The sum of three numbers is 505. If the ratio of the first number to the second number is 3 : 5 and that of the second number to the third number is 7 : 9, then what is the second number?

Options

A. 135
B. 170
C. 175
D. 140

Answer

To solve the problem step by step, we will follow the given information and derive the second number systematically.

### Step 1: Define the Variables
Let the three numbers be represented as:
- First number = \( a \)
- Second number = \( b \)
- Third number = \( c \)

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the three numbers is:
\[ a + b + c = 505 \]
2. The ratio of the first number to the second number is:
\[ \frac{a}{b} = \frac{3}{5} \quad \Rightarrow \quad a = \frac{3}{5}b \]
3. The ratio of the second number to the third number is:
\[ \frac{b}{c} = \frac{7}{9} \quad \Rightarrow \quad c = \frac{9}{7}b \]

### Step 3: Substitute the Ratios into the Sum Equation
Substituting the expressions for \( a \) and \( c \) in terms of \( b \) into the sum equation:
\[ \frac{3}{5}b + b + \frac{9}{7}b = 505 \]

### Step 4: Find a Common Denominator
To simplify the equation, we need a common denominator for the fractions. The least common multiple of 5 and 7 is 35. Thus, we rewrite each term:
- \( \frac{3}{5}b = \frac{21}{35}b \)
- \( b = \frac{35}{35}b \)
- \( \frac{9}{7}b = \frac{45}{35}b \)

Now, substituting these into the equation gives:
\[ \frac{21}{35}b + \frac{35}{35}b + \frac{45}{35}b = 505 \]

### Step 5: Combine the Terms
Combining the fractions:
\[ \frac{21 + 35 + 45}{35}b = 505 \]
\[ \frac{101}{35}b = 505 \]

### Step 6: Solve for \( b \)
To isolate \( b \), multiply both sides by \( \frac{35}{101} \):
\[ b = 505 \times \frac{35}{101} \]
Calculating this gives:
\[ b = 505 \times \frac{35}{101} = 175 \]

### Conclusion
Thus, the second number \( b \) is:
\[ \boxed{175} \]