A gear 12 cm in diameter is turning a gear 18 cm in diameter. When the smaller gear has 42 revolutions, how many has the larger one made ?

Options

A. 28
B. 20
C. 15
D. 24

Answer

To solve the problem of how many revolutions the larger gear makes when the smaller gear has made 42 revolutions, we can follow these steps:

### Step 1: Find the Circumference of Both Gears
The circumference \( C \) of a circle (or gear in this case) can be calculated using the formula:
\[ C = \pi \times d \]
where \( d \) is the diameter of the gear.

- For the smaller gear (12 cm in diameter):
\[ C_{\text{small}} = \pi \times 12 = 12\pi \text{ cm} \]

- For the larger gear (18 cm in diameter):
\[ C_{\text{large}} = \pi \times 18 = 18\pi \text{ cm} \]

### Step 2: Calculate the Distance Traveled by the Smaller Gear
When the smaller gear makes 42 revolutions, the total distance it travels can be calculated as:
\[ \text{Distance}_{\text{small}} = \text{Number of Revolutions} \times C_{\text{small}} \]
Substituting the values:
\[ \text{Distance}_{\text{small}} = 42 \times 12\pi = 504\pi \text{ cm} \]

### Step 3: Determine the Number of Revolutions of the Larger Gear
Now, we need to find out how many revolutions the larger gear makes when it travels the same distance. The number of revolutions \( x \) of the larger gear can be calculated using the formula:
\[ x = \frac{\text{Distance}_{\text{small}}}{C_{\text{large}}} \]
Substituting the values:
\[ x = \frac{504\pi}{18\pi} \]

### Step 4: Simplify the Expression
The \( \pi \) cancels out:
\[ x = \frac{504}{18} \]
Now, perform the division:
\[ x = 28 \]

### Conclusion
Thus, the larger gear makes **28 revolutions** when the smaller gear has made 42 revolutions.

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