A small wheel with 6 cogs is meshed with a larger wheel that has 14 cogs. If the smaller wheel makes 21 complete revolutions, how many revolutions will the larger wheel make while the teeth remain in continuous mesh?

Introduction / Context:
This is a classic gear and cogs question which appears frequently in aptitude tests. Two wheels with different numbers of teeth mesh together, and the number of revolutions of one wheel is related to the number of revolutions of the other wheel. The idea is based on the fact that the number of tooth engagements must be the same for both wheels, so the total teeth passing a contact point must match on each wheel.

Given Data / Assumptions:
- Smaller wheel has 6 cogs or teeth.
- Larger wheel has 14 cogs or teeth.
- The smaller wheel makes 21 revolutions.
- Wheels are meshed without slipping, so tooth engagements match exactly.
- We need the number of revolutions made by the larger wheel.

Concept / Approach:
When two gears mesh, the total number of teeth that pass through the point of contact must be equal for both gears. If the smaller wheel has fewer teeth, it will make more revolutions than the larger wheel to keep the teeth count matched. Mathematically, we equate teeth passed by each wheel: teeth per revolution multiplied by number of revolutions for one wheel equals the same product for the other wheel.

Step-by-Step Solution:
Step 1: Teeth passed by the smaller wheel in one revolution = 6. Step 2: Teeth passed by the larger wheel in one revolution = 14. Step 3: If the smaller wheel makes 21 revolutions, total teeth passed by it = 6 * 21. Step 4: Let the revolutions of the larger wheel be R. Then total teeth passed by the larger wheel = 14 * R. Step 5: Since teeth engagements must match, set 6 * 21 = 14 * R. Step 6: Calculate 6 * 21 = 126, so 126 = 14 * R. Step 7: Solve for R to get R = 126 / 14 = 9.

Verification / Alternative check:
As another check, we can also consider the inverse proportionality between revolutions and number of teeth when tooth flow is equal. The ratio of revolutions is the inverse of the ratio of teeth: revolutions of smaller : revolutions of larger = 14 : 6. Given that revolutions of smaller are 21, we can set 21 : R = 14 : 6 and solve for R. This cross multiplication also leads to R = 9, confirming the initial calculation.

Why Other Options Are Wrong:
7 revolutions would result in only 14 * 7 = 98 teeth passing on the larger wheel, which does not match 6 * 21 = 126 teeth from the smaller wheel. 11 or 13 revolutions produce even more mismatched total teeth counts. Only 9 revolutions gives 14 * 9 = 126, which matches exactly the small wheel tooth count and is therefore the only consistent answer.

Common Pitfalls:
Students sometimes confuse the direct and inverse ratios, mistakenly thinking that the wheel with more teeth must make more revolutions. In reality, for the same tooth flow, the wheel with more teeth makes fewer revolutions. Another pitfall is to divide the number of teeth incorrectly or forget that both wheels must pass the same total number of teeth at the contact point.

Final Answer:
The larger wheel will make 9 complete revolutions while the smaller wheel makes 21 revolutions.