To solve for the value of [tex]\( x \)[/tex] in the given list of numbers [tex]\( 20, 30, \frac{3x + 5}{2}, 50, 60 \)[/tex] where the median is 40, follow these steps:
1. Identify the Median Position: Since the list has 5 numbers, the median is the middle number. Therefore, the median here is the value at the third position in the list, which is [tex]\(\frac{3x + 5}{2}\)[/tex].
2. Set the Median Equal to 40: According to the problem, the median value is given as 40. Hence we set up the equation:
[tex]\[
\frac{3x + 5}{2} = 40
\][/tex]
3. Solve the Equation:
- First, eliminate the fraction by multiplying both sides of the equation by 2:
[tex]\[
2 \times \frac{3x + 5}{2} = 40 \times 2
\][/tex]
Simplifying this gives:
[tex]\[
3x + 5 = 80
\][/tex]
- Next, isolate [tex]\( 3x \)[/tex] by subtracting 5 from both sides:
[tex]\[
3x = 80 - 5
\][/tex]
Simplifying this gives:
[tex]\[
3x = 75
\][/tex]
- Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 3:
[tex]\[
x = \frac{75}{3}
\][/tex]
Simplifying this gives:
[tex]\[
x = 25
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{25} \)[/tex].
