Results 1 to 11 of 11

Math Help - Annuity problem

  1. #1
    Junior Member
    Joined
    Nov 2006
    Posts
    42

    Annuity problem

    Hi guys. Would be much appreciated if somebody could give me a start on this problem, I'm really stuck on it.

    A washing machine, cash price $ 850 is available on the following terms:
    A deposit of $ 100 followed by equal payments at the end of each month for the next 18 months.
    If interest is 6% pa compounded monthly for the first 6 months and 8% pa compounded monthly thereafter, determine the size of the regular monthly payments.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    Amount borrowed = 750
    Future Value of 750 (6% 6 months; 8% 12 months) = Future Value of 18 payments (similar interest)

    Formula for FV of amount A: FV = A(1 + i)^n
    Formula for FV of payment P: FV = P[(1 + i)^n - 1] / i

    Let x = .06/12 and y = .08/12

    FV of 750: [750(1 + x)^6](1 + y)^12

    FV of P: {P[(1 + x)^6 - 1](1 + y)^12} / x + {P[(1 + y)^12 - 1]} / y

    {P[(1 + x)^6 - 1](1 + y)^12} / x + {P[(1 + y)^12 - 1]} / y = [750(1 + x)^6](1 + y)^12

    Py[(1 + x)^6 - 1](1 + y)^12 + Px[(1 + y)^12 - 1] = [750xy(1 + x)^6](1 + y)^12

    P = {[750xy(1 + x)^6](1 + y)^12} / {y[(1 + x)^6 - 1](1 + y)^12 + x[(1 + y)^12 - 1]}

    Substitute x = .06/12 and y = .08/12 back in to get P = 43.97989... or 43.98 rounded.

    If you're confused with this: P[(1 + x)^6 - 1](1 + y)^12
    it means the accumulation of 6 payments at 6% earns 8% during next 12 months.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2006
    Posts
    42
    Quote Originally Posted by Wilmer View Post

    If you're confused with this: P[(1 + x)^6 - 1](1 + y)^12
    it means the accumulation of 6 payments at 6% earns 8% during next 12 months.
    Thanks, that little bit down the end made it all very clear. I was having trouble making everything fit together (annuities, compound interest etc) but something in that last sentence made everything make sense.

    Thanks for the "AHA" moment.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    Glad you followed...

    I find it is a good idea with these (or anything similarly "longish") to go this way:

    x = .06/12, u = (1 + x)^12
    y = .08/12, v = (1 + y)^18

    P = 750 x u y v / [y v (u - 1) + x (v - 1)]

    Letting A = amount borrowed, n = months at 1st rate, m = months at 2nd rate,
    p = 1st rate, q = 2nd rate, then you can get a general case formula:

    P = A x u y v / [y v (u - 1) + x (v - 1)]
    where
    x = p / 12
    u = (1 + x)^n
    y = q / 12
    v = (1 + y)^m

    Of course that's restricted to monthly compounding and 2 rates,
    but you get the idea, right?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Apr 2008
    Posts
    83
    Just thought I'd present the equivalent alternative "bird's eye view"
    Comparison date at beginning:
    <br />
850 - 100 = {\rm{R}}\frac{{{\rm{1 - }}\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{ - \left( {{\textstyle{1 \over 2}} \times 12} \right)} }}{{{\textstyle{{0.06} \over {12}}}}} + {\rm{R}}\frac{{{\rm{1 - }}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{ - \left( {1 \times 12} \right)} }}{{{\textstyle{{0.08} \over {12}}}}}\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{ - \left( {{\textstyle{1 \over 2}} \times 12} \right)} <br />

    Comparison date at end:
    <br />
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}  = <br />
{\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)}  - 1}}{{{\textstyle{{0.06} \over {12}}}}}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}  + {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}  - 1}}{{{\textstyle{{0.08} \over {12}}}}}<br />
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    In the 2nd one, you didn't leave a "space" between the = sign and the R.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Apr 2008
    Posts
    83
    Roger that.
    I hope this one's more picturesque?

    <br />
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)} \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)} <br />
<br />
 = {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)}  - 1}}{{{\textstyle{{0.06} \over {12}}}}}\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}  + {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{\left( {1 \times 12} \right)}  - 1}}{{{\textstyle{{0.08} \over {12}}}}}<br />
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    Much better; you get a little green star
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Apr 2008
    Posts
    83
    Simpler perhaps might be the end of six months as the comprison date

    <br />
\left( {850 - 100} \right)\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)}  = {\rm{R}}\frac{{\left( {1 + {\textstyle{{0.06} \over {12}}}} \right)^{\left( {{\textstyle{1 \over 2}} \times 12} \right)}  - 1}}{{{\textstyle{{0.06} \over {12}}}}} + {\rm{R}}\frac{{{\rm{1}} - \left( {1 + {\textstyle{{0.08} \over {12}}}} \right)^{ - \left( {1 \times 12} \right)} }}{{{\textstyle{{0.08} \over {12}}}}}<br />
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    I'd like to see a space on left and right of the + sign on the right;
    ------ + R. Thanking you in advance, I remain sincerely yours.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Apr 2008
    Posts
    83
    As soon as my keyboard error gets fixed (copy and paste from old files and other sources,as opposed to good old keyboard output, seems to be generating undesirable effects with MathType output - I have no wish to learn/master latex code)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Annuity Problem
    Posted in the Business Math Forum
    Replies: 5
    Last Post: September 3rd 2011, 07:11 PM
  2. Another annuity problem?
    Posted in the Business Math Forum
    Replies: 5
    Last Post: July 22nd 2009, 04:11 AM
  3. Annuity Due Problem
    Posted in the Business Math Forum
    Replies: 0
    Last Post: March 4th 2009, 08:56 PM
  4. Annuity problem? Please Help!
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 6th 2008, 06:35 AM
  5. Annuity problem
    Posted in the Business Math Forum
    Replies: 1
    Last Post: September 5th 2007, 11:56 AM

Search Tags


/mathhelpforum @mathhelpforum