SOLUTION: Steve averaged 70 in his first m number of exams. After taking n more exams, he had an overall average of 75. In terms of m and n, his average for the last n exams was?

Algebra ->  Average -> SOLUTION: Steve averaged 70 in his first m number of exams. After taking n more exams, he had an overall average of 75. In terms of m and n, his average for the last n exams was?      Log On


   

Question 1015907: Steve averaged 70 in his first m number of exams. After taking n more exams, he had an overall average of 75. In terms of m and n, his average for the last n exams was?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
overall average is equal to overall total divided by total number of elements.

overall total is equal to overall average multiplied by total number of elements.

total for m elements is 70 * m.

total for n elements is x * n.

we use x because we don't know what it is yet.

total for all elements is (m + n) * 75.

formula for totals becomes:

(70 * m) + (x * n) = 75 * (m + n)

distribute the multiplication on the right side of the equation to get:

(70 * m) + (x * n) = (75 * m) + (75 * n)

subtract (70 * m) from both sides of the equation to get:

(x * n) = (75 * m) - (70 * m) + (75 * n)

factor the m out of the first 2 terms on the right side of the equation to get:

(x * n) = (75 - 70) * m + (75 * n)

simplify to get:

(x * n) = 5 * m + 75 * n

divide both sides of the equation by n to get:

x = ((5 * m) + (75 * n)) / n

that's your solution.

you can confirm by assuming any value of m and any value of n and solving for x.

as an example, i'll assume m = 7 and n = 12.

equation of x = ((5 * m) + (75 * n)) / n becomes:

x = ((5 * 7) + (75 * 12)) / 12 which becomes:

x = (35 + 900) / 12 which becomes:

x = 935 / 12 which becomes:

x = 77.9166666667

if this is correct, then:

70 * 7 + 77.9166666667 * 12 should be equal to 75 * 19.

19 is used on the right side of the equation because m + n = 7 + 12 = 19.

i do the math and i get:

1425 = 1425

that's the overall total.

divide that by 19 and you get an average of 75, as you should.

this confirms the formula is good.

the formula is:

x = %28%285+%2A+m%29+%2B+%2875+%2A+n%29%29+%2F+n

is works regardless of what the value of m or n is as long as those values are positive whole numbers.

i tested with m = 22 and n = 33 and i got 75 as an overall average.

you can test yourself with any value of m and any value of n that you can think of.

it should work for those as well.