A wheel that has $6$ cogs is meshed with a larger wheel of $14$ cogs. When the smaller wheel has made $21$ revolutions, then the number of revolutions made by the larger wheel is

### Concept & Chain Rule The number of cogs on a wheel is inversely proportional to the number of revolutions it makes when meshed with another wheel. $$ N_1 \times R_1 = N_2 \times R_2 $$ where $N$ is the number of cogs and $R$ is the number of revolutions. ### Step-by-Step Solution * **Given**: - Smaller wheel: $N_1 = 6$ cogs, $R_1 = 21$ revolutions. - Larger wheel: $N_2 = 14$ cogs, $R_2 = x$ revolutions. * **Calculation**: Using the formula: $$ 6 \times 21 = 14 \times x $$ $$ 126 = 14x $$ $$ x = \frac{126}{14} = 9 $$ ### Exam Strategy & Shortcut Observe the ratio of cogs: $6 : 14 = 3 : 7$. Because revolutions are inversely proportional to size, the revolutions will be in the ratio $7 : 3$. Since the smaller wheel turns $21$ times ($7 \times 3$), the larger will turn $3 \times 3 = 9$ times. ### Common Pitfall Setting up a direct proportion (which would yield $49$) instead of an inverse proportion, forgetting that a larger wheel needs fewer turns to cover the same distance. ### Final Answer Therefore, the correct answer is **9**.