The sum of three numbers is 490. 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. 50
B. 100
C. 150
D. 200

Answer

To solve the problem step by step, we need to find the second number given the conditions of the ratios and the total sum of the numbers.

### Step 1: Set up the ratios
We are given:
- The ratio of the first number (let's call it A) to the second number (B) is \(2:3\).
- The ratio of the second number (B) to the third number (C) is \(5:8\).

From the first ratio, we can express A and B in terms of a variable \(x\):
- \(A = 2x\)
- \(B = 3x\)

From the second ratio, we can express B and C in terms of another variable \(y\):
- \(B = 5y\)
- \(C = 8y\)

### Step 2: Equalize the expressions for B
Since B is represented in both ratios, we can set the expressions equal to each other:
\[ 3x = 5y \]

### Step 3: Solve for one variable in terms of the other
From the equation \(3x = 5y\), we can express \(y\) in terms of \(x\):
\[ y = \frac{3x}{5} \]

### Step 4: Substitute y into the expression for C
Now we can express C in terms of x:
\[ C = 8y = 8 \left(\frac{3x}{5}\right) = \frac{24x}{5} \]

### Step 5: Write the sum of A, B, and C
We know the sum of the three numbers is 490:
\[ A + B + C = 490 \]
Substituting the expressions we have:
\[ 2x + 3x + \frac{24x}{5} = 490 \]

### Step 6: Combine the terms
To combine the terms, we need a common denominator:
- The common denominator for \(2x\) and \(3x\) is 5.
- So, we rewrite \(2x\) and \(3x\):
\[ 2x = \frac{10x}{5} \]
\[ 3x = \frac{15x}{5} \]

Now we can rewrite the equation:
\[ \frac{10x}{5} + \frac{15x}{5} + \frac{24x}{5} = 490 \]
Combining the fractions gives:
\[ \frac{49x}{5} = 490 \]

### Step 7: Solve for x
To solve for \(x\), multiply both sides by 5:
\[ 49x = 2450 \]
Now divide by 49:
\[ x = \frac{2450}{49} = 50 \]

### Step 8: Find the second number B
Now that we have \(x\), we can find B:
\[ B = 3x = 3 \times 50 = 150 \]

### Conclusion
The second number is **150**.

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