To solve the problem, we need to find the sixth number in a set of six numbers, given that the mean of the entire set is 60 and the mean of the first five numbers is 50. ### Step-by-Step Explanation 1. **Understanding the Mean**: The mean (or average) of a set of numbers is calculated by dividing the sum of all the numbers by the count of the numbers. In mathematical terms, if we have a set of numbers \( x_1, x_2, x_3, \ldots, x_n \), the mean \( \mu \) is given by: \[ \mu = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \] 2. **Calculating the Total Sum of the Six Numbers**: We know that the mean of the six numbers is 60. Since there are six numbers, we can find the total sum of these numbers: \[ \text{Total Sum of Six Numbers} = \text{Mean} \times \text{Count} = 60 \times 6 = 360 \] 3. **Calculating the Total Sum of the First Five Numbers**: We are also given that the mean of the first five numbers is 50. Therefore, we can calculate the total sum of these five numbers: \[ \text{Total Sum of First Five Numbers} = \text{Mean} \times \text{Count} = 50 \times 5 = 250 \] 4. **Finding the Sixth Number**: Let’s denote the sixth number as \( x_6 \). We know that the total sum of all six numbers is the sum of the first five numbers plus the sixth number: \[ \text{Total Sum of Six Numbers} = \text{Total Sum of First Five Numbers} + x_6 \] Substituting the values we calculated: \[ 360 = 250 + x_6 \] To find \( x_6 \), we can rearrange the equation: \[ x_6 = 360 - 250 = 110 \] ### Conclusion The sixth number in the set is **110**. ### Explanation of Other Options - **Option A: 105** - This is incorrect because if the sixth number were 105, the total sum would be \( 250 + 105 = 355 \), which does not equal 360. - **Option B: 100** - This is also incorrect. If the sixth number were 100, the total sum would be \( 250 + 100 = 350 \), which is again not equal to 360. - **Option C: 95** - This option is incorrect as well. If the sixth number were 95, the total sum would be \( 250 + 95 = 345 \), which does not match the required total of 360. - **Option D: 110** - This is the correct answer, as shown in our calculations. ### Revision Summary - The mean is calculated by dividing the total sum by the number of items. - To find the sixth number, calculate the total sum of the six numbers and subtract the sum of the first five. - Always check each option against the total sum to confirm correctness. - The correct sixth number in this case is 110.