To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that the mean and median of the five numbers [tex]\( 9, 10, x, y, 13 \)[/tex] are the same, we can follow these steps:
1. Arrange the numbers:
The numbers given are already in order: 9, 10, [tex]\( x \)[/tex], [tex]\( y \)[/tex], 13.
2. Determine the mean:
The mean of the numbers is calculated by adding all the numbers and then dividing by the total number of numbers.
[tex]\[
\text{Mean} = \frac{9 + 10 + x + y + 13}{5}
\][/tex]
3. Determine the median:
For an odd number of ordered values, the median is the middle number. Here, the five numbers are already in order, and the middle number is [tex]\( y \)[/tex].
Therefore:
[tex]\[
\text{Median} = y
\][/tex]
4. Set up the equation:
Since the problem states that the mean and the median are the same, we can set the mean equal to [tex]\( y \)[/tex].
[tex]\[
\frac{9 + 10 + x + y + 13}{5} = y
\][/tex]
5. Simplify the equation:
[tex]\[
\frac{32 + x + y}{5} = y
\][/tex]
6. Eliminate the fraction by multiplying both sides by 5:
[tex]\[
32 + x + y = 5y
\][/tex]
7. Rearrange the terms:
[tex]\[
32 + x + y = 5y \implies x + 32 = 4y \implies x = 4y - 32
\][/tex]
8. Use the position for median and additional information:
Given that the third number in the ordered list should be the median to maintain the ascending order, thus, in context to the order [tex]\(x = 10.0\)[/tex] and [tex]\(y = 10.5\)[/tex].
9. Calculate [tex]\( y \)[/tex] using the median position information:
[tex]\[
y = 10.5
\][/tex]
10. Finally, substitute [tex]\( y = 10.5 \)[/tex] into the equation [tex]\( x = 4y - 32 \)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[
x = 4(10.5) - 32
\][/tex]
[tex]\[
x = 42 - 32
\][/tex]
[tex]\[
x = 10
\][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( x = 10 \)[/tex] and [tex]\( y = 10.5 \)[/tex].
