A toy T jumps forward or backward. In each forward jump, it moves 5' forward whereas in each backward jump, it moves 2' backward. If in 31 jumps, T moves exactly 15' forward, then what is the difference of the number of forward and backward jumps?

Correct Option: D (9)

This problem evaluates an aspirant's proficiency in formulating and solving simultaneous linear equations, a fundamental topic in UPSC CSAT aptitude tests. Let the number of forward jumps be x and the number of backward jumps be y. Based on the problem's parameters, we can establish two mathematical conditions:

1. The total number of jumps is 31: x + y = 31
2. The net movement is exactly 15 feet forward. Since each forward jump yields +5 feet and each backward jump yields -2 feet: 5x - 2y = 15

To solve the system, multiply the first equation by 2, yielding 2x + 2y = 62. Adding this to the second equation eliminates y, resulting in 7x = 77, which gives x = 11. Substituting x = 11 back into the first equation provides y = 20. The question asks for the numerical difference between the number of forward and backward jumps: |20 - 11| = 9. Thus, Option D is rigorously proven to be correct.

Why incorrect options are wrong:

• Option A (6): Incorrect because if the difference were 6 (y - x = 6 and x + y = 31), solving it would yield 2y = 37, meaning y = 18.5. A fractional number of jumps is mathematically invalid here.
• Option B (7): Incorrect because if the difference were 7, solving y - x = 7 and x + y = 31 yields y = 19 and x = 12. The net movement would then be 5(12) - 2(19) = 60 - 38 = 22 feet, which contradicts the mandated 15 feet.
• Option C (8): Incorrect as an even difference (8) combined with an odd sum (31) again results in fractional jumps (2y = 39, so y = 19.5), violating the discrete integer nature of jumps.

Concluding Takeaway: For 'net effect' word problems, systematically translate the conditions into linear equations. A helpful shortcut is to verify the parity (odd/even nature) of sums and differences; countable entities like 'jumps' must result in integers, immediately eliminating options that produce fractions.