Question

4/15 of 5/7 of a number is greater than 4/9 of 2/5 of the same number by 8. What is half of that number?

• A. 112
• B. 216
• C. 315
• D. 412

Hint:

Always read the final question carefully: "What is half of that number?". It is easy to accidentally select the full number 630 if it were an option.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires setting up a single-variable linear equation representing fractional parts of an unknown number.
Detailed Explanation:
Let the unknown number be $x$.
First, express the two fractional parts of $x$:
- First part: \[ \frac{4}{15} \times \frac{5}{7} \times x = \frac{4}{21} x \] - Second part: \[ \frac{4}{9} \times \frac{2}{5} \times x = \frac{8}{45} x \] According to the problem, the first part is greater than the second part by 8: \[ \frac{4}{21} x - \frac{8}{45} x = 8 \] To solve, find the Least Common Multiple (LCM) of 21 and 45: - $21 = 3 \times 7$
- $45 = 3^2 \times 5$
- $\text{LCM} = 3^2 \times 5 \times 7 = 315$
Convert the fractions: \[ \left( \frac{4 \times 15}{315} - \frac{8 \times 7}{315} \right) x = 8 \] \[ \left( \frac{60 - 56}{315} \right) x = 8 \] \[ \frac{4}{315} x = 8 \] \[ x = 2 \times 315 = 630 \] The problem asks for half of that number: \[ \text{Half of the number} = \frac{x}{2} = \frac{630}{2} = 315 \]

Step 2: Final Answer:
Half of the number is 315, which corresponds to Option (C).