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:

which is what you have.

You could also write that as

showing just how the difference between "H", the desired average, and "G", the current average, influences the needed future average.