principal} \times (1 + r_1)(1 + r_2)(1 + r_3)$ .
For a fixed annual compound rate, use $\text{Principal} \times (1 + r)^n$ .

Common Mistakes

Adding the percentages together ( $1.1 + 1.2 + 1.4 = 3.7\%$ ) and computing simple interest instead of compounding sequentially.
Choosing Account A incorrectly without showing full comparative working.

Things to Be Careful About

Give the final money answer to two decimal places representing dollars and cents.

Techniques used

calculate compounding with varying interest ratesuse standard compound interest formulacompare financial outcomes

The rest of this paper

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