Mathematics Standard • Year 11 • Module 1 • Lesson 4

Rearranging Formulas

Build fluency making a different variable the subject of a formula using inverse operations, and check by substituting clean numbers.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Define "subject of a formula" in your own words.

Q1.2 Complete the rearranged forms.

From d = st: s = ____________ t = ____________

From A = bh: b = ____________ h = ____________

Q1.3 To make r the subject of C = 2πr you divide both sides by ____________ (write the full divisor, not just "2").

Stuck? Revisit lesson § Worked Example 2 — divide by the WHOLE multiplier 2π, not just 2.

2. Worked example — rearrange d = st for s, then use it

Follow each step. Notice that "rearrange" and "solve for a number" use the same balancing logic — the only difference is what we keep on the other side.

Problem. A car travels d = 180 km in t = 3 h. Find the average speed s by first making s the subject of d = st.

Step 1 — Identify the multiplier on s.

d = s × t (s is multiplied by t)

Reason: to isolate s, undo the multiplication by t.

Step 2 — Divide both sides by t.

d / t = st / t ⇒ s = d / t

Reason: division is the inverse of multiplication; t / t = 1 on the right.

Step 3 — Substitute and calculate.

s = 180 / 3 = 60

Reason: now the formula matches what we need to find.

Step 4 — Number check.

Use s = 60, t = 3 in original: d = 60 × 3 = 180 ✓

Conclusion. The car's average speed is 60 km/h.

3. Faded example — fill in the missing steps

Make h the subject of A = bh, then find h when A = 96 m² and b = 12 m. Fill in each blank. 4 marks

Step 1 — Identify the multiplier on h:

A = b × h, so divide both sides by ____________.

Step 2 — Rearranged formula:

h = ____________

Step 3 — Substitute A = ____________ and b = ____________:

h = ____________ / ____________ = ____________ m

Step 4 — Check with the original formula:

A = b × h = ____________ × ____________ = ____________ m² ✓

Conclusion. The height is ____________ m.

Stuck? Revisit lesson § Worked Example 3 — Rearrange and substitute.

4. Graduated practice — rearrange, then use

Show each rearrangement on the first line and the substitution on the second.

Foundation — one-step rearrangements (4 questions)

Q	Formula	Rearrange for...
4.1 1	d = st	t = ____________
4.2 1	A = lw	w = ____________
4.3 1	F = ma	a = ____________
4.4 1	C = 2πr	r = ____________

Standard — rearrange THEN substitute (6 questions)

Write the rearranged formula first, then substitute the numbers. Always include units.

4.5 Use d = st. Find t when d = 240 km and s = 80 km/h. 2 marks

4.6 Use d = st. Find s when d = 150 km and t = 2.5 h. 2 marks

4.7 Use A = lw. Find l when A = 72 cm² and w = 8 cm. 2 marks

4.8 Use V = lwh. Find h when V = 360 cm³, l = 10 cm, w = 6 cm. 2 marks

4.9 Use C = 2πr. Find r when C = 31.4 cm. Use π ≈ 3.14. 2 marks

4.10 Use F = ma. Find a when F = 240 N and m = 60 kg. 2 marks

Extension — two-step rearrange (2 questions)

4.11 Rearrange C = 8 + 1.50k to make k the subject. Then find k when C = 23. 3 marks

4.12 The perimeter of a rectangle is P = 2l + 2w. Rearrange to make w the subject. Then find w when P = 36 cm and l = 11 cm. 3 marks

Stuck on 4.11? Subtract 8 first, then divide by 1.50. Same as solving a two-step equation.

5. Self-check the easy 3

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

For 4.1 I wrote t = d/s (I divided both sides of d = st by s, not by t).

For 4.4 I divided by the WHOLE multiplier 2π, getting r = C / (2π) — not r = C/2π (with π left out).

For 4.7 (A = lw) I checked: l × w = 9 × 8 = 72 ✓, confirming the rearranged formula gave the right value.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Subject of a formula

The variable that appears by itself on one side of the equals sign — e.g. d in d = st.

Q1.2 — Rearranged forms

From d = st: s = d/t, t = d/s. From A = bh: b = A/h, h = A/b.

Q1.3 — Divisor for C = 2πr

Divide both sides by 2π (the whole product, not just 2). So r = C / (2π).

Q3 — Faded h = A/b example

Step 1: divide both sides by b. Step 2: h = A/b. Step 3: A = 96, b = 12, h = 96/12 = 8 m. Step 4: A = 12 × 8 = 96 m² ✓. Conclusion: height = 8 m.

Q4.1-4.4 — Rearranged formulas

4.1: t = d/s. 4.2: w = A/l. 4.3: a = F/m. 4.4: r = C/(2π).

Q4.5 — Time from d = st

t = d/s = 240/80 = 3 h.

Q4.6 — Speed from d = st

s = d/t = 150/2.5 = 60 km/h.

Q4.7 — l from A = lw

l = A/w = 72/8 = 9 cm.

Q4.8 — h from V = lwh

h = V/(lw) = 360/(10 × 6) = 360/60 = 6 cm.

Q4.9 — r from C = 2πr

r = C / (2π) = 31.4 / (2 × 3.14) = 31.4 / 6.28 = 5 cm.

Q4.10 — a from F = ma

a = F/m = 240/60 = 4 m/s².

Q4.11 — k from C = 8 + 1.50k

Subtract 8: C − 8 = 1.50k. Divide by 1.50: k = (C − 8) / 1.50.
Substitute C = 23: k = (23 − 8)/1.50 = 15/1.50 = 10 km.

Q4.12 — w from P = 2l + 2w

Subtract 2l: P − 2l = 2w. Divide by 2: w = (P − 2l)/2.
Substitute P = 36, l = 11: w = (36 − 22)/2 = 14/2 = 7 cm.