Mathematics Standard • Year 11 • Module 1 • Lesson 4
Rearranging Formulas
Apply formula rearrangement to real Australian contexts — road trips, garden planning, circle geometry, simple interest and recipe scaling.
Problem 1 — Road-trip timing
A family is planning a road trip from Sydney to Coffs Harbour, a distance of about 540 km. They want to know how long the trip will take at different cruising speeds.
Set up: What are we solving for?
(i) Starting from d = st, rearrange to make t the subject. 1 mark
(ii) Use the rearranged formula to find the time at s = 90 km/h. Give your answer in hours and minutes. 2 marks
(iii) Repeat for s = 100 km/h. How many minutes does the family save by driving at 100 km/h instead of 90? 2 marks
Stuck on minutes? Multiply the decimal part of the hours by 60 (e.g. 0.4 h → 0.4 × 60 = 24 min).Problem 2 — Garden bed planning
A landscaper is designing a rectangular garden bed that must have an area of A = 24 m². The owner has space for a maximum width of 4 m.
Set up: What are we solving for?
(i) Starting from A = lw, rearrange to make l the subject. 1 mark
(ii) Find l when w = 4 m and A = 24 m². 1 mark
(iii) The owner changes the width to 3 m to leave room for a path. Find the new length required to keep the same 24 m² area, and state in one sentence how the bed's shape has changed. 2 marks
Stuck? With a fixed area, narrower width must mean a longer length.Problem 3 — Pizza tray radius
A pizza shop sells pizzas described by the trim length of crust around the outside. A "medium" pizza has 75.4 cm of crust circumference. Use C = 2πr with π ≈ 3.14.
Set up: What are we solving for?
(i) Rearrange C = 2πr to make r the subject. 1 mark
(ii) Find the radius r of the medium pizza. 2 marks
(iii) A "large" pizza has C = 100.5 cm. Calculate its radius and find by how many centimetres the large pizza's diameter (2r) exceeds the medium's. 2 marks
Stuck? Compute r for both pizzas using r = C/(2π), then compare 2r values.Problem 4 — Simple interest rate
The simple interest formula is I = PRT, where I is interest in dollars, P is the principal in dollars, R is the annual interest rate (as a decimal), and T is time in years. A friend lends $4,000 for 2 years and earns $360 in interest.
Set up: What are we solving for?
(i) Rearrange I = PRT to make R the subject. 1 mark
(ii) Find R for this loan, then convert to a percentage. 2 marks
(iii) The friend wants to earn $540 instead of $360 by changing the time period, with the same principal and rate. Rearrange for T and find the new time required. 2 marks
Stuck? R = I / (PT). For (iii), T = I / (PR) — use the R you found in (ii).Problem 5 — Catering for a party
A muffin recipe says "use F = 2.5p grams of flour", where p is the number of people being served. A baker has 600 g of flour available.
Set up: What are we solving for?
(i) Rearrange F = 2.5p to make p the subject. 1 mark
(ii) Calculate the maximum whole number of people p the baker can serve with 600 g of flour. 2 marks
(iii) An updated recipe is F = 25 + 2.5p (the 25 g is a fixed allowance for testing the mixture). Rearrange this new formula for p, then find the maximum people the baker can serve with the same 600 g. State the new answer in a sentence. 3 marks
Stuck on (iii)? Same as a two-step rearrange — subtract 25 first, then divide by 2.5.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Road-trip timing
Set up. Rearrange d = st for t, then substitute the two speeds and compare.
(i) t = d/s.
(ii) t = 540/90 = 6 hours exactly → 6 h 0 min.
(iii) t = 540/100 = 5.4 h = 5 h 24 min. The faster speed saves 6 h − 5 h 24 min = 36 minutes.
Problem 2 — Garden bed
Set up. Rearrange A = lw for l, then use the area constraint with two different widths.
(i) l = A/w.
(ii) l = 24/4 = 6 m (bed is 6 m × 4 m).
(iii) l = 24/3 = 8 m. The bed is now 8 m × 3 m — narrower and longer (more rectangular and elongated) while keeping the same 24 m² area.
Problem 3 — Pizza tray
Set up. Rearrange C = 2πr for r, then compute and compare.
(i) r = C / (2π).
(ii) r = 75.4 / (2 × 3.14) = 75.4 / 6.28 = 12 cm. Diameter = 24 cm.
(iii) Large: r = 100.5 / 6.28 = 16 cm. Diameter = 32 cm. Difference in diameters = 32 − 24 = 8 cm bigger across.
Problem 4 — Simple interest
Set up. Rearrange I = PRT for R, then again for T.
(i) R = I / (PT).
(ii) R = 360 / (4000 × 2) = 360/8000 = 0.045 = 4.5% per annum.
(iii) T = I / (PR) = 540 / (4000 × 0.045) = 540/180 = 3 years. To earn $540 at the same rate and principal, the loan needs to run for 3 years instead of 2.
Problem 5 — Catering / flour
Set up. Rearrange the simple recipe formula for p, then redo for the updated formula with a fixed allowance.
(i) p = F / 2.5.
(ii) p = 600 / 2.5 = 240 → up to 240 people (whole-number serving).
(iii) Rearrange F = 25 + 2.5p: p = (F − 25)/2.5. Substitute F = 600: p = (600 − 25)/2.5 = 575/2.5 = 230. The baker can now serve up to 230 people — the 25 g testing allowance has cost 10 servings.