A gear 15 cm in diameter is turning a gear 21 cm in diameter. When the smaller gear has 105 revolutions, how many has the larger one made?

The correct answer is 75.

To find the number of revolutions the larger gear makes, we can analyze the relationship between the gear diameters and their revolutions.

When two gears are in mesh and turning each other, the total distance traveled along their outer edges (the pitch circles) must be the same. The distance a gear's edge travels in one revolution is equal to its circumference. Therefore, the total distance traveled by a gear is equal to its circumference multiplied by the number of revolutions it makes.

Let:
- $D_1$ be the diameter of the smaller gear = 15 cm
- $R_1$ be the number of revolutions of the smaller gear = 105
- $D_2$ be the diameter of the larger gear = 21 cm
- $R_2$ be the number of revolutions of the larger gear

The circumference of a gear is given by $\pi D$. Since the distance traveled by both gears is equal:

$\pi D_1{R}_1=\pi D_2{R}_2$

We can simplify this equation by dividing both sides by $\pi$:

$D_1{R}_1=D_2{R}_2$

This shows that the number of revolutions is inversely proportional to the diameter of the gear. Now, we substitute the known values into the formula to solve for $R_2$:

$15\times 105=21\times R_2$

To isolate $R_2$, we divide both sides by 21:

$R_2=\frac{15\times 105}{21}$

We can simplify the fraction by dividing 105 by 21, which equals 5:

$R_2=15\times 5$

$R_2=75$

Thus, the larger gear makes 75 revolutions.