If difference between them `4/5 " of " 3/4` of a number and `2/5 " of " 1/6` of the same number is 648, then number is
A)1110
B)1215
C)1325
D)1440

Options

A. 1110
B. 1215
C. 1325
D. 1440

Answer

To solve the problem, we need to find a number \( x \) such that the difference between \( \frac{4}{5} \) of \( \frac{3}{4} \) of \( x \) and \( \frac{2}{5} \) of \( \frac{1}{6} \) of \( x \) equals 648. Let's break this down step by step.

### Step 1: Express the two parts mathematically
1. Calculate \( \frac{4}{5} \) of \( \frac{3}{4} \) of \( x \):
\[ \frac{4}{5} \times \frac{3}{4} \times x = \frac{4 \times 3}{5 \times 4} \times x = \frac{3}{5} x \]

2. Calculate \( \frac{2}{5} \) of \( \frac{1}{6} \) of \( x \):
\[ \frac{2}{5} \times \frac{1}{6} \times x = \frac{2 \times 1}{5 \times 6} \times x = \frac{2}{30} x = \frac{1}{15} x \]

### Step 2: Set up the equation based on the problem statement
The difference between these two expressions is given as 648:
\[ \frac{3}{5} x - \frac{1}{15} x = 648 \]

### Step 3: Find a common denominator and simplify
To combine the fractions, we need a common denominator. The least common multiple of 5 and 15 is 15. Rewrite \( \frac{3}{5} x \):
\[ \frac{3}{5} x = \frac{9}{15} x \]
Now, substitute this back into the equation:
\[ \frac{9}{15} x - \frac{1}{15} x = 648 \]
Combine the fractions:
\[ \frac{9 - 1}{15} x = 648 \]
\[ \frac{8}{15} x = 648 \]

### Step 4: Solve for \( x \)
To isolate \( x \), multiply both sides by \( \frac{15}{8} \):
\[ x = 648 \times \frac{15}{8} \]
Calculating the right-hand side:
\[ x = 648 \times 1.875 = 1215 \]

### Final Answer
Thus, the number is \( \boxed{1215} \).