`1/2 of 3/4` of a number is `2 (1)/(2)` of 10.what is the number ?

Options

A. 50
B. 60
C. `66 (2)/(3)`
D. 56

Answer

To solve the question step by step, let's break it down:

### Step 1: Understand the problem
The problem states that \( \frac{1}{2} \) of \( \frac{3}{4} \) of a number \( x \) is equal to \( 2 \frac{1}{2} \) of 10.

### Step 2: Convert mixed number to improper fraction
First, we need to convert \( 2 \frac{1}{2} \) into an improper fraction.
\[ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

### Step 3: Set up the equation
Now we can set up the equation based on the problem statement:
\[ \frac{1}{2} \times \frac{3}{4} \times x = \frac{5}{2} \times 10 \]

### Step 4: Simplify the right side
Calculate \( \frac{5}{2} \times 10 \):
\[ \frac{5}{2} \times 10 = \frac{5 \times 10}{2} = \frac{50}{2} = 25 \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:
\[ \frac{1}{2} \times \frac{3}{4} \times x = 25 \]

### Step 6: Simplify the left side
Calculate \( \frac{1}{2} \times \frac{3}{4} \):
\[ \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \]
So the equation becomes:
\[ \frac{3}{8} x = 25 \]

### Step 7: Solve for \( x \)
To find \( x \), multiply both sides by the reciprocal of \( \frac{3}{8} \):
\[ x = 25 \times \frac{8}{3} \]
Calculating this gives:
\[ x = \frac{25 \times 8}{3} = \frac{200}{3} \]

### Step 8: Convert to mixed number
Now convert \( \frac{200}{3} \) into a mixed number:
\[ 200 \div 3 = 66 \quad \text{(remainder 2)} \]
Thus, \( \frac{200}{3} = 66 \frac{2}{3} \).

### Final Answer
The number is \( 66 \frac{2}{3} \).

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