This question requires us to find a specific value within a series of numbers when given overall averages and averages of subsets. We need to determine the value of the seventh result.

Average Calculation: Find the Seventh Result

We are given the following information:

• The total number of results is 13.
• The average of these 13 results is 68.
• The average of the first seven results is 63.
• The average of the last seven results is 70.

Average Concept Explained

The average (or mean) of a set of numbers is calculated by summing all the numbers and then dividing by the count of numbers in the set. The formula is:

$$ \text{Average} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} $$

From this, we can derive the sum of observations:

$$ \text{Sum of Observations} = \text{Average} \times \text{Number of Observations} $$

Solving the Average Problem Step-by-Step

Total Sum Calculation for 13 Results

First, let's find the total sum of all 13 results.

Sum of 13 results = Average of 13 results $\times$ 13

$$ \text{Sum}_{1-13} = 68 \times 13 $$

$$ \text{Sum}_{1-13} = 884 $$

Sum Calculation for First Seven Results

Next, we calculate the sum of the first seven results.

Sum of first 7 results = Average of first 7 results $\times$ 7

$$ \text{Sum}_{1-7} = 63 \times 7 $$

$$ \text{Sum}_{1-7} = 441 $$

Sum Calculation for Last Seven Results

Now, let's calculate the sum of the last seven results.

Sum of last 7 results = Average of last 7 results $\times$ 7

$$ \text{Sum}_{7-13} = 70 \times 7 $$

$$ \text{Sum}_{7-13} = 490 $$

Seventh Result Calculation Method

The seventh result is included in both the 'first seven' and the 'last seven' sets. Therefore, if we add the sum of the first seven results and the sum of the last seven results, the seventh result gets counted twice.

Sum of (first 7 results + last 7 results) = Sum$_{1-7}$ + Sum$_{7-13}$

$$ \text{Sum}_{1-7} + \text{Sum}_{7-13} = 441 + 490 $$

$$ \text{Sum}_{1-7} + \text{Sum}_{7-13} = 931 $$

This sum (931) includes all results from 1 to 13 once, plus the seventh result an additional time.

To find the value of the seventh result, we subtract the total sum of all 13 results (which includes each number just once) from this combined sum.

Seventh Result = (Sum of first 7 + Sum of last 7) - Total Sum of 13 results

$$ \text{Seventh Result} = (\text{Sum}_{1-7} + \text{Sum}_{7-13}) - \text{Sum}_{1-13} $$

$$ \text{Seventh Result} = 931 - 884 $$

$$ \text{Seventh Result} = 47 $$

Final Answer Verification

The calculation shows that the value of the seventh result is 47. This aligns with the options provided.