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Cycling Maths

Maths in cycling: gear changes

For older juniors, introduce easier and harder gears as ratios.

Quick answer

Maths in cycling: gear changes help you pedal more easily or go faster by changing the ratio between how many times your pedals turn and how many times your wheels turn. When you're going uphill, you use a smaller gear (fewer teeth on the front chainring) so you can pedal easier. When you're going downhill or flat, you use a larger gear (more teeth on the front chainring) so you can go faster.

Learning outcome

By the end of this page, you will understand how gear ratios in cycling help you pedal more easily or go faster, and you will be able to compare simple gear combinations to choose the best one for your ride. Success signs:
  • You can explain why you might change gears when cycling
  • You can identify which gear combination makes it easier to pedal
  • You can compare two gear ratios to decide which one is better for a specific situation
  • You can play a game to practice gear combinations

Why this matters

Gear changes are a real-life way to use maths in cycling. When you ride a bike, you're using ratios and comparisons to make your journey easier or faster. Understanding gear ratios helps you:
  • Ride up hills without getting too tired
  • Go faster on flat roads
  • Choose the right gear for the situation
  • Improve your cycling skills and enjoyment
Whether you're learning to ride a bike or training for a race, gear changes are a useful skill that uses maths to help you do better.

How it works

Gear changes in cycling use ratios. A gear ratio is a comparison between the number of teeth on the front chainring (the big gear attached to your pedals) and the number of teeth on the back cog (the smaller gear attached to your rear wheel). To work out a gear ratio, you divide the number of teeth on the front chainring by the number of teeth on the back cog. Example: If your front chainring has 42 teeth and your back cog has 14 teeth, your gear ratio is 42 ÷ 14 = 3. This means that for every one turn of the pedals, your wheel turns 3 times. When the gear ratio is:
  • Low (like 3:1), it's easier to pedal but you go slower
  • High (like 5:1), it's harder to pedal but you go faster

Examples you can see

Let's imagine you're learning to ride your bike and you're trying to go up a hill. Your bike has a front chainring with 42 teeth and a back cog with 14 teeth. Gear ratio = 42 ÷ 14 = 3 This means for every one turn of your pedals, your wheel turns 3 times. It's a low gear, so it's easy to pedal but you won't go very fast. Now imagine you're on a flat road and you want to go fast. You switch to a front chainring with 48 teeth and a back cog with 16 teeth. Gear ratio = 48 ÷ 16 = 3 Even though the numbers are different, the ratio is still 3, so you're still going 3 times as fast as your pedals turn. But if you switch to a front chainring with 48 teeth and a back cog with 12 teeth: Gear ratio = 48 ÷ 12 = 4 This is a higher gear ratio, so you'll go faster but it will be harder to pedal.

Try it as a mini adventure

Adventure: The Gear Challenge You're on a cycling adventure! You start at the bottom of a hill, and you need to get to the top. You have 3 gear options to choose from: 1. Front chainring with 36 teeth, back cog with 12 teeth 2. Front chainring with 42 teeth, back cog with 14 teeth 3. Front chainring with 48 teeth, back cog with 16 teeth Challenge: Which gear combination will make it easiest to pedal up the hill? Which one will help you go fastest on the flat road? Step-by-step: 1st time: Calculate the gear ratio for option 1: 36 ÷ 12 = 3 2nd time: Calculate the gear ratio for option 2: 42 ÷ 14 = 3 3rd time: Calculate the gear ratio for option 3: 48 ÷ 16 = 3 Answer: All three options give the same gear ratio of 3, so they're all equally easy to pedal. But if you want to go faster on flat roads, you'd want a higher gear ratio.

Common mistakes

  • Mistake: Thinking that a higher number in the front chainring always means a harder gear. Why it happens: Confusing the numbers in the ratio without dividing. Correct answer: You need to divide the front teeth by the back teeth to get the gear ratio. A 48-tooth front with a 16-tooth back gives a 3:1 ratio, which is easier than a 42-tooth front with a 14-tooth back (also 3:1), but a 48-tooth front with a 12-tooth back gives a 4:1 ratio, which is harder.
  • Mistake: Believing that a gear ratio of 4 is always better than a gear ratio of 3. Why it happens: Not understanding that a higher ratio means faster but harder pedaling. Correct answer: A gear ratio of 4 is harder to pedal but makes you go faster. A gear ratio of 3 is easier to pedal but makes you go slower. You choose based on what you need.
  • Mistake: Ignoring the back cog when calculating gear ratios. Why it happens: Focusing only on the front chainring and forgetting the back cog. Correct answer: Gear ratios are always calculated by dividing the front teeth by the back teeth.

Parent and carer guide

As a parent or carer, you can help your child understand gear changes by:
  • Explaining that gear ratios are like comparing numbers to see how hard or easy it is to pedal
  • Using a simple example with a toy bike or drawing a gear ratio chart
  • Letting your child practice calculating ratios with real numbers
  • Encouraging them to try different gear combinations on a bike or even a video game
  • Noticing when they change gears and asking them to explain why they chose that gear
  • Practicing at home with a simple chart or table to help them remember
What to say:
  • "Why do you think you changed to that gear?"
  • "How did that gear make pedaling feel?"
  • "Let's work out the gear ratio together."
What not to worry about:
  • Your child may not get it right away – that's normal
  • Don't pressure them to memorize all the ratios
  • Focus on understanding the concept, not just the numbers

Practice questions

  1. What is the gear ratio if the front chainring has 36 teeth and the back cog has 12 teeth?
  2. What is the gear ratio if the front chainring has 42 teeth and the back cog has 14 teeth?
  3. Which gear ratio is easier to pedal: 3:1 or 5:1?
  4. Which gear ratio is faster: 3:1 or 5:1?
  5. What is the gear ratio if the front chainring has 48 teeth and the back cog has 16 teeth?
Answers:
  1. 36 ÷ 12 = 3
  2. 42 ÷ 14 = 3
  3. 3:1 is easier to pedal
  4. 5:1 is faster
  5. 48 ÷ 16 = 3

FAQ

What is a gear ratio in cycling?

A gear ratio in cycling is the comparison between how many teeth are on the front chainring (attached to the pedals) and how many teeth are on the back cog (attached to the wheel). It tells you how many times the wheel turns for every one turn of the pedals.

Why do cyclists change gears?

Cyclists change gears to make pedaling easier or to go faster. When going uphill, they use a smaller gear (lower ratio) to make it easier to pedal. When going downhill or on flat ground, they use a larger gear (higher ratio) to go faster.

How do I calculate a gear ratio?

You calculate a gear ratio by dividing the number of teeth on the front chainring by the number of teeth on the back cog. For example, if the front chainring has 42 teeth and the back cog has 14 teeth, the gear ratio is 42 ÷ 14 = 3.

What does a gear ratio of 3 mean?

A gear ratio of 3 means that for every one turn of the pedals, the wheel turns 3 times. It's a medium gear that's good for general riding.

Are higher gear ratios always better?

No, higher gear ratios are not always better. They make the bike go faster but are harder to pedal. Lower gear ratios are easier to pedal but make the bike go slower. The best gear depends on the situation – like going uphill or on flat ground.