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OG Problem Solving Q 78

This topic has 2 expert replies and 5 member replies
melguy Master | Next Rank: 500 Posts
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OG Problem Solving Q 78

Post Wed Oct 12, 2011 7:46 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Hello All

    I tried searching explanation for the below question in the forum but could not find a good answer. I would appreciate if anyone could plz help.

    Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity?

    (A) $10,464
    (B) $ 864
    (C) $ 816
    (D) $ 800
    (E) $ 480

    I am just curious to know that the Q states "8 percent compounded semiannually" so why do we need to multiply by .04% two times.

    I understand the process but I am curious as to why don't we multiply by .08% two times instead of .04% since the interest rate is 8% not 4%. Be it semi annually or annually the interest rate is the same?

    Thanks

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    Post Wed Oct 12, 2011 8:30 pm
    Interest rates are almost universally quoted on an annual basis. So when they say the interest rate is 8%, that is an annual interest rate. So, if we invest $100 at 8% compounded annually, at the end of one year we would have $108. Halfway through the year we would have $104. Now, when we change it to 8% compounded semiannually, that is still an annual interest rate. We are just compounding it more frequently. The phrase "8 percent compounded semiannually" really means an 8 percent annual interest rate compounded semi-annually. They just don't include the part about it being an annual interest rate because it is understood.

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    fcabanski Master | Next Rank: 500 Posts
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    Post Wed Oct 12, 2011 9:21 pm
    Not sure if I'm reading this correctly, but a formula for compound interest is:

    A = P(1+r/n)^nt

    A is the principal plus accumulated interest.

    P is the principal.

    A-P then answers this question - the amount of interest accumulated.

    r is the annual interest rate.

    n is the number of compound periods in a year.

    t is the number of years.

    This one is simple. So you can solve it without knowing the compound interest formula as long as you know:

    - The principal applied in each compound period is the annual rate divided by the number of compound periods.
    - Compound interest means the interest in each period applies to the total of principal and interest accumulated in all previous periods.

    The balance in each period is the principal plus ...the principal times the annual rate divided by the number of compound periods.

    1st period. 10,000+ 10,000*.08/2 = 10,400
    2nd period. 10,400 + 10,400*.08/2 = 10816.

    Interest accumulated = 10,816 - 10,000 = 816



    Last edited by fcabanski on Wed Oct 12, 2011 9:29 pm; edited 1 time in total

    Post Wed Oct 12, 2011 9:26 pm
    Interest rates for semiannually is similar to how you would normally calculate interests.

    2 things to remember:

    Interest rates would be R%/n
    (R% - rate of interest)
    (n - number of times interest is calculated)

    Here, you have 8% compounded semiannually, so r - 8% and n - 2 (1 year - semiannually)

    Hence rate for calculation would be R%/n - 8/2 = 4% and number of terms would be 2.

    Amt = P[1 + (r/100)]^n

    = 10000[(1 + 0.04]^2
    = 10816

    Interest would be 10816 - 10000 = 816

    Post Thu Oct 13, 2011 2:59 am
    melguy wrote:
    Hello All

    I tried searching explanation for the below question in the forum but could not find a good answer. I would appreciate if anyone could plz help.

    Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity?

    (A) $10,464
    (B) $ 864
    (C) $ 816
    (D) $ 800
    (E) $ 480

    I am just curious to know that the Q states "8 percent compounded semiannually" so why do we need to multiply by .04% two times.

    I understand the process but I am curious as to why don't we multiply by .08% two times instead of .04% since the interest rate is 8% not 4%. Be it semi annually or annually the interest rate is the same?

    Thanks
    When the time period is brief, the compounded interest will be just a bit more than the simple interest.
    Simple interest = .08(10,000) = 800.
    The correct answer must be just a bit more than 800.

    The correct answer is C.

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    fcabanski Master | Next Rank: 500 Posts
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    Post Thu Oct 13, 2011 8:43 am
    That's a good way to look at it and to solve it quickly.

    Post Wed May 20, 2015 3:06 pm
    melguy wrote:
    Hello All

    I tried searching explanation for the below question in the forum but could not find a good answer. I would appreciate if anyone could plz help.

    Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity?

    (A) $10,464
    (B) $ 864
    (C) $ 816
    (D) $ 800
    (E) $ 480

    I am just curious to know that the Q states "8 percent compounded semiannually" so why do we need to multiply by .04% two times.

    I understand the process but I am curious as to why don't we multiply by .08% two times instead of .04% since the interest rate is 8% not 4%. Be it semi annually or annually the interest rate is the same?

    Thanks
    We could have also looked at this problem a bit more conceptually. We know that when an investment has a rate of 8% ANNUAL interest and it compounds SEMI-ANNUALLY (twice a year), the investment earns 4% interest every SIX MONTHS. So in this case we know:

    Interest earned for the first six months = 0.04 x $10,000 = $400

    Her investment is now worth ($400 + $10,000) = $10,400

    Interest earned for the next six months = 0.04 x $10,400 = $416

    Thus, the total interest earned = $400 + $416 = $816

    The answer is C

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    Post Fri May 22, 2015 4:40 am
    In simple terms-

    Principal+ Interest for in first Term of 6 months becomes principal for next six moths.

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