Problem from the chapter named Average

Question

The mean of the 5 smallest numbers from a group is 15 while the mean of all the numbers of the group taken together is 17.If the mean of the numbers leaving the smallest five oi is 18.25.How many numbers were there in the group in all?

Answer 1

Let's set up equations to describe our number set then solve them.

First: "The mean of the 5 smallest numbers from a group is 15"

If we set the sum of the five smallest numbers to equal x, we can say x/5 = 15, or x = 75.

Next: "the mean of all the numbers of the group taken together is 17"

Let's call the sum of all of the numbers equal to y, and n can be equal to the number of digits in the set. That allows us to say y/n = 17, or solving for y, y = 17n.

Finally: "If the mean of the numbers leaving the smallest five oi is 18.25"

Leaving out the smallest five can be expressed as y - x. Taking the mean of them means we'd have to divide the the number of digits, minus the smallest five: n - 5. This tells us the average is (y-x)/(n-5) = 18.25.

To solve our problem, we'll use x = 75 and y = 17n in the last equation to solve for n, the number of digits in the number set:

(y - x)/(n - 5) = 18.25 Our equation.

(17n - 75)/(n - 5) = 18.25 Substitute values for x and y.

17n - 75 = 18.25 (n - 5) Multiply both sides by n - 5 to get rid of that denominator.

17n - 75 = 18.25n - 91.25 Distribute 18.25 to both terms inside the parentheses.

17n + 16.25 = 18.25n Add 91.25 to both sides.

16.25 = 1.25n Subtract 17n from both sides.

n = 13 Divide both sides by 1.25 to solve for n.

There were 13 numbers in the set.

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