Question:
The average of the first 7 numbers in a series is 60. When the 8th number is added, the average of the first 8 numbers becomes 63. The 9th number is 11 more than the 8th number. It is also given that the average of the 2nd to the 9th numbers is 66. Find the value of the 1st number in the series.
The average of the first 7 numbers in a series is 60. When the 8th number is added, the average of the first 8 numbers becomes 63. The 9th number is 11 more than the 8th number. It is also given that the average of the 2nd to the 9th numbers is 66. Find the value of the 1st number in the series.
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In problems involving consecutive averages, the difference in sums directly gives the value of the newly added member. For example, Sum(8 numbers) - Sum(7 numbers) = 8th number.
Updated On: Nov 30, 2025
- 68
- 70
- 71
- 75
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We are given a series of 9 numbers and information about the averages of different subsets of these numbers. Our goal is to use this information to find the value of the first number.
Step 2: Key Formula or Approach:
The fundamental relationship used here is: Sum of observations = Average × Number of observations.
Step 3: Detailed Explanation:
Let the numbers in the series be \(N_1, N_2, \ldots, N_9\).
Information 1: The average of the first 7 numbers is 60.
Sum of the first 7 numbers (\(S_7\)) = \(7 \times 60 = 420\).
\[ N_1 + N_2 + N_3 + N_4 + N_5 + N_6 + N_7 = 420 \] Information 2: The average of the first 8 numbers is 63.
Sum of the first 8 numbers (\(S_8\)) = \(8 \times 63 = 504\).
\[ N_1 + N_2 + \ldots + N_8 = 504 \] From this, we can find the 8th number, \(N_8\):
\[ N_8 = S_8 - S_7 = 504 - 420 = 84 \] Information 3: The 9th number is 11 more than the 8th number.
\[ N_9 = N_8 + 11 = 84 + 11 = 95 \] Information 4: The average of the 2nd to the 9th numbers is 66.
There are 8 numbers from the 2nd to the 9th (\(N_2, \ldots, N_9\)).
Sum of numbers from 2nd to 9th (\(S_{2-9}\)) = \(8 \times 66 = 528\).
\[ N_2 + N_3 + \ldots + N_9 = 528 \] Now we have two expressions for sums involving \(N_2\) to \(N_7\):
From \(S_8\), we have \(N_1 + (N_2 + \ldots + N_8) = 504\).
We know \(S_{2-9}\) can be written as \((N_2 + \ldots + N_8) + N_9 = 528\).
Using \(N_9 = 95\), we can find the sum \((N_2 + \ldots + N_8)\):
\[ (N_2 + \ldots + N_8) + 95 = 528 \] \[ (N_2 + \ldots + N_8) = 528 - 95 = 433 \] Now substitute this back into the equation for \(S_8\):
\[ N_1 + 433 = 504 \] \[ N_1 = 504 - 433 = 71 \] Step 4: Final Answer
The value of the 1st number in the series is 71.
We are given a series of 9 numbers and information about the averages of different subsets of these numbers. Our goal is to use this information to find the value of the first number.
Step 2: Key Formula or Approach:
The fundamental relationship used here is: Sum of observations = Average × Number of observations.
Step 3: Detailed Explanation:
Let the numbers in the series be \(N_1, N_2, \ldots, N_9\).
Information 1: The average of the first 7 numbers is 60.
Sum of the first 7 numbers (\(S_7\)) = \(7 \times 60 = 420\).
\[ N_1 + N_2 + N_3 + N_4 + N_5 + N_6 + N_7 = 420 \] Information 2: The average of the first 8 numbers is 63.
Sum of the first 8 numbers (\(S_8\)) = \(8 \times 63 = 504\).
\[ N_1 + N_2 + \ldots + N_8 = 504 \] From this, we can find the 8th number, \(N_8\):
\[ N_8 = S_8 - S_7 = 504 - 420 = 84 \] Information 3: The 9th number is 11 more than the 8th number.
\[ N_9 = N_8 + 11 = 84 + 11 = 95 \] Information 4: The average of the 2nd to the 9th numbers is 66.
There are 8 numbers from the 2nd to the 9th (\(N_2, \ldots, N_9\)).
Sum of numbers from 2nd to 9th (\(S_{2-9}\)) = \(8 \times 66 = 528\).
\[ N_2 + N_3 + \ldots + N_9 = 528 \] Now we have two expressions for sums involving \(N_2\) to \(N_7\):
From \(S_8\), we have \(N_1 + (N_2 + \ldots + N_8) = 504\).
We know \(S_{2-9}\) can be written as \((N_2 + \ldots + N_8) + N_9 = 528\).
Using \(N_9 = 95\), we can find the sum \((N_2 + \ldots + N_8)\):
\[ (N_2 + \ldots + N_8) + 95 = 528 \] \[ (N_2 + \ldots + N_8) = 528 - 95 = 433 \] Now substitute this back into the equation for \(S_8\):
\[ N_1 + 433 = 504 \] \[ N_1 = 504 - 433 = 71 \] Step 4: Final Answer
The value of the 1st number in the series is 71.
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