The sum of three numbers is 116. The ratio of second to the third is `9 : 16` and the first to the third is `1 : 4`. The second number is

Options

A. 30
B. 32
C. 34
D. 36

Answer

To solve the problem step by step, let's denote the three numbers as \( x \), \( y \), and \( z \). According to the problem:

1. The sum of the three numbers is given by:
\[ x + y + z = 116 \]

2. The ratio of the second number to the third number is:
\[ \frac{y}{z} = \frac{9}{16} \]
This implies:
\[ y = \frac{9}{16} z \]

3. The ratio of the first number to the third number is:
\[ \frac{x}{z} = \frac{1}{4} \]
This implies:
\[ x = \frac{1}{4} z \]

Now, we can substitute the expressions for \( x \) and \( y \) in terms of \( z \) into the sum equation.

### Step 1: Substitute \( x \) and \( y \) in terms of \( z \)
Substituting \( x \) and \( y \) into the sum equation:
\[ \frac{1}{4} z + \frac{9}{16} z + z = 116 \]

### Step 2: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple of 4, 16, and 1 is 16. So, we rewrite the equation:
\[ \frac{4}{16} z + \frac{9}{16} z + \frac{16}{16} z = 116 \]

### Step 3: Combine the fractions
Now, combine the fractions:
\[ \left( \frac{4 + 9 + 16}{16} \right) z = 116 \]
This simplifies to:
\[ \frac{29}{16} z = 116 \]

### Step 4: Solve for \( z \)
To isolate \( z \), multiply both sides by \( \frac{16}{29} \):
\[ z = 116 \times \frac{16}{29} \]
Calculating this gives:
\[ z = 4 \times 16 = 64 \]

### Step 5: Find \( y \)
Now that we have \( z \), we can find \( y \):
\[ y = \frac{9}{16} z = \frac{9}{16} \times 64 = 36 \]

### Conclusion
Thus, the second number \( y \) is:
\[ \boxed{36} \]