A small wheel with 6 cogs is meshed with a larger wheel with 14 cogs. When the smaller wheel makes 21 revolutions, how many revolutions does the larger wheel make?

Introduction / Context:
This question deals with meshed gears or toothed wheels, a practical application of inverse proportion. When two cog wheels mesh, the number of teeth and the number of revolutions they make are inversely proportional, because the same number of teeth must pass through the point of contact. This concept is used in mechanical design and appears in aptitude tests under chain rule or ratio topics.

Given Data / Assumptions:

• The small wheel has 6 cogs (teeth).
• The large wheel has 14 cogs.
• The wheels are meshed without slipping.
• The small wheel makes 21 complete revolutions.
• We need the number of revolutions of the large wheel.

Concept / Approach:
When gears mesh, teeth passing a given point are the same for both wheels, so total teeth moved = number of revolutions * number of cogs. For the same contact, the product of cogs and revolutions is equal for both wheels. From this we can see that revolutions are inversely proportional to the number of cogs: more teeth mean fewer revolutions for the same tooth travel.

Step-by-Step Solution:
Step 1: Let R1 be the revolutions of the small wheel and R2 for the large wheel.Step 2: Number of cogs on the small wheel = 6, on the large wheel = 14.Step 3: Teeth moved by small wheel = R1 * 6.Step 4: Teeth moved by large wheel = R2 * 14.Step 5: Since they are meshed, R1 * 6 = R2 * 14.Step 6: Given R1 = 21, so 21 * 6 = R2 * 14, which gives R2 = (21 * 6) / 14.Step 7: Simplify: 21 / 7 = 3, 14 / 7 = 2, so R2 = (3 * 6) / 2 = 18 / 2 = 9 revolutions.

Verification / Alternative check:
Use direct inverse proportion. Revolutions are in the ratio of the opposite number of cogs: R1 : R2 = 14 : 6. Since R1 is 21, we set 21 : R2 = 14 : 6. Cross multiply to get 21 * 6 = 14 * R2, which again leads to R2 = 9. This quick ratio approach confirms the detailed calculation.

Why Other Options Are Wrong:
Values like 4 or 6 are too small and do not satisfy the equality of tooth travel. Values like 12 or 14 are too large and would correspond to fewer teeth per wheel than actually given. Only 9 revolutions matches the correct inverse proportion relationship.

Common Pitfalls:

• Assuming revolutions are directly proportional to the number of cogs instead of inversely proportional.
• Multiplying the number of cogs instead of forming the correct ratio.
• Simple arithmetic mistakes in simplifying (21 * 6) / 14.

Final Answer:
The larger wheel makes 9 revolutions when the smaller wheel makes 21 revolutions.