If 4/5 of 6/7 of a number is equal to 216, then what is 8/9 of that number?

Introduction / Context:
This question checks understanding of fractions of a number and the ability to work backward from a given fractional value. You are given that a composite fraction of a number equals 216, and you must find another fraction of the same number. It is a typical algebraic fractions problem appearing in aptitude tests.

Given Data / Assumptions:
- 4/5 of 6/7 of a number N is equal to 216.
- We must find the value of 8/9 of N.

Concept / Approach:
The statement 4/5 of 6/7 of N equals 216 can be written as:
(4 / 5) * (6 / 7) * N = 216. From this, we solve for N. After obtaining N, we compute (8 / 9) * N. The key is careful manipulation of fractions and simplification to avoid arithmetic mistakes.

Step-by-Step Solution:
We have (4 / 5) * (6 / 7) * N = 216. First simplify the fraction (4 / 5) * (6 / 7) = 24 / 35. So (24 / 35) * N = 216. To solve for N, multiply both sides by 35 / 24. N = 216 * (35 / 24). Simplify 216 / 24 = 9. Thus N = 9 * 35 = 315. Now compute 8 / 9 of N. 8 / 9 of N = (8 / 9) * 315. Compute 315 / 9 = 35. So 8 / 9 of N = 8 * 35 = 280.

Verification / Alternative check:
We can verify by plugging N = 315 back into the original condition. Compute 6/7 of 315: since 315 / 7 = 45, 6/7 of 315 = 6 * 45 = 270. Then 4/5 of 270 is (4 / 5) * 270 = 4 * 54 = 216, which matches the given value. This confirms that N = 315 is correct, and therefore 8/9 of N = 280 is also correct.

Why Other Options Are Wrong:
179, 160, 149, and 200: None of these values matches 8/9 of 315. They result from incorrect handling of fractions or miscalculation while solving for N. Only 280 is consistent with both the given condition and the derived value of N.

Common Pitfalls:
A common mistake is to misinterpret "4/5 of 6/7 of a number" and either add or average the fractions instead of multiplying them. Another error is to make arithmetic mistakes when simplifying 216 * 35 / 24. It helps to cancel common factors early to keep numbers small and manageable, such as reducing 216 / 24 to 9 before multiplying by 35.

Final Answer:
The value of 8/9 of the number is 280.