The sum of three numbers is 162. If the ratio of the first number to the second number is `5 : 7 `and that of the second number to the third number is `5: 3`, then what is the second number?

Options

A. 35
B. 70
C. 80
D. 40

Answer

To solve the problem step by step, we will follow the given ratios and the sum of the numbers.

### Step 1: Define the Variables
Let the first number be \( A \), the second number be \( B \), and the third number be \( C \). According to the problem:
- The ratio of the first number to the second number is \( 5:7 \).
- The ratio of the second number to the third number is \( 5:3 \).

### Step 2: Express the Numbers in Terms of a Common Variable
From the ratio \( 5:7 \), we can express:
\[ A = 5x \quad \text{and} \quad B = 7x \]
From the ratio \( 5:3 \), we can express:
\[ B = 5y \quad \text{and} \quad C = 3y \]

### Step 3: Equate the Two Expressions for \( B \)
Since both expressions represent \( B \), we can set them equal to each other:
\[ 7x = 5y \]
From this equation, we can express \( y \) in terms of \( x \):
\[ y = \frac{7}{5}x \]

### Step 4: Substitute \( y \) into the Expression for \( C \)
Now, substitute \( y \) into the expression for \( C \):
\[ C = 3y = 3 \left(\frac{7}{5}x\right) = \frac{21}{5}x \]

### Step 5: Write the Sum of the Numbers
According to the problem, the sum of the three numbers is 162:
\[ A + B + C = 162 \]
Substituting the expressions we have:
\[ 5x + 7x + \frac{21}{5}x = 162 \]

### Step 6: Combine Like Terms
Combine the terms on the left side:
\[ 12x + \frac{21}{5}x = 162 \]
To combine these, we can convert \( 12x \) into a fraction with a denominator of 5:
\[ \frac{60}{5}x + \frac{21}{5}x = 162 \]
This simplifies to:
\[ \frac{81}{5}x = 162 \]

### Step 7: Solve for \( x \)
To solve for \( x \), multiply both sides by 5:
\[ 81x = 810 \]
Now divide by 81:
\[ x = 10 \]

### Step 8: Find the Value of \( B \)
Now that we have \( x \), we can find \( B \):
\[ B = 7x = 7 \times 10 = 70 \]

### Conclusion
The second number \( B \) is \( 70 \).

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