Math worksheet for practicing ratio and proportion problems.

Let’s solve this step by step.

We are given two gear trains: Gear Train 1 and Gear Train 2. We need to find the overall gear ratio for each, then answer questions about speed and torque.

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Gear Train 1:
It has three gears:
- Input gear (A) = 8 teeth
- Middle gear (B) = 32 teeth
- Output gear (C) = 40 teeth

The formula for gear ratio between two gears is:

> Gear Ratio = Number of Teeth on Driven Gear ÷ Number of Teeth on Driving Gear

In a train with multiple gears, we multiply the ratios of each stage.

Stage 1: A drives B → Ratio = B/A = 32/8 = 4
Stage 2: B drives C → But note: B is now driving C, so Ratio = C/B = 40/32 = 5/4 = 1.25

Wait — actually, in a simple gear train like this, if gear A turns gear B, and gear B turns gear C, then the overall ratio from A to C is:

> Overall Ratio = (Teeth on C) / (Teeth on A) — because intermediate gears cancel out!

Why? Because:

Ratio A→B = B/A
Ratio B→C = C/B
Multiply them: (B/A) × (C/B) = C/A ← B cancels!

So for Gear Train 1:

Overall Ratio = C / A = 40 / 8 = 5

That means for every 1 turn of input gear A, output gear C turns 1/5 of a turn. So it reduces speed but increases torque.

But let’s double-check using stages:

Stage 1: A (8T) drives B (32T) → Ratio = 32/8 = 4 → So B turns slower than A by factor of 4.

Stage 2: B (32T) drives C (40T) → Ratio = 40/32 = 1.25 → So C turns slower than B by factor of 1.25.

Total reduction = 4 × 1.25 = 5 → Same as before.

✔ So Gear Train 1 overall ratio = 5:1 (output turns once for every 5 turns of input)

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Gear Train 2:
Input gear (D) = 16 teeth
Middle gear (E) = 24 teeth
Output gear (F) = 48 teeth

Again, overall ratio = F / D = 48 / 16 = 3

Check via stages:

Stage 1: D→E = 24/16 = 1.5
Stage 2: E→F = 48/24 = 2
Total = 1.5 × 2 = 3 ✔

So Gear Train 2 overall ratio = 3:1

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Now answer the questions:

Question 1: Which gear train gives you more mechanical advantage?

Mechanical advantage in gears usually refers to how much torque is multiplied. Higher gear ratio = more torque at output (but less speed).

Gear Train 1: 5:1 → multiplies torque by 5
Gear Train 2: 3:1 → multiplies torque by 3

So Gear Train 1 gives more mechanical advantage.

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Question 2: If you want to increase speed, which gear train should you use?

To increase speed, you want a gear ratio LESS than 1 — meaning output turns faster than input.

But both our ratios are greater than 1 — they reduce speed.

Wait — maybe the question means “which one reduces speed LESS?” so that output is relatively faster?

Gear Train 2 has ratio 3:1 → output turns 1/3 speed of input
Gear Train 1 has ratio 5:1 → output turns 1/5 speed of input

So Gear Train 2 output is faster than Gear Train 1 output.

Therefore, if you must choose between these two, Gear Train 2 will give you higher output speed (though still slower than input).

But technically, neither increases speed — both decrease it. However, since the question asks “which should you use” implying choice between them, we pick the one with smaller reduction → Gear Train 2.

Alternatively, if the system allowed reversing input/output, but the diagram shows fixed direction.

Based on standard interpretation: lower gear ratio = less speed reduction = relatively faster output.

✔ So answer: Gear Train 2

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Question 3: What is the purpose of an idler gear?

An idler gear is a gear placed between input and output that doesn’t change the overall gear ratio — it only changes the direction of rotation.

In both gear trains above, the middle gear (B in Train 1, E in Train 2) is an idler — it affects direction but not the final ratio (since its tooth count cancels out).

Purpose: To reverse direction without changing speed/torque ratio.

✔ Answer: To change the direction of rotation without affecting the gear ratio.

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Final Answers:

Gear Train 1 Overall Ratio: 5
Gear Train 2 Overall Ratio: 3

More mechanical advantage: Gear Train 1
Increase speed (relatively): Gear Train 2
Purpose of idler gear: Change direction without changing ratio

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Final Answer:
Gear Train 1 overall ratio: 5
Gear Train 2 overall ratio: 3
More mechanical advantage: Gear Train 1
To increase speed (between the two): Gear Train 2
Purpose of idler gear: To change the direction of rotation without affecting the gear ratio.