The number that should be added to both the so that the numerator and denominator of `(4^(2))/(9^(2))`, so that the fraction becomes `(4)/(9)` is

Options

A. 0
B. 16
C. 36
D. 81

Answer

To solve the problem, we need to find the number \( x \) that should be added to both the numerator and denominator of the fraction \( \frac{4^2}{9^2} \) so that it becomes \( \frac{4}{9} \).

### Step-by-Step Solution:

1. **Identify the original fraction**:
The original fraction is \( \frac{4^2}{9^2} \).
\[ \frac{4^2}{9^2} = \frac{16}{81} \]

2. **Set up the equation**:
We want to add \( x \) to both the numerator and the denominator. This gives us the new fraction:
\[ \frac{16 + x}{81 + x} \]
We want this fraction to equal \( \frac{4}{9} \):
\[ \frac{16 + x}{81 + x} = \frac{4}{9} \]

3. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives us:
\[ 9(16 + x) = 4(81 + x) \]

4. **Distribute both sides**:
Expanding both sides:
\[ 144 + 9x = 324 + 4x \]

5. **Rearrange the equation**:
To isolate \( x \), we move all terms involving \( x \) to one side and constant terms to the other side:
\[ 9x - 4x = 324 - 144 \]
This simplifies to:
\[ 5x = 180 \]

6. **Solve for \( x \)**:
Divide both sides by 5:
\[ x = \frac{180}{5} = 36 \]

7. **Conclusion**:
The number that should be added to both the numerator and denominator is \( 36 \).

### Final Answer:
The number that should be added is **36**.

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