If 6/7 of 8/5 of a number is 192, what is 3/4 of that number?

Introduction / Context:
This problem combines fractions and algebra. It asks the learner to work with fractional multiples of an unknown number and then find another specific fraction of the same number. It is typical in aptitude tests to check fraction manipulation skills.

Given Data / Assumptions:

• (6/7) of (8/5) of a number equals 192.
• We must find 3/4 of that number.
• The number is assumed to be positive.

Concept / Approach:
Let the number be N. The given condition becomes (6/7) * (8/5) * N = 192. We can simplify the product of fractions and then solve for N. After that, we compute 3/4 of N directly.

Step-by-Step Solution:
Let the number be N.Given (6/7) * (8/5) * N = 192.Simplify the fraction: (6/7) * (8/5) = 48 / 35.So (48 / 35) * N = 192.Multiply both sides by 35: 48N = 192 * 35.Compute 192 * 35 = 6720.Thus N = 6720 / 48 = 140.We need 3/4 of N = 3/4 * 140 = 105.

Verification / Alternative check:
Check the initial condition with N = 140: (6/7) of (8/5) of 140 is (6/7) * (8/5) * 140. First (8/5) * 140 = 8 * 28 = 224. Then (6/7) * 224 = 6 * 32 = 192, which matches the given value, confirming N = 140 and 3/4 of N = 105.

Why Other Options Are Wrong:
Values like 77, 36 or 80 correspond to other fractions of 140 or wrong intermediate computations. Only 105 is exactly 3/4 of 140 and consistent with all given conditions.

Common Pitfalls:
Common errors include inverting fractions incorrectly, cancelling wrong terms or forgetting to multiply the right side by the denominator when solving for N. Careful stepwise handling of fractions avoids these issues.

Final Answer:
Three fourths of the number is 105.