the sum of three number is 116. the ratio of the second to the third is 9: 16 and the first to 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, we will define the variables based on the ratios provided and then use the information to find the second number.

### Step 1: Define the Variables
Let the first number be \( x \). According to the problem:
- The ratio of the first number to the third number is \( 1:4 \). Therefore, the third number can be expressed as \( 4x \).
- The ratio of the second number to the third number is \( 9:16 \). We can express the second number in terms of the third number.

### Step 2: Express the Second Number
Since the third number is \( 4x \), we can express the second number \( y \) using the ratio:
\[ \frac{y}{4x} = \frac{9}{16} \]
From this, we can find \( y \):
\[ y = \frac{9}{16} \times 4x = \frac{36x}{16} = \frac{9x}{4} \]

### Step 3: Write the Equation for the Sum of the Numbers
Now we have expressions for all three numbers:
- First number: \( x \)
- Second number: \( \frac{9x}{4} \)
- Third number: \( 4x \)

According to the problem, the sum of these three numbers is 116:
\[ x + \frac{9x}{4} + 4x = 116 \]

### Step 4: Combine the Terms
To combine the terms, we need a common denominator. The common denominator for \( x \), \( \frac{9x}{4} \), and \( 4x \) is 4. We can rewrite \( x \) as \( \frac{4x}{4} \) and \( 4x \) as \( \frac{16x}{4} \):
\[ \frac{4x}{4} + \frac{9x}{4} + \frac{16x}{4} = 116 \]
Now combine the fractions:
\[ \frac{4x + 9x + 16x}{4} = 116 \]
\[ \frac{29x}{4} = 116 \]

### Step 5: Solve for \( x \)
To eliminate the fraction, multiply both sides by 4:
\[ 29x = 116 \times 4 \]
\[ 29x = 464 \]
Now divide by 29:
\[ x = \frac{464}{29} = 16 \]

### Step 6: Find the Second Number
Now that we have \( x \), we can find the second number \( y \):
\[ y = \frac{9x}{4} = \frac{9 \times 16}{4} = \frac{144}{4} = 36 \]

### Final Answer
The second number is \( 36 \).

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