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Math Help - Generalizing with Variables

  1. #1
    Junior Member
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    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!!!
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  2. #2
    MHF Contributor

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    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 \frac{nG+ mX}{m+ n}. Set that equal to H and solve for X:
    \frac{nG+ mX}{m+n}= H
    nG+ mX= (m+ n)H
    mX= (m+ n)H- nG
    X= \frac{(m+n)H- nG}{m} which is what you have.

    You could also write that as X=\frac{mH+ n(H-G)}{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.
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