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

Options

A. A) 10
B. B) 20
C. C) 30
D. D) 40

Answer

To solve the problem step by step, we will denote the three numbers as A, B, and C. We are given the following information:

1. The sum of the three numbers is 98:
\[ A + B + C = 98 \]

2. The ratio of the first number (A) to the second number (B) is 2:3:
\[ \frac{A}{B} = \frac{2}{3} \]
This implies:
\[ A = \frac{2}{3}B \]

3. The ratio of the second number (B) to the third number (C) is 5:8:
\[ \frac{B}{C} = \frac{5}{8} \]
This implies:
\[ C = \frac{8}{5}B \]

Now, we will substitute the expressions for A and C in terms of B into the sum equation.

### Step 1: Substitute A and C in terms of B
Substituting \( A \) and \( C \) into the sum equation:
\[ \frac{2}{3}B + B + \frac{8}{5}B = 98 \]

### Step 2: Find a common denominator
To combine the terms, we need a common denominator. The denominators are 3 and 5, so the least common multiple is 15.

Rewriting each term:
\[ \frac{2}{3}B = \frac{10}{15}B, \quad B = \frac{15}{15}B, \quad \frac{8}{5}B = \frac{24}{15}B \]

Now, substituting these back into the equation:
\[ \frac{10}{15}B + \frac{15}{15}B + \frac{24}{15}B = 98 \]

### Step 3: Combine the fractions
Now combine the fractions:
\[ \frac{10 + 15 + 24}{15}B = 98 \]
\[ \frac{49}{15}B = 98 \]

### Step 4: Solve for B
To isolate B, multiply both sides by \( \frac{15}{49} \):
\[ B = 98 \times \frac{15}{49} \]

### Step 5: Simplify the multiplication
Calculating the right side:
\[ B = 2 \times 15 = 30 \]

Thus, the second number \( B \) is 30.

### Final Answer
The second number is:
\[ \boxed{30} \]

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