Mathematics Standard • Year 12 • Module 7 • Lesson 1

Simple Interest — Skill Drill

Build fluency with I = P r n and A = P(1 + rn): convert rates, calculate interest, rearrange to find any unknown.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write each percentage rate as a decimal.

5% = ____________ 4.5% = ____________ 7.2% = ____________ 0.6% = ____________

Q1.2 Complete the simple-interest formula and the total-amount formula.

I = ______ × ______ × ______ A = P + ______ = P(1 + ______ )

Q1.3 Rearrange I = P r n to make each variable the subject:

P = ____________ r = ____________ n = ____________

Stuck? Revisit lesson § Key Ideas — Simple Interest formula and Rearranging the Formula.

2. Worked example — find interest and total amount

Follow each line carefully. Every step has a reason on the right.

Problem. $5,000 is invested at 4% p.a. simple interest for 3 years. Calculate the interest earned and the total amount at the end.

Step 1 — Identify P, r, n.

P = $5,000 r = 0.04 (per year) n = 3 years

Reason: write the rate as a decimal and keep time in the same unit as the rate (both per year).

Step 2 — Substitute into I = P r n.

I = 5,000 × 0.04 × 3 = $600

Reason: simple interest is the rate × principal applied for each year.

Step 3 — Total amount A = P + I (or P(1 + rn)).

A = 5,000 + 600 = $5,600 (check: 5,000 × 1.12 = $5,600 ✓)

Reason: A and P(1 + rn) must agree — the second is a quick check.

Conclusion. Interest = $600, total amount = $5,600.

3. Faded example — fill in the missing steps

A loan of $8,000 is taken at 7.5% p.a. simple interest for 5 years. Find the interest and the total repayment. Fill in each blank. 4 marks

Step 1 — Identify P, r, n:

P = $ ____________ r = ____________ n = ____________ years

Step 2 — Substitute into I = P r n:

I = ________ × ________ × ________ = $ ____________

Step 3 — Find A = P + I:

A = $ ________ + $ ________ = $ ____________

Conclusion. Interest = $ ____________, total repayment = $ ____________.

Stuck? Revisit lesson § Worked Example — $8,000 at 7.5% over 5 years.

4. Graduated practice — Simple-interest calculations

Show your working below each part. Keep dollar amounts to 2 decimal places unless told otherwise.

Foundation — single-step substitution (4 questions)

Q	Problem	Answer
4.1 1	P = $2,000, r = 5% p.a., n = 4 years. Find I.
4.2 1	P = $3,500, r = 6% p.a., n = 2 years. Find I.
4.3 1	P = $1,200, r = 4% p.a., n = 5 years. Find A.
4.4 1	P = $6,000, r = 3.2% p.a., n = 6 years. Find I.

Standard — typical HSC difficulty (6 questions)

Show at least one line of substitution and clearly label your final answer with units.

4.5 Calculate the simple interest on $3,500 invested at 4.5% p.a. for 4 years. 2 marks

4.6 $6,000 is invested at 3.2% p.a. simple interest. Find the total amount after 6 years. 2 marks

4.7 What rate of simple interest is needed for $4,000 to earn $960 interest over 5 years? 2 marks

4.8 How long will $2,500 take to grow to $3,250 at 6% p.a. simple interest? 2 marks

4.9 Find the principal that earns $540 interest at 4.5% p.a. simple interest over 4 years. 2 marks

4.10 A loan of $P at 6% p.a. simple interest amounts to $6,440 after 4 years. Find P. 2 marks

Extension — convert units / compare (2 questions)

4.11 $4,800 is invested at 6% p.a. simple interest for 18 months. Calculate the interest earned. (Hint: convert 18 months to years first.) 3 marks

4.12 Two investments both start at $5,000. Investment X earns 8% p.a. simple interest. Investment Y earns 6% p.a. simple interest. After how many years will X be worth exactly $1,000 more than Y? 3 marks

Stuck on 4.12? Write a difference expression in n and set it equal to 1,000.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I converted 5% to 0.05 before multiplying, not left it as "5".

For 4.3 I gave the total amount A, not just the interest I.

For 4.4 I used n = 6 (years, matching r per year), not 72 (months).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Decimal rates

5% = 0.05, 4.5% = 0.045, 7.2% = 0.072, 0.6% = 0.006.

Q1.2 — Formulas

I = P × r × n. A = P + I = P(1 + rn).

Q1.3 — Rearrangements

P = I / (rn). r = I / (Pn). n = I / (Pr).

Q3 — Faded example ($8,000 at 7.5% over 5 years)

Step 1: P = $8,000, r = 0.075, n = 5.
Step 2: I = 8,000 × 0.075 × 5 = $3,000.
Step 3: A = $8,000 + $3,000 = $11,000.
Conclusion: Interest = $3,000, total repayment = $11,000.

Q4.1 — Simple interest

I = 2,000 × 0.05 × 4 = $400.00.

Q4.2 — Simple interest

I = 3,500 × 0.06 × 2 = $420.00.

Q4.3 — Total amount

I = 1,200 × 0.04 × 5 = $240. A = 1,200 + 240 = $1,440.00.

Q4.4 — Simple interest

I = 6,000 × 0.032 × 6 = $1,152.00.

Q4.5 — $3,500 at 4.5% for 4 years

I = 3,500 × 0.045 × 4 = $630.00.

Q4.6 — Total amount on $6,000 at 3.2% for 6 years

A = 6,000 × (1 + 0.032 × 6) = 6,000 × 1.192 = $7,152.00.

Q4.7 — Find rate

r = I / (Pn) = 960 / (4,000 × 5) = 960 / 20,000 = 0.048 = 4.8% p.a.

Q4.8 — Find time

Interest needed = $3,250 − $2,500 = $750. n = I / (Pr) = 750 / (2,500 × 0.06) = 750 / 150 = 5 years.

Q4.9 — Find principal

P = I / (rn) = 540 / (0.045 × 4) = 540 / 0.18 = $3,000.00.

Q4.10 — Loan: amount given, find P

A = P(1 + rn): 6,440 = P(1 + 0.06 × 4) = P(1.24). P = 6,440 / 1.24 = $5,200.00.

Q4.11 — $4,800 at 6% for 18 months

n = 18 / 12 = 1.5 years. I = 4,800 × 0.06 × 1.5 = $432.00. (Common slip: using n = 18 gives I = $5,184 — twelve times too big.)

Q4.12 — When does X beat Y by $1,000?

X − Y = 5,000(1 + 0.08n) − 5,000(1 + 0.06n) = 5,000 × 0.02n = 100n.
Set 100n = 1,000, so n = 10 years.