Math Detours · Grade 7 · Connections · 1

Why a bike has gears

48 teeth at the front and 16 at the back turn the wheel three times for each turn of the pedals. Ratios you can feel in your legs.

Name ____________________Date ____________

Tonight's idea

Count the teeth. This bike has 48 on the front ring, by the pedals, and 16 on the sprocket on the back wheel. A chain joins them and cannot slip. Each tooth that passes at the front pulls one link along, and each link moves one tooth at the back.

16 teeth back wheel: 2.1 m around 48 teeth chain pedal
One turn of the pedals pulls 48 links of chain past the back sprocket, which has only 16 teeth.

Turn the pedals once. 48 teeth pass, so 48 links move. The back sprocket has only 16 teeth, so it must turn \(48 \div 16 = 3\) times. The wheel is fixed to it and turns 3 times too.

Gear ratio = front teeth ÷ back teeth = turns of the wheel for one turn of the pedals.

Each turn of the wheel rolls the bike forward by the distance around the tyre, here about 2.1 m. So one pedal turn carries you \(3 \times 2.1 = 6.3\) m.

Why not always use a big ratio? You pay for distance with push. A high gear goes far each turn and is hard to push. A low gear goes a short way and is easy: right for a hill.

Watch one: 40 teeth in front, 20 behind

Gear ratio \(40 \div 20 = 2\). The wheel turns twice for each pedal turn, so you travel \(2 \times 2.1 = 4.2\) m.

How many pedal turns to ride 420 m? \(420 \div 4.2 = 100\).

Now change to a back sprocket with 25 teeth. Ratio \(40 \div 25 = 1.6\), distance \(1.6 \times 2.1 = 3.36\) m, and \(420 \div 3.36 = 125\) turns. More turns, each one easier.

Why a wheel rolls once around its rim each turn: Roll a wheel once. Ratios in grade 6: sheet 2.

Your turn

The wheel is 2.1 m around in every question.

  1. Find the gear ratio and the distance for one pedal turn: (a) 48 front, 24 back; (b) 36 front, 12 back; (c) 32 front, 32 back.
  2. A bike has one front ring with 42 teeth and back sprockets with 14, 21 and 28 teeth. (a) Find the three gear ratios. (b) Which back sprocket would you pick for a steep hill, and why?
  3. A rider in the 48 and 16 gear turns the pedals 80 times a minute. (a) How far does she go in one minute? (b) What is her speed in km per hour?
  4. A bike has two front rings, 48 and 36, and four back sprockets, 12, 16, 18 and 24. That makes 8 pairs. (a) Find two pairs with the same gear ratio. (b) How many different ratios does the bike really have?

Stretch: build the gear you want

You want exactly 5.25 m for each pedal turn. (a) What gear ratio is that? (b) A front ring has between 30 and 50 teeth. Find every front and back pair that gives this ratio. Teeth come in whole numbers.

Answers

  1. (a) Ratio 2, distance \(2 \times 2.1 = 4.2\) m. (b) Ratio 3, 6.3 m. (c) Ratio 1, 2.1 m: the wheel turns exactly as fast as the pedals.
  2. (a) \(42 \div 14 = 3\), \(42 \div 21 = 2\), \(42 \div 28 = 1.5\). (b) The 28. It has the lowest ratio, so each pedal turn moves the bike the shortest way (3.15 m) and needs the least push.
  3. (a) One turn is 6.3 m, so \(80 \times 6.3 = 504\) m. (b) \(504 \times 60 = 30{,}240\) m in an hour, about 30 km per hour.
  4. (a) \(48 \div 16 = 3\) and \(36 \div 12 = 3\). Also \(48 \div 24 = 2\) and \(36 \div 18 = 2\). (b) The 48 ring gives 4, 3, \(2\tfrac23\), 2. The 36 ring gives 3, 2, 2.25, 1.5. Two repeats, so 6 different ratios: 4, 3, \(2\tfrac23\), 2.25, 2, 1.5.

Stretch. (a) \(5.25 \div 2.1 = 2.5\). (b) A ratio of 2.5 means 5 front teeth for every 2 back teeth, so the front count is a multiple of 5: 30 and 12, 35 and 14, 40 and 16, 45 and 18, 50 and 20.

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