[JMO Mentoring Feb2012 Q5] On the second to last test of the school year, Barbara scored 98 and her average score so far then increased by 1. On the last test she scored 70 at which her average score then decreased by 2. How many tests has she taken through the school year?

Question

Please help!

Answer 1

If she has taken x tests, then
the average of the first x-2 tests was m
and thus we have, adding up total points earned,
m(x-2) + 98 = (m+1)(x-1)
(m+1)(x-1) + 70 = x(m-1)
solve for x and get x=10

Answer 2

After n tests, let her average be A
Then the sum (S) of all the scores is
S = nA

after 2nd last test: (S + 98)/(n+1) = A + 1
S + 98 = nA + n + A + 1
S = nA + n + A - 97 , but S = nA
nA = nA + n+ A - 97
n + A = 97 **

after last test: (S + 98 + 70)/(n+2) = A+1 - 2
S + 168 = (n+2)(A-1)
S + 168 = nA - n + 2A - 2
S = nA - n + 2A - 170
nA = nA - n +2A - 170
-n + 2A = 170 ***
add ** and ***
3A = 267
A = 89

in ** , n + A = 97
n = 98-89 = 9

So she took 11 tests

check:
after 9 tests, A = 89, S = 801
on 10th test, sum = 801+98 = 899 , A = 899/10 = 89.9 or 90, it increased by 1
on 11th test, sum = 969, A = 969/11 = 88 , yes it decreased by 2
My answer is correct

Answer 3

Thank you so much.