Mathematics Standard • Year 11 • Module 1 • Lesson 4

Rearranging Formulas

Practise HSC-style writing on rearranging formulas — three multi-mark short answers and one extended response with marking criteria.

Master · Past-Paper Style

1. Short-answer questions

1.1 Rearrange d = st to make t the subject. Then find t when d = 150 km and s = 50 km/h. State your answer with units. 3 marks Band 3

1.2 Rearrange C = 2πr to make r the subject. Use it to find r when C = 31.4 cm. Take π ≈ 3.14. 4 marks Band 3-4

1.3 The perimeter of a rectangle is P = 2l + 2w.
(a) Rearrange the formula to make l the subject.
(b) A rectangular garden has perimeter 42 m and width 8 m. Find the length.
(c) The owner doubles the width to 16 m but keeps the perimeter at 42 m. Use the rearranged formula to find the new length and explain in one sentence whether this is physically possible. 4 marks Band 4

Stuck on 1.3(c)? If 2l + 2(16) = 42 gives a negative l, the rectangle cannot physically exist with that width — the perimeter is too small.

2. Extended response

2.1 The simple-interest formula I = PRT is used by a credit union.

I = simple interest earned ($)

P = principal invested ($)

R = annual interest rate (as a decimal, e.g. 0.04 means 4% p.a.)

T = time invested (years)

(a) Rearrange I = PRT to make T the subject.
(b) An investor puts $12,000 into an account paying 3.5% p.a. simple interest. How long must they invest to earn $1,260 in interest? Show your full working.
(c) Rearrange I = PRT to make R the subject. A different investor wants to earn $960 from a $6,000 investment held for 4 years. What annual interest rate (as a percentage) is required?
(d) Write one sentence comparing the two investments, identifying which delivers a higher annual percentage return. 7 marks Band 5-6

Explicit marking criteria

Part (a) — 1 mark

• 1 mark — T = I / (PR), with R and P together in the denominator.

Part (b) — 2 marks

• 1 mark — correct substitution: T = 1260 / (12000 × 0.035).

• 1 mark — correct T = 3 years (with units).

Part (c) — 2 marks

• 1 mark — rearranges to R = I / (PT) and substitutes correctly: R = 960 / (6000 × 4).

• 1 mark — R = 0.04 = 4% per annum.

Part (d) — 2 marks

• 1 mark — identifies that Investor 2 earns 4% p.a. vs Investor 1's 3.5% p.a.

• 1 mark — clear comparison sentence: Investor 2 has the higher annual percentage return (by 0.5 percentage points), even though the total dollars earned differ.

Your response:

Stuck on (d)? Compare RATES (3.5% vs 4%), not total dollars — a smaller deposit earning a higher rate can still be the better deal per dollar invested.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — sample responses + marking notes

1.1 — Time from d = st (3 marks)

Sample response.
d = st ⇒ divide both sides by s ⇒ t = d/s.
Substitute: t = 150 / 50 = 3 hours.

Marking notes. 1 mark — rearranged formula t = d/s. 1 mark — correct substitution. 1 mark — final answer with units. A bare "3" without "hours" loses the units mark.

1.2 — Radius from C = 2πr (4 marks)

Sample response.
C = 2πr ⇒ divide both sides by 2π ⇒ r = C / (2π).
Substitute C = 31.4, π ≈ 3.14:
r = 31.4 / (2 × 3.14) = 31.4 / 6.28 = 5 cm.

Marking notes. 1 mark — rearranged correctly with WHOLE divisor 2π (a response showing r = C/2π without brackets is awarded the mark if it then computes 31.4/6.28 correctly; r = C/2 × π = 49.298 loses both rearrange and substitution marks). 1 mark — correct denominator 2 × 3.14 = 6.28. 1 mark — correct value 5. 1 mark — units "cm".

1.3 — Perimeter rearrangement (4 marks)

(a) Sample response. P = 2l + 2w ⇒ P − 2w = 2l ⇒ l = (P − 2w)/2.

(b) Sample response. l = (42 − 16)/2 = 26/2 = 13 m.

(c) Sample response. l = (42 − 32)/2 = 10/2 = 5 m. A 16 m wide, 5 m long rectangle is physically possible — just unusual (wider than long). Note: if the width had been 22 m, l would have come out as −1 m, which is NOT physically possible because lengths cannot be negative.

Marking notes. 1 mark — correct rearrangement. 1 mark — correct l = 13 m. 1 mark — correct l = 5 m for w = 16. 1 mark — sentence noting it IS physically possible (or correctly noting when it would NOT be).

2.1 — Simple interest rearrangement (7 marks): sample Band-6 response with annotations

Sample Band-6 response.

(a) Rearrange for T.

I = PRT ⇒ divide both sides by PR ⇒ T = I / (PR). [1 mark — correct rearrangement with PR in the denominator.]

(b) Time required.

T = 1260 / (12000 × 0.035) = 1260 / 420 = 3 years. [1 mark — substitutes correctly using R as decimal 0.035. 1 mark — correct T with units.]

(c) Rearrange for R and find rate.

I = PRT ⇒ R = I / (PT). [1 mark — rearrangement and substitution: R = 960 / (6000 × 4) = 960 / 24000.]
R = 0.04 = 4% per annum. [1 mark — correct rate as a percentage.]

(d) Comparison.

Investor 1 earns at 3.5% p.a.; Investor 2 earns at 4% p.a. [1 mark — identifies the two rates.]
Investor 2's account delivers the higher annual percentage return (4% vs 3.5%, a margin of 0.5 percentage points), even though Investor 1 earns more total dollars by depositing twice as much for almost as long. [1 mark — clear comparison sentence distinguishing rate from dollars.]

Total: 7/7.

Band descriptors for marker.

Band 3: Rearranges for T correctly and gets T = 3 but does not attempt (c) or (d). ≈ 3 marks.

Band 4: All rearrangements correct and both numerical answers (3 years, 4%) correct; no comparison or a confused one. ≈ 5 marks.

Band 5: Comparison made but compares only dollar amounts ("Investor 1 earned more, so theirs is better") — misses that 4% > 3.5%. ≈ 6 marks.

Band 6: Complete, rearrangements correct, both rates calculated, AND a comparison sentence that correctly identifies the higher annual percentage as the better rate. 7/7.