The average of 13 numbers is 48.5. The average of the first five numbers is 56.4 and that of the last nine numbers is 45.6. what is the `5^(th)` number ?

Options

A. 61.7
B. 51.9
C. 61.9
D. 60.7

Answer

To find the 5th number in the given problem, we will follow these steps:

### Step 1: Calculate the total sum of the 13 numbers.
The average of the 13 numbers is given as 48.5. To find the total sum, we multiply the average by the number of items:

\[ \text{Total Sum} = \text{Average} \times \text{Number of Items} = 48.5 \times 13 \]

Calculating this gives:

\[ \text{Total Sum} = 48.5 \times 13 = 630.5 \]

### Step 2: Calculate the sum of the first 5 numbers.
The average of the first 5 numbers is given as 56.4. To find the sum of these numbers, we again multiply the average by the number of items:

\[ \text{Sum of First 5 Numbers} = 56.4 \times 5 \]

Calculating this gives:

\[ \text{Sum of First 5 Numbers} = 56.4 \times 5 = 282 \]

### Step 3: Calculate the sum of the last 9 numbers.
The average of the last 9 numbers is given as 45.6. To find the sum of these numbers, we multiply the average by the number of items:

\[ \text{Sum of Last 9 Numbers} = 45.6 \times 9 \]

Calculating this gives:

\[ \text{Sum of Last 9 Numbers} = 45.6 \times 9 = 410.4 \]

### Step 4: Set up the equation to find the 5th number.
Now, we know the total sum of all 13 numbers, the sum of the first 5 numbers, and the sum of the last 9 numbers. The 5th number is included in both the sum of the first 5 numbers and the sum of the last 9 numbers. Therefore, we can express the total sum as:

\[ \text{Total Sum} = \text{Sum of First 5 Numbers} + \text{Sum of Last 9 Numbers} - \text{5th Number} \]

Substituting the values we have:

\[ 630.5 = 282 + 410.4 - \text{5th Number} \]

### Step 5: Solve for the 5th number.
Now, we simplify the equation:

\[ 630.5 = 692.4 - \text{5th Number} \]

Rearranging gives:

\[ \text{5th Number} = 692.4 - 630.5 \]

Calculating this gives:

\[ \text{5th Number} = 61.9 \]

### Final Answer:
The 5th number is **61.9**.

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