The sum of three numbers is 540. The ratio of second to third is `9: 13` and that of first to third is `2 : 7`. The third number is :

Options

A. 273
B. 280
C. 250
D. 286

Answer

To solve the problem step by step, we will first define the three numbers based on the given ratios and then find the third number.

### Step 1: Define the Variables
Let the three numbers be:
- First number = A
- Second number = B
- Third number = C

### Step 2: Set Up the Ratios
From the problem, we know:
1. The ratio of the second number to the third number (B:C) is 9:13.
2. The ratio of the first number to the third number (A:C) is 2:7.

### Step 3: Express B and A in Terms of C
From the ratios, we can express B and A in terms of C:
- From the ratio B:C = 9:13, we can write:
\[ B = \frac{9}{13}C \]

- From the ratio A:C = 2:7, we can write:
\[ A = \frac{2}{7}C \]

### Step 4: Set Up the Equation for the Sum
According to the problem, the sum of the three numbers is 540:
\[ A + B + C = 540 \]
Substituting the expressions for A and B in terms of C:
\[ \frac{2}{7}C + \frac{9}{13}C + C = 540 \]

### Step 5: Find a Common Denominator
To combine the fractions, we need a common denominator. The least common multiple of 7 and 13 is 91. We will convert each term:
- \(\frac{2}{7}C = \frac{26}{91}C\)
- \(\frac{9}{13}C = \frac{63}{91}C\)
- \(C = \frac{91}{91}C\)

Now, we can rewrite the equation:
\[ \frac{26}{91}C + \frac{63}{91}C + \frac{91}{91}C = 540 \]

### Step 6: Combine the Terms
Combining the terms gives:
\[ \frac{26 + 63 + 91}{91}C = 540 \]
\[ \frac{180}{91}C = 540 \]

### Step 7: Solve for C
To find C, multiply both sides by \(\frac{91}{180}\):
\[ C = 540 \times \frac{91}{180} \]
Calculating this gives:
\[ C = 540 \times \frac{91}{180} = 540 \times 0.505555 \approx 273 \]

### Conclusion
Thus, the third number \(C\) is:
\[ \boxed{273} \]