If eight fifth of four third of a number is equal to 192, then what is five sixth of that number?

Introduction / Context:
This question checks understanding of fractional multipliers applied to an unknown number. Instead of giving the number directly, the problem tells us what happens when the number is multiplied by certain fractions. From that information we must recover the original number and then compute another fractional part of it. This is a standard type of question in school mathematics and competitive exams to test comfort with fractions and algebra.

Given Data / Assumptions:

• Let the unknown number be N.
• Eight fifth of four third of N equals 192.
• This means (8 / 5) * (4 / 3) * N = 192.
• We must find five sixth of N, that is (5 / 6) * N.

Concept / Approach:
The idea is to simplify the combined fractional multiplier (8 / 5) * (4 / 3), solve the resulting equation for N, and then multiply N by 5 / 6. Doing this step by step avoids mistakes with the fractions. This approach is much faster than guessing or trying random values for N.

Step-by-Step Solution:
We are given (8 / 5) * (4 / 3) * N = 192.First simplify the product of the fractions: (8 / 5) * (4 / 3) = (8 * 4) / (5 * 3) = 32 / 15.So (32 / 15) * N = 192.To solve for N, multiply both sides by the reciprocal 15 / 32.N = 192 * (15 / 32).Simplify 192 / 32 = 6.Thus N = 6 * 15 = 90.We need five sixth of N, which is (5 / 6) * N.Compute (5 / 6) * 90 = 5 * 15 = 75.Therefore, five sixth of the number is 75.

Verification / Alternative check:
Check with N = 90.Four third of 90 is (4 / 3) * 90 = 4 * 30 = 120.Eight fifth of 120 is (8 / 5) * 120 = 8 * 24 = 192, which matches the given condition.Next, five sixth of 90 is (5 / 6) * 90 = 75, confirming the final result.

Why Other Options Are Wrong:
192 is the value of the combined fractional expression, not five sixth of the number, so option 192 is misleading.180 and 65 do not equal (5 / 6) * 90 and arise from incorrect fraction handling or arithmetic errors.Only 75 is consistent with both the given equality and the required fraction.

Common Pitfalls:
A frequent mistake is to add the fractions instead of multiplying them.Some learners invert the fraction incorrectly when solving for N, which leads to N = 192 * (32 / 15) instead of 192 * (15 / 32).Another issue is failing to simplify 192 and 32 before multiplying, which makes the arithmetic harder and more error prone.

Final Answer:
Five sixth of the number is 75.