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Operations on Rational Numbers OG-13, Q199

This topic has 3 expert replies and 0 member replies
AkiB Junior | Next Rank: 30 Posts Default Avatar
Joined
27 Apr 2013
Posted:
16 messages

Operations on Rational Numbers OG-13, Q199

Post Sat Jun 14, 2014 6:23 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    0.99999999/ 1.0001 - 0.99999991 /1.0003 =
    (A) 10^-8
    (B) 3(10^-8)
    (C) 3(10^-4)
    (D) 2(10^-4)
    (E) 10^-4

    What is the Easiest way to solve this

    OA D

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    Post Sat Jun 14, 2014 6:38 am
    Quote:
    0.99999999/1.0001 - 0.99999991/1.0003 =

    A. 10^-8
    B. 3(10^-8)
    C. 3(10^-4)
    D. 2(10^-4)
    E. 10^-4
    One approach is to recognize that both 9999.9999 and 9999.9991 can be rewritten as differences of squares.

    First, 0.99999999 = 1 - 0.00000001
    = (1 - 0.0001)(1 + 0.0001)

    Similarly, 9999.9991 = 1 - 0.00000009
    = (1 - 0.0003)(1 + 0.0003)

    Original question: 0.99999999/1.0001 - 0.99999991/1.0003
    = (1 - 0.0001)(1 + 0.0001)/(1.0001) - (1 - 0.0003)(1 + 0.0003)/(1.0003)
    = (1 - 0.0001)(1.0001)/(1.0001) - (1 - 0.0003)(1.0003)/(1.0003)
    = (1 - 0.0001) - (1 - 0.0003)
    = 1 - 0.0001 - 1 + 0.0003
    = 0.0002
    = 2 x 10^(-4) = D

    Cheers,
    Brent

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    Post Sat Jun 14, 2014 6:40 am
    Quote:
    0.99999999/1.0001 - 0.99999991/1.0003=

    A. 10^-8
    B. 3(10^-8)
    C. 3(10^-4)
    D. 2(10^-4)
    E. 10^-4
    Another approach is to combine the fractions and then use some APPROXIMATION.

    First combine the fractions by finding a common denominator.
    (9999.9999)/(10001) - (9999.9991)/(10003)
    = (9999.9999)(10003)/(10001)(10003) - (9999.9991)(10001) /(10003)(10001)
    = [(10003)(9999.9999) - (10001)(9999.9991)] / (10001)(10003)
    ≈ [(10003)(10^4) - (10001)(10^4)] / (10^4)(10^4) ... (approximately)
    ≈ [(10003) - (10001)] / (10^4) ... (divided top and bottom by 10^4)
    ≈ 2/(10^4)
    ≈ 2(10^-4)
    = D

    Cheers,
    Brent

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    Post Sun Jun 15, 2014 2:57 am
    AkiB wrote:
    0.99999999/ 1.0001 - 0.99999991 /1.0003 =
    (A) 10^-8
    (B) 3(10^-8)
    (C) 3(10^-4)
    (D) 2(10^-4)
    (E) 10^-4
    Alternate approach:
    The answer choices imply that 1.0001 must divide easily into 0.99999999 and that 1.0003 must divide easily into 0.99999991.

    x = 0.99999999/1.0001.
    1.0001x = 0.99999999.
    Notice the values in red.
    For the product on the left to yield 0.99999999 -- a value with a rightmost digit of 9 -- the rightmost digit of x must be 9.

    y = 0.99999991/1.0003.
    1.0003y = 0.99999991.
    Notice the values in red.
    For the product on the left to yield 0.99999991 -- a value with a rightmost digit of 1 -- the rightmost digit of y must be 7.

    Thus:
    x-y = (value with a rightmost digit of 9) - (value with a rightmost digit of 7) = (value with a rightmost digit of 2).

    Only one answer choice includes a digit of 2:
    D: 2(10^-4) = 0.0002.

    The correct answer is D.

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