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The average of 15 numbers is 56. The ave...

The average of 15 numbers is 56. The average of first 8 numbers is 54.4 and that of last 8 numbers is 56.6. If `8^(th)` number is excluded then what is the average of the remaining numbers? (correct to one decimal place)

A

56.6

B

53.8

C

53 .6

D

56 .2

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will follow the mathematical principles of averages and sums. ### Step 1: Calculate the sum of the 15 numbers Given that the average of 15 numbers is 56, we can calculate the total sum of these numbers using the formula: \[ \text{Sum} = \text{Average} \times \text{Number of quantities} \] So, \[ \text{Sum of 15 numbers} = 56 \times 15 = 840 \] ### Step 2: Calculate the sum of the first 8 numbers The average of the first 8 numbers is given as 54.4. Therefore, we can calculate their total sum: \[ \text{Sum of first 8 numbers} = 54.4 \times 8 = 435.2 \] ### Step 3: Calculate the sum of the last 8 numbers The average of the last 8 numbers is given as 56.6. Thus, we find their total sum: \[ \text{Sum of last 8 numbers} = 56.6 \times 8 = 452.8 \] ### Step 4: Identify the 8th number Since the first 8 numbers and the last 8 numbers overlap at the 8th number, we can find the 8th number by using the sums calculated above. The total of the first 8 and last 8 numbers includes the 8th number twice. Therefore, we can set up the equation: \[ \text{Sum of first 8 numbers} + \text{Sum of last 8 numbers} - \text{Sum of 15 numbers} = \text{8th number} \] Substituting the values we calculated: \[ 435.2 + 452.8 - 840 = \text{8th number} \] Calculating this gives: \[ 888 - 840 = 48 \] So, the 8th number is 48. ### Step 5: Calculate the new sum after excluding the 8th number Now, we need to find the sum of the remaining numbers after excluding the 8th number: \[ \text{New Sum} = \text{Sum of 15 numbers} - \text{8th number} = 840 - 48 = 792 \] ### Step 6: Calculate the new average Now, we find the average of the remaining 14 numbers: \[ \text{New Average} = \frac{\text{New Sum}}{\text{Number of remaining quantities}} = \frac{792}{14} \] Calculating this gives: \[ \text{New Average} = 56.57142857 \approx 56.6 \text{ (to one decimal place)} \] ### Final Answer The average of the remaining numbers, correct to one decimal place, is **56.6**. ---
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