A fitness center purchased a number of exercise machines: 4 costing $1700 each, 8 costing 1300, and X costing 1200 each, where X is a positive odd integer. If the median price for all the exercise machines purchased by the fitness center was 1300, what is the greatest possible value of X? Pls help solve it! Thx a lot!!!
The easiest way to get this one might be to recognize that the median is when you have the same number of machines at a greater or equal value and at a lesser or equal value.
If you want the median to be $1300, well, you already know all the machines that are of a greater or equal value. There are 8 $1300 machines, and 4 $1700 machines. That’s 12 machines. If your median is going to be one of the $1300 machines, there will be 11 at a greater or equal value. Which means you can purchase 11 $1200 machines (lesser value) and still have a $1300 median.
If that’s not working for you, the other, almost-as-fast way is just to list things out. Start by listing the ones you know:
$1300 $1300 $1700$1300 $1300 $1700$1300 $1300 $1700$1300 $1300 $1700
Now start writing $1200s, crossing off the list as you go:
$1200 $1300 $1300 $1700$1200 $1300 $1300 $1700$1200 $1300 $1300 $1700$1200 $1300 $1300 $1700
$1200 $1200 $1300 $1300 $1700 $1200 $1200 $1300 $1300 $1700 $1200 $1200 $1300 $1300 $1700 $1200 $1200 $1300 $1300 $1700
Almost there, right? Remember, for each $1200 you list, you have to cross off one $1300. How many more can you list before there’s only one $1300 left?
$1200 $1200 $1200 $1300 $1300 $1700 $1200 $1200 $1200 $1300 $1300 $1700 $1200 $1200 $1200 $1300 $1300 $1700 $1200 $1200 $1300 $1300 $1700
Count up your $1200s. There are 11 of them.