For a certain set of five numbers, the mean of all but the largest is 80, and the mean of all but the smallest number is 90. What is the range of the set of five numbers?
if the numbers a,b,c,d,e are sorted in increasing order, then
(a+b+c+d)/4 = 80
(b+c+d+e)/4 = 90
equating the expressions for (b+c+d)/4, we have
80 - a/4 = 90 - e/4
e/4 - a/4 = 10
e-a = 40
the range is thus 40
320,360
To find the range of a set of numbers, we need to know the smallest and largest number in the set. In this case, we are given the mean (average) of all but the largest number and the mean of all but the smallest number. Let's call the set of numbers A, B, C, D, and E.
We are given that the mean of A, B, C, and D is 80, so we can write their sum as:
A + B + C + D = 80 * 4 = 320 (1)
We are also given that the mean of B, C, D, and E is 90, so we can write their sum as:
B + C + D + E = 90 * 4 = 360 (2)
Now, let's subtract equation (1) from equation (2) to cancel out B, C, and D:
(B + C + D + E) - (A + B + C + D) = 360 - 320
Simplifying, we get:
E - A = 40
Therefore, E is the largest number, and A is the smallest number in the set.
Now, we know that the range of a set is the difference between the largest and smallest numbers. So, the range in this case is:
Range = E - A = 40
Thus, the range of the set of five numbers is 40.