Generalizing with Variables
Generalize the average test grade. Suppose Jane has an average of G after n tests. What average score does she need on the next m tests to average H for all m + n tests?
t1 = test #1, t2 = test #2, ...
(t1 + t2 + ... + tn)/n = G
(t1 + t2 + ... + tn) = n*G
((t1 + t2 + ... + tn) + (tn+1 + tn+2 + ... + tm))/(m+n) = H
(n*G + (tn+1 + tn+2 + ... + tm))/(m+n) = H
n*G + (tn+1 + tn+2 + ... + tm) = (m+n)*H
(tn+1 + tn+2 + ... + tm) = (m+n)*H - n*G
(tn+1 + tn+2 + ... + tm)/m = ((m+n)*H - n*G)/m
((m+n)*H - n*G)/m is my answer but I'm a little unsure of this. Any help would be greatly appreciated!!!
Re: Generalizing with Variables
Yes, that is correct. If she averaged "G" on n tests, then her total score is nG. If she averages "X" on the next m tests, her total score for those tests would be mX so here total score over all m+ n tests would be nG+ mX. Her average over all the tests would be 
. Set that equal to H and solve for X:
H)
H- nG)
H- nG}{m})
which is what you have.
You could also write that as }{m}= H+ \frac{n}{m}(H- G))
showing just how the difference between "H", the desired average, and "G", the current average, influences the needed future average.
