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Math Help - Interest/Discount Rate

  1. #1
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    Interest/Discount Rate

    Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his bank account, and Robbie deposits 50 into his. Interest is credited to each account at a discount rate of d. The amount of interest earned in Bruce’s account during the 11th year is equal to X. The amount of interest earned in Robbie’s account during the 17th year is also equal to X. Calculate X.

    I know the answer in 38.9, I just can't remember how to get there .... Thanks!
    Last edited by mr fantastic; September 1st 2009 at 05:10 AM. Reason: Restored deleted question.
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  2. #2
    Member garymarkhov's Avatar
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    Quote Originally Posted by jlt1209 View Post
    Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his bank account, and Robbie deposits 50 into his. Interest is credited to each account at a discount rate of d. The amount of interest earned in Bruce’s account during the 11th year is equal to X. The amount of interest earned in Robbie’s account during the 17th year is also equal to X. Calculate X.

    I know the answer in 38.9, I just can't remember how to get there .... Thanks!
    Cool question.

    Bruce's interest earned between t=10 and t=11 is equal to Robbie's interest earned between t=16 and t=17, so we can write

    100(1+r)^11 - 100(1+r)^10 = 50(1+r)^17 - 100(1+r)^16

    We need some common terms, so rewrite as

    100(1+r)^10(1+r) - 100(1+r)^10 = 50(1+r)^16(1+r) - 100(1+r)^16

    Pull some terms out to get

    100(1+r)^10((1+r) - 1) = 50(1+r)^16((1+r) - 1)

    Hmm, delicious

    2(1+r)^10((1+r) - 1) = (1+r)^16((1+r) - 1)

    2(1+r)^10/((1+r)^16) = ((1+r) - 1)/((1+r) - 1)

    Shaping up!

    2(1+r)^10/((1+r)^16) = 1

    2/((1+r)^6) = 1

    2 = (1+r)^6

    2^(1/6) = 1+r

    1.12246 = 1+r

    r = 0.12246

    Now that we've got r, just plug it into one side of

    100(1+r)^11 - 100(1+r)^10 = 50(1+r)^17 - 100(1+r)^16

    to find X.

    100(1+0.12246)^11 - 100(1+0.12246)^10 = 38.87792

    ROCK AND ROLL!
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  3. #3
    Member jonah's Avatar
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    Quote Originally Posted by jlt1209 View Post
    Interest is credited to each account at a discount rate of d.
    OR

    If d is the effective discount rate, then

    <br />
100\left( {1 - d} \right)^{ - 11}  - 100\left( {1 - d} \right)^{ - 10}  = 50\left( {1 - d} \right)^{ - 17}  - 50\left( {1 - d} \right)^{ - 16} <br />
<br />
 \Leftrightarrow d = 1 - \left( {{\textstyle{1 \over 2}}} \right)^{{\textstyle{1 \over 6}}}  \approx 0.1091012819...<br />

    which then implies that

    <br />
X \approx 38.87927686...<br />

    Quote Originally Posted by jlt1209 View Post

    i know the answer in 38.9, i just can't remember how to get there .... Thanks!
    Thanks my ass! Judging by the lack of any kind of work you've shown so far, seems more like you've never been there at all and just want others to do your work for you.
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