Non-compound gear train diagram showing the relationship between driving, idler, and driven gears with 12, 36, and 60 teeth respectively.

Let’s solve this step by step.

First, remember the rule:
Gear Ratio = Driven Gear Teeth : Driving Gear Teeth
And always reduce to lowest terms (like simplifying a fraction).

Also:
- If Gear A is driving and has fewer teeth than Gear B (driven), then Gear B turns slower — so if A turns once, B turns *less* than one time.
- If Gear A has more teeth than Gear B, then Gear B turns faster — so if A turns once, B turns *more* than one time.
The number of times the driven gear turns = (Teeth of Driving Gear) ÷ (Teeth of Driven Gear)

Wait — actually, let’s think carefully:

If Driving Gear (A) has 10 teeth, and Driven Gear (B) has 20 teeth…
That means for every full turn of A, it moves 10 teeth. But B needs 20 teeth to make one full turn. So B only turns half a turn.
So: Turns of Driven Gear = Teeth of Driving Gear / Teeth of Driven Gear

Yes! That’s correct.

So formula:
> Turns of Driven Gear = (Teeth of Driving Gear) ÷ (Teeth of Driven Gear)

But note: sometimes the question asks “If B turns one time, how many times does A turn?” — that’s the reverse. Then you do:
> Turns of Driving Gear = (Teeth of Driven Gear) ÷ (Teeth of Driving Gear)

Got it? Let’s go row by row.

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Part 1: Simple Gear Pairs (A = driving, B = driven)

#### Row 1: A=10, B=20
Ratio = Driven : Driving = B:A = 20:10 → simplify → 2:1
If A turns one time, B turns = 10 ÷ 20 = 0.5 times

#### Row 2: A=40, B=10
Ratio = B:A = 10:40 → simplify → 1:4
If A turns one time, B turns = 40 ÷ 10 = 4 times

#### Row 3: A=36, B=60
Ratio = B:A = 60:36 → divide both by 12 → 5:3
Question says: “If B turns one time, A will turn” → now B is turning, so we treat B as driver? Wait no — the setup still says A is driving, but the question is asking what happens if B turns. So we reverse.

Actually, in this case, since they’re asking “if B turns one time”, we need to find how many times A turns when B is turned manually. So:

Turns of A = Teeth of B ÷ Teeth of A = 60 ÷ 36 = 5/3 ≈ 1.666... → but better as fraction: 5/3 or 1⅔

But let’s keep it as simplified fraction unless told otherwise.

Wait — maybe they want decimal? Looking at other problems, probably fraction or mixed number is fine. But let’s see context.

Actually, in worksheets like this, often they expect reduced fractions or decimals. Let’s use fractions where needed.

So: 60 ÷ 36 = 5/3 → so A turns 5/3 times

But let me double-check logic:

If B (60 teeth) turns once, it moves 60 teeth. A (36 teeth) must rotate enough to match those 60 teeth. So rotations of A = 60 / 36 = 5/3. Yes.

#### Row 4: A=12, B=60
Ratio = B:A = 60:12 → simplify → 5:1
If A turns one time, B turns = 12 ÷ 60 = 1/5 or 0.2 times

#### Row 5: A=84, B=36
Ratio = B:A = 36:84 → divide by 12 → 3:7 → wait, 36÷12=3, 84÷12=7 → so 3:7? But usually we write ratio as smaller first? No — ratio is Driven:Driving, so B:A = 36:84 → reduce by dividing by 12 → 3:7. But 3:7 is already reduced.

Alternatively, divide by GCF(36,84)=12 → yes, 3:7.

If A turns one time, B turns = 84 ÷ 36 = 7/3 ≈ 2.333... → 7/3 or 2⅓ times

Wait — 84 ÷ 36 = 7/3? Let's check: 36 × 2 = 72, 84 - 72 = 12, so 2 + 12/36 = 2 + 1/3 = 7/3. Yes.

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Now, Multiple Gears in a Train:

Rule given: Ignore idler gears. Only care about first (driving) and last (driven). So ratio is C:A (since A drives, C is driven, B is idler).

Example given: A=10, B=50, C=40 → ratio A:B is 5:1? Wait no — example says: “ratio for A:B is 5:1” — wait that can’t be right if A=10, B=50.

Wait, re-read example:

“Example: A has 10 teeth; B has 50 teeth, C has 40 teeth, the ratio for A:B is 5:1”

Wait — that seems backwards. If A is driving (10 teeth), B is driven (50 teeth), ratio should be Driven:Driving = B:A = 50:10 = 5:1. Oh! They wrote “ratio for A:B is 5:1” — but that would mean A:B = 5:1, which implies A is larger? Confusing.

Actually, looking again: “the ratio for A:B is 5:1” — but if A=10, B=50, then B:A = 5:1. Probably typo in explanation.

Then it says: “ratio for B:C is 4:5” — B=50, C=40 → C:B = 40:50 = 4:5. Again, they say “B:C is 4:5” — which would be B:C = 4:5, meaning B=4 parts, C=5 parts — but B=50, C=40, so actually C:B = 4:5.

I think they meant: ratio from A to B is 5:1 (meaning B turns 1/5 of A), and from B to C is 4:5 (meaning C turns 5/4 of B?). This is messy.

But then they say: Multiply 5:1 x 4:5 and get 4:1. And simpler to ignore idlers: C:A = 40:10 = 4:1.

