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Decimal Fraction

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Decimal Fraction [#permalink] New post 23 Feb 2011, 08:52
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If t= 1/2^9x5^3 is expressed as a terminating decimal, how many zeros will t have between the decimal point and the first nonzero digit to the right of the decimal point/
a. three
b. four
c. five
d. six
c. nine

suggest shortest and easier approach please.
[Reveal] Spoiler: OA

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Re: Decimal Fraction [#permalink] New post 23 Feb 2011, 09:12
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t= \frac{1}{2^9 * 5^3}

t= \frac{1}{2^6 * 2^3 * 5^3}

t= \frac{1}{2^6 * (2*5)^3}

t= \frac{1}{64 * (10)^3}

Multiplying numerator and denominator by 10^2

t= \frac{10^2}{64 * (10)^5}

t= \frac{1.something}{(10)^5}

t= 1.something * 10^{-5}

To remove 10^{-5} we need to move decimal point 5 digits to the left

t= .00001something

4 zeros between decimal and first non-zero digit.

Ans: "B"
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Re: Decimal Fraction [#permalink] New post 23 Feb 2011, 13:30
I think there is no easier way than this.

fluke wrote:
t= \frac{1}{2^9 * 5^3}

t= \frac{1}{2^6 * 2^3 * 5^3}

t= \frac{1}{2^6 * (2*5)^3}

t= \frac{1}{64 * (10)^3}

Multiplying numerator and denominator by 10^2

t= \frac{10^2}{64 * (10)^5}

t= \frac{1.something}{(10)^5}

t= 1.something * 10^{-5}

To remove 10^{-5} we need to move decimal point 5 digits to the left

t= .00001something

4 zeros between decimal and first non-zero digit.

Ans: "B"
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Re: Decimal Fraction [#permalink] New post 23 Feb 2011, 21:46
1/1000 * 1/2^6 = 5^6/5^6 * 1/2^6 * 1/1000 = 5^6/10^9 = 5 digits/10^9

=> Answer is B (4 zeroes)
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Re: Decimal Fraction [#permalink] New post 30 Mar 2011, 19:52
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What fluke did above is great. Let me just add here that if you are stuck with how to proceed, don't shy away from quick and easy calculations.

\frac{1}{2^9*5^3} = \frac{1}{2^6*1000}

Now, I can divide 1 by 64 to get the decimal point: .01

If I divide this further by 1000, the decimal moves 3 places to the left and I get four 0s before the 1.

Sometimes, under pressure in the exam, Math will fail you. Go with your instincts and use logic. (Except if your instincts tell you to multiply a four digit number with a five digit number - then you are definitely missing the point!)
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Re: Decimal Fraction   [#permalink] 30 Mar 2011, 19:52
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