The sum of three numbers is 98. If the ratio of the first to the second is 2 : 3 and that of the second to the third is 5 : 8, then the second number is:

Options

A. 20
B. 30
C. 38
D. 48

Answer

To solve the problem step by step, we will follow these instructions:

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

### Step 2: Set Up the Equations
According to the problem, we have the following information:
1. The sum of the three numbers is 98:
\[ a + b + c = 98 \]
2. The ratio of the first number to the second number is \( 2:3 \):
\[ \frac{a}{b} = \frac{2}{3} \implies a = \frac{2}{3}b \]
3. The ratio of the second number to the third number is \( 5:8 \):
\[ \frac{b}{c} = \frac{5}{8} \implies c = \frac{8}{5}b \]

### Step 3: Substitute the Values
Now, substitute the expressions for \( a \) and \( c \) in terms of \( b \) into the sum equation:
\[ \frac{2}{3}b + b + \frac{8}{5}b = 98 \]

### Step 4: Find a Common Denominator
To simplify the equation, we need a common denominator for the fractions. The least common multiple of 3 and 5 is 15. Rewrite the equation:
\[ \frac{10}{15}b + \frac{15}{15}b + \frac{24}{15}b = 98 \]

### Step 5: Combine the Terms
Combine the terms on the left side:
\[ \left(\frac{10 + 15 + 24}{15}\right)b = 98 \]
\[ \frac{49}{15}b = 98 \]

### Step 6: Solve for \( b \)
Now, multiply both sides by \( \frac{15}{49} \):
\[ b = 98 \times \frac{15}{49} \]
\[ b = 2 \times 15 = 30 \]

### Step 7: Conclusion
The second number \( b \) is \( 30 \).