Answer :
Certainly! Let's break down the steps to find the sum of the remaining numbers when the average of 48 numbers is given and one of the numbers is explicitly known.
### Step-by-Step Solution:
1. Understanding the given information:
- The average of 48 different numbers is 64.
- One of the numbers is 64.
2. Calculate the total sum of all 48 numbers:
- The formula to find the total sum of numbers when the average is known is:
[tex]\[ \text{Total Sum} = \text{Average} \times \text{Number of Items} \][/tex]
- Here, the average is 64, and the number of items is 48.
- Therefore:
[tex]\[ \text{Total Sum} = 64 \times 48 \][/tex]
3. Given the condition:
- One of these numbers is 64.
4. Calculate the remaining sum:
- To find the sum of the remaining numbers, we need to subtract the given single number (64) from the total sum we calculated.
- Hence:
[tex]\[ \text{Remaining Sum} = \text{Total Sum} - \text{Given Number} \][/tex]
5. Performing the calculations:
- The total sum of the 48 numbers is:
[tex]\[ 64 \times 48 = 3072 \][/tex]
- The remaining sum, after subtracting the given number (64), is:
[tex]\[ 3072 - 64 = 3008 \][/tex]
### Final Answer:
The sum of the remaining numbers, excluding the one number which is 64, is 3008.
### Step-by-Step Solution:
1. Understanding the given information:
- The average of 48 different numbers is 64.
- One of the numbers is 64.
2. Calculate the total sum of all 48 numbers:
- The formula to find the total sum of numbers when the average is known is:
[tex]\[ \text{Total Sum} = \text{Average} \times \text{Number of Items} \][/tex]
- Here, the average is 64, and the number of items is 48.
- Therefore:
[tex]\[ \text{Total Sum} = 64 \times 48 \][/tex]
3. Given the condition:
- One of these numbers is 64.
4. Calculate the remaining sum:
- To find the sum of the remaining numbers, we need to subtract the given single number (64) from the total sum we calculated.
- Hence:
[tex]\[ \text{Remaining Sum} = \text{Total Sum} - \text{Given Number} \][/tex]
5. Performing the calculations:
- The total sum of the 48 numbers is:
[tex]\[ 64 \times 48 = 3072 \][/tex]
- The remaining sum, after subtracting the given number (64), is:
[tex]\[ 3072 - 64 = 3008 \][/tex]
### Final Answer:
The sum of the remaining numbers, excluding the one number which is 64, is 3008.
