Worked Solutions
Financial Mathematics — Worked Solutions (Preliminary Maths Standard)
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Worked examples for Preliminary Maths Standard financial mathematics. Each shows where the marks are awarded, the key idea, and the full solution explained by your choice of tutor — Stella, Ella or Cassie.
How to use these
Try each question first, then check your working. Use the tutor tabs to read the full solution in the style that suits you: Stella is direct and challenging, Ella is warm and explains the why, and Cassie is concise and analytical.
Example 1 — Earning money with overtime
Question
Maya is paid $\$24.00$ per hour for a normal week of $38$ hours. Any extra hours are paid at time-and-a-half. In one week she works $44$ hours. Calculate her total pay for that week.
Solution
Split the week into normal hours and overtime hours, then pay each at its own rate.
Normal pay: $38 \times \$24.00 = \$912.00$.
Overtime hours: $44 - 38 = 6$ hours at time-and-a-half, so the rate is $1.5 \times \$24.00 = \$36.00$.
Overtime pay: $6 \times \$36.00 = \$216.00$.
Total: $\$912.00 + \$216.00 = \$1128.00$.
Don't pay all $44$ hours at the overtime rate — only the hours past $38$ count as overtime.
The trick here is that not every hour is paid the same. The first $38$ hours are normal time, and only the hours beyond that are overtime, so we work out the two parts separately.
Normal hours first: $38$ hours at $\$24.00$ gives $38 \times 24 = \$912.00$.
Now the overtime. She worked $44$ hours, so the extra is $44 - 38 = 6$ hours. "Time-and-a-half" means the rate is one and a half times normal: $1.5 \times \$24.00 = \$36.00$ per hour. Those $6$ hours earn $6 \times 36 = \$216.00$.
Add the two parts: $\$912.00 + \$216.00 = \$1128.00$.
Splitting normal time from overtime is the whole idea — overtime is a reward for the extra hours, so only those get the higher rate.
Separate normal hours from overtime.
- Normal: $38 \times 24 = \$912.00$
- Overtime hours: $44 - 38 = 6$
- Overtime rate: $1.5 \times 24 = \$36.00$
- Overtime pay: $6 \times 36 = \$216.00$
- Total: $912 + 216 = \$1128.00$
Where the marks go
- 1 mark: Correct normal pay ($38 \times \$24 = \$912$)
- 1 mark: Correct overtime hours and rate ($6$ hours at $\$36$)
- 1 mark: Correct total pay ($\$1128.00$)
Key idea
Overtime applies only to the hours beyond the normal week — pay normal hours and overtime hours at their own rates, then add.
Example 2 — Simple versus compound interest
Question
$\$5000$ is invested for $3$ years at $6\%$ per annum. Calculate the interest earned under simple interest, and the interest earned under interest compounded annually. State which earns more and by how much, to the nearest cent.
Solution
Simple interest first: $I = Prn = 5000 \times 0.06 \times 3 = \$900.00$.
Compound: the final amount is $A = P(1+r)^n = 5000(1.06)^3$.
$1.06^3 = 1.191016$, so $A = 5000 \times 1.191016 = \$5955.08$.
Compound interest earned $= A - P = 5955.08 - 5000 = \$955.08$.
Compound wins by $\$955.08 - \$900.00 = \$55.08$.
Compound interest is interest on interest — never just compare the two amounts, subtract the principal to get the interest first.
We're comparing two ways the same money can grow. Simple interest is calculated only on the original $\$5000$ every year, while compound interest is calculated on a balance that keeps growing.
Simple interest uses $I = Prn$: $I = 5000 \times 0.06 \times 3 = \$900.00$. That's the interest directly.
Compound interest grows the whole balance by $6\%$ each year, so we use $A = P(1+r)^n = 5000(1.06)^3$. Working out $1.06^3 = 1.191016$, the balance becomes $A = \$5955.08$. But that's the total, so to get just the interest we subtract the original $\$5000$: $5955.08 - 5000 = \$955.08$.
Comparing: compound earns more, by $955.08 - 900.00 = \$55.08$.
The difference appears because compound interest earns interest on previous interest — the extra $\$55.08$ is the interest the earlier interest itself generated.
Simple: $I = Prn$.
- $I = 5000 \times 0.06 \times 3 = \$900.00$
Compound: $A = P(1+r)^n$.
- $A = 5000(1.06)^3 = 5000 \times 1.191016 = \$5955.08$
- Interest $= 5955.08 - 5000 = \$955.08$
Compound earns more by $955.08 - 900.00 = \$55.08$.
Where the marks go
- 1 mark: Correct simple interest ($\$900.00$)
- 1 mark: Correct compound amount $A = 5000(1.06)^3 = \$5955.08$
- 1 mark: Correct compound interest ($A - P = \$955.08$)
- 1 mark: States compound earns more, by $\$55.08$
Key idea
Simple interest is charged only on the principal ($I = Prn$); compound interest grows the whole balance ($A = P(1+r)^n$), so subtract the principal to find the interest earned.