Ah! So final ratio is Driven (C) : Driving (A) = 40:10 = 4:1.

So rule: For multiple gears in line, just take Last Driven : First Driving, ignore middle idlers.

So for all these, Ratio = C : A (reduced)

And “How many times”: If A turns one time, C turns = Teeth of A ÷ Teeth of C

Because: Turns of Driven (C) = Teeth of Driving (A) / Teeth of Driven (C)

Same as before.

But note: some questions ask “if C turns one time, how many times does A turn?” — then it’s reverse: Teeth of C / Teeth of A

Or “if B turns one time, how many times does C turn?” — but B is idler, so if B turns, it affects C directly? Actually, since B is between A and C, if you turn B, then C turns based on B:C ratio.

But the worksheet says: “In all of the following examples Gear A is the driving gear, Gear B is the idler, Gear C is the driven gear.” — so normally A drives, but some questions change who is turning.

So we have to read each question carefully.

Let’s go row by row.

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Part 2: Multiple Gears (A=driving, B=idler, C=driven)

#### Row 1: A=10, B=20, C=10
Ratio = C:A = 10:10 = 1:1
If A turns one time, C turns = 10 ÷ 10 = 1 time

#### Row 2: A=10, B=40, C=5
Ratio = C:A = 5:10 = 1:2
If A turns one time, C turns = 10 ÷ 5 = 2 times

#### Row 3: A=12, B=36, C=60
Ratio = C:A = 60:12 = 5:1
Question: “If C turns one time, A will turn” → so now C is turning, find how many times A turns.
Turns of A = Teeth of C ÷ Teeth of A = 60 ÷ 12 = 5 times

#### Row 4: A=84, B=12, C=36
Ratio = C:A = 36:84 → simplify: divide by 12 → 3:7
Question: “If B turns one time, C will turn” → B is idler, so if B turns, it drives C. So we look at B and C.
B=12 (now acting as driver?), C=36 (driven).
So turns of C = Teeth of B ÷ Teeth of C = 12 ÷ 36 = 1/3 time

Is that correct? Since B is connected to C, yes — if B turns once, it moves 12 teeth, C has 36 teeth, so C turns 12/36 = 1/3.

Even though normally A drives B, here the question specifies “if B turns”, so we consider B as the input.

#### Row 5: A=36, B=60, C=84
Ratio = C:A = 84:36 → simplify: divide by 12 → 7:3
Question: “If A turns one time, C will turn” → turns of C = Teeth of A ÷ Teeth of C = 36 ÷ 84 = 3/7 → 3/7 time

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Last question: What are the purposes of an idler gear?

From knowledge: Idler gears don’t change the overall gear ratio (since we ignore them), but they can:
- Change the direction of rotation (so output spins same direction as input, instead of opposite)
- Allow distance between driving and driven gears to be increased
- Sometimes used to connect multiple gears in a train without affecting speed ratio

So answer: To change the direction of rotation or to bridge space between gears without changing the gear ratio.

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Now, let’s compile all answers neatly.

Final Answer:

Simple Gear Pairs:

1. A=10, B=20 → Ratio: 2:1 → If A turns one time, B turns: 0.5 or 1/2
2. A=40, B=10 → Ratio: 1:4 → If A turns one time, B turns: 4
3. A=36, B=60 → Ratio: 5:3 → If B turns one time, A turns: 5/3 or 1⅔
4. A=12, B=60 → Ratio: 5:1 → If A turns one time, B turns: 1/5 or 0.2
5. A=84, B=36 → Ratio: 3:7 → If A turns one time, B turns: 7/3 or 2⅓

Multiple Gears in a Train:

1. A=10, B=20, C=10 → Ratio: 1:1 → If A turns one time, C turns: 1
2. A=10, B=40, C=5 → Ratio: 1:2 → If A turns one time, C turns: 2
3. A=12, B=36, C=60 → Ratio: 5:1 → If C turns one time, A turns: 5
4. A=84, B=12, C=36 → Ratio: 3:7 → If B turns one time, C turns: 1/3
5. A=36, B=60, C=84 → Ratio: 7:3 → If A turns one time, C turns: 3/7

Purpose of idler gear:
To change the direction of rotation or to allow gears to be spaced apart without changing the overall gear ratio.

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Wait — in the multiple gears section, for row 4, I said ratio is 3:7 — but let me confirm: C:A = 36:84 = 3:7, yes.

And for row 5: C:A = 84:36 = 7:3, yes.

Also, in simple gears row 3: ratio B:A = 60:36 = 5:3, correct.

Row 5: B:A = 36:84 = 3:7, correct.

All good.

Now, for the "how many times" answers, I’ll write them as fractions unless whole numbers, to be precise.

Some might prefer decimals, but fractions are exact.

In the worksheet, blanks are lines, so either is fine, but fractions are safer.

For example, 1/2, 4, 5/3, 1/5, 7/3, etc.

Similarly for multiple gears.

One thing: in row 3 of multiple gears, question is “If C turns one time, A will turn” — I said 5 times. Is that correct?

C=60, A=12. If C turns once, it moves 60 teeth. A has 12 teeth, so A must turn 60/12 = 5 times. Yes.

In row 4: “If B turns one time, C will turn” — B=12, C=36. B turns once → moves 12 teeth → C turns 12/36 = 1/3. Correct.

Okay.

Final Answer formatted as requested.