Question 3
The diameter of the rear wheel of a bicycle is 725 mm. A gear with 50 teeth is attached to the rear wheel and driven by a pedal gear with 150 teeth. The pedals complete four rotations.
What is the distance travelled by the bicycle, rounded to one decimal place?
Not quite. 3.0 m turns the gear ratio upside down. Each pedal turn drives 150 teeth past the 50-tooth wheel gear, so the wheel turns 3 times per pedal turn, not a third of a time. In 2025, 18.8% of Queensland students chose (A).
Not quite. 6.8 m is the distance for one pedal turn, $3\times\pi\times0.725$. The pedals complete four rotations, so the wheel turns 12 times. In 2025, 13.3% of Queensland students chose (B).
Not quite. $12\times0.725=8.7\text{ m}$ leaves out the $\pi$. One turn of the wheel moves the bicycle one circumference, $\pi d$, not one diameter. In 2025, 27.7% of Queensland students chose (C).
That is it. Step through the hints below to see why. In 2025, 39.5% of Queensland students chose (D).
The pedal gear is the driver and the wheel gear is driven. Each pedal turn moves 150 teeth of chain past the 50-tooth gear, so the wheel turns 3 times.
$$\text{wheel turns}=4\times\frac{150}{50}=12$$
The formula book’s ratio by teeth is driven over driver, $50\div150=\frac{1}{3}$. A VR below 1 is right for a bicycle built for speed. Multiplying the pedal turns by $\frac{1}{3}$ is how you land on (A).
One turn of the wheel rolls the bicycle forward by one circumference. Convert the diameter to metres first.
$$\begin{aligned}\text{distance}&=12\times\pi\times0.725\\[10pt]&=27.33\text{ m}\approx27.3\text{ m}\end{aligned}$$
Using the diameter in place of the circumference, $12\times0.725$, is how you land on (C).
Each wrong option is one slip away from the key. (A) inverts the ratio, (B) uses one pedal turn instead of four, and (C) drops the $\pi$.
Check the size of your answer too. One turn of this wheel is about 2.3 m, so twelve turns must be well over 20 m.
$$\begin{aligned}\text{distance}&=4\times\frac{150}{50}\times\pi\times0.725\\[10pt]&=12\times2.2777\\[10pt]&=27.33\text{ m}\approx27.3\text{ m}\end{aligned}$$
Cross the drive from the pedals to the wheel, then turn wheel revolutions into distance with the circumference.
The most common wrong answer was (C), which leaves out the $\pi$. Fewer than two in five students found the key. Write “distance = turns × πd” as its own line every time a wheel, drum or pulley turns.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and any figures are from the 2025 QCAA Engineering external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted, except where a third-party source is named. Tangent Tuition is not affiliated with the QCAA.
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