[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?
Please help!
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
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
Thank you so much.
To solve this problem, we can work backwards.
First, let's assume that Barbara has taken 'x' tests so far in the school year.
We are given that on the second to last test, Barbara scored 98 and her average score increased by 1. This means that the sum of her scores before the second to last test was (x - 1)*(average score) and the sum of her scores after the second to last test was (x - 1)*(average score) + 98.
Since the average score increased by 1, we can set up the equation:
(x - 1)*(average score) + 98 = x*(average score) -----> Equation 1
Next, we are given that on the last test, Barbara scored 70 and her average score decreased by 2. This means that the sum of her scores before the last test was x*(average score) and the sum of her scores after the last test was x*(average score) + 70.
Since the average score decreased by 2, we can set up the equation:
x*(average score) + 70 = (x + 1)*(average score) -----> Equation 2
Now we have a system of equations with two variables (x and average score).
Let's solve this system of equations to find the values of x and average score.
From Equation 1, we expand the equation to get:
(x - 1)*(average score) + 98 = x*(average score)
x*(average score) - average score + 98 = x*(average score)
- average score + 98 = 0
average score = 98
Substituting this value of average score into Equation 2:
x*(98) + 70 = (x + 1)*(98)
98x + 70 = 98x + 98
70 = 98
Oops! We've hit a contradiction. This means that there is no solution to the system of equations.
Hence, there is no answer to the question and the information given seems to be inconsistent.