Mathematics (Syllabus D) 4024/22 — October/November 2019

Walkthrough

A worker is away from work for $7\frac{3}{4}$ hours each day, which is $7.75$ hours. Over the 5 working days, one employee works $7.75 \times 5 = 38.75$ hours. At an hourly rate of $15.20, one employee earns $589.00 a week. The four employees together therefore earn $4 \times 589.00 = 2356.00$ dollars, so the total is $2356.00.

Key Takeaways

This is a repeated multiplication problem based on a hourly rate. Convert the mixed number to a decimal before multiplying, then keep each step simple so all factors are clearly used.

Common Mistakes

The most common error is finding only one employee’s weekly pay and forgetting to multiply by 4. Another common error is converting $7\frac{3}{4}$ incorrectly as 7.3 or “7.45” instead of 7.75.

Things to Be Careful About

Use consistent units: $7.75$ hours per day, 5 days per week, $15.20 per hour and 4 employees. Show the multiplications in steps so the method is clear. The calculator is allowed on this component, but a clear product keeps all method marks available.

Walkthrough

The original profit was $25,700 in 2017 and the later profit was $22,102. The decrease is the difference between them. To write the decrease as a percentage, divide the decrease by the original amount and multiply by 100.

The original profit is the denominator, not the 2018 profit.

Key Takeaways

A percentage decrease always compares the amount lost with the original amount, not with the later amount. The result can also be seen by calculating what fraction of the original profit was kept and then subtracting that from 100.

Common Mistakes

• Using $22,102 as the denominator gives approximately $16.3\%$, the percentage fall relative to the new profit, which is wrong.
• Writing 86 is not the percentage decrease: 86 is the percentage of the original profit that was retained. It is given as a special-case mark in the scheme only when no working is awarded.

Things to Be Careful About

Preserve the order of subtraction: $25\,700 - 22\,102$, not the reverse. Divide by the original 2017 amount and do not start from the later 2018 amount. The answer should be shown as a percentage.

Walkthrough

The quoted $465.75 is not the original price; it is the price after two percentage additions. The first 8% addition can be written as multiplying by $1.08$; the second 15% is applied to the new amount, not to the original, so it multiplies by $1.15$. The combined multiplier is therefore $1.08 \times 1.15 = 1.242$, not $1.23$. The initial amount is found by dividing the final amount by the combined multiplier.

Key Takeaways

Two successive percentage increases are done by multiplying the two multipliers together. To undo them, divide the final amount by the product of the multipliers. This is a reverse-percentage problem rather than a direct percentage problem, because we are working backwards to the original amount.

Common Mistakes

The most common mistake in this question is adding the two percentages as $8\% + 15\% = 23\%$ and dividing by $1.23$. That gives an answer such as $378.65 or $379, which the mark scheme treats as a special case score, not full marks, because the 15% is being applied to the amount including sales tax.

Answer

Make sure the multiplier for 8% is $1.08$, not $0.08$, and the multiplier for the second increase is $1.15$, not $0.15$. The combined multiplier has to be exactly $1.242$; the initial cost should be quoted as a money amount in dollars.

Walkthrough

The account earns compound interest, so the multiplier $(1 + \frac{3.1}{100})$ is applied once each year. With an annual rate of $3.1\%$, the yearly multiplier is $1.031$. At the end of 5 years the account contains:

Evaluating this gives $9901.7567\ldots$, so after taking out $9300 the amount left is:

Rounded to the nearest 10 the amount is $601.76.

Key Takeaways

Compound interest repeats a multiplier for each time period, so the exponent is the number of years. The withdrawal is taken after all interest has been added, not from the original $8500. The answer requires the compound interest approach rather than simple interest.

Common Mistakes

• Using $r = 0.31$ instead of $r = 0.031$.
• Applying the withdrawal before calculating the 5 years interest, which gives the wrong amount left.
• Using simple interest instead of compound interest, so ignoring the power of 5.
• Rounding too early and losing the exact final cent. The mark scheme accepts both 601.75 and 601.76 for different rounding choices, but it is best to evaluate the balance fully before subtracting.

Things to Be Careful About

Use the multiplier $1 + \frac{3.1}{100} = 1.031$. The full compound formula $R \left(1+\frac{r}{100}\right)^n$ should be shown. The 5 in the exponent and the subtraction of $9300 are both essential; losing either drops marks. The final amount is a money amount, so give it to two decimal places with a dollar sign.