# Math Help - Interest/Discount Rate

1. ## 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!

2. Originally Posted by jlt1209
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!

3. Originally Posted by jlt1209
Interest is credited to each account at a discount rate of d.
OR

If d is the effective discount rate, then

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

which then implies that

$
X \approx 38.87927686...
$

Originally Posted by jlt1209

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.