This problem requires us to find a specific value (the fifth result) when we know the average of a group of numbers and the average of a subset of that group. We can solve this by understanding how averages relate to the total sum of the numbers.

Average Concept Explained

The fundamental concept of an average, also known as the mean, is calculated by summing up all the values in a dataset and then dividing by the total count of those values. The formula is:

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

From this basic formula, we can derive a way to calculate the sum of the values if we know the average and the number of values:

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

Calculating Total Sums

We are given two key pieces of information:

• The average of 5 results is 46.
• The average of the first 4 results is 45.

Using the formula derived above, we can calculate the total sum for both scenarios.

Sum of the 5 results:

$$ \text{Sum}_{5 \text{ results}} = 46 \times 5 $$

$$ \text{Sum}_{5 \text{ results}} = 230 $$

Sum of the first 4 results:

$$ \text{Sum}_{4 \text{ results}} = 45 \times 4 $$

$$ \text{Sum}_{4 \text{ results}} = 180 $$

Determining the Fifth Result

The total sum of the 5 results includes the sum of the first 4 results plus the fifth result itself. Therefore, to find the value of the fifth result, we simply subtract the sum of the first 4 results from the total sum of the 5 results.

$$ \text{Fifth Result} = \text{Sum}_{5 \text{ results}} - \text{Sum}_{4 \text{ results}} $$

$$ \text{Fifth Result} = 230 - 180 $$

$$ \text{Fifth Result} = 50 $$

So, the value of the fifth result is 50.