If t = 1/(2^9*5^3) is expressed as a terminating decimal, ho

Question

If  t = \frac{1}{(2^9*5^3)}  is expressed as a terminating decimal, how many zeros will t have between the decimal point and the fist nonzero digit to the right of the decimal point?

A. Three
B. Four
C. Five
D. Six
E. Nine

Bunuel · Accepted Answer

If  t = \frac{1}{(2^9*5^3)}  is expressed as a terminating decimal, how many zeros will t have between the decimal point and the fist nonzero digit to the right of the decimal point?

A. Three
B. Four
C. Five
D. Six
E. Nine

Given:  t=\frac{1}{2^9*5^3} .

Multiply by  \frac{5^6}{5^6} :

t=\frac{5^6}{(2^9*5^3)*5^6}=\frac{25*625}{10^9}=\frac{15625}{10^9}=0.000015625 .

Hence  t  will have 4 zeroes between the decimal point and the fist nonzero digit.

Answer: B.

Or another way:

t=\frac{1}{2^9*5^3}=\frac{1}{(2^3*5^3)*2^6}=\frac{1}{10^3*64}=\frac{1}{64000} .

Now,  \frac{1}{64,000}  is greater than  \frac{1}{100,000}=0.00001  and less than  \frac{1}{10,000}=0.0001 , so  \frac{1}{64,000}  is something like  0.0000xxxx .

Answer: B.

abhishekkpv · Answer

t= 1 / (2^9 * 5^3) 
or t=1/(2^3*5^3)*2^6

t=1/(10^3)*64

1/64 will be 0.01 and shifting the decimal point by three places to account for 1/10^3..

we get 4 zeros followed by 1..
Ans:B)

catty2004 · Answer

Can someone please explain how is multiplying 5^6 to the denominator (2^9 * 5^3) get 10^9?

kraizada84 · Answer

5^6*(2^9*5^3) = 2^9*5^(6+3)= 2^9 *5^9 = (2*5)^9 = 10^9

Hope this helps...!!!

olwan · Answer

Please I would like to know why you multiplied by 5^6. and I did not understand your second approach.

Bunuel · Answer

We are multiplying by 5^6/5^6 to convert denominator to the base of 10, so to simplify it to the decimal form: 0.xxxx.

shahir16 · Answer

Hi, I still dont understand why we have to multiply by 5^6/5^6 i understand that this equals one but what is the general rule for this? how did you know to pick 5^6?

Thanks

Bunuel · Answer

Welcome to GMAT Club shahir16.

We want the denominator of the fraction to be written as some power of 10. We need that in order to transform the fraction into decimal easily.

Now, the denominator = 2^9 * 5^3, hence we need to multiply it by 5^6 to get 10^9.

Hope it's clear.

PareshGmat · Answer

\frac{1}{2^9 5^3}

= \frac{5^6}{2^9 5^3 5^6}

= \frac{125^2}{10^9}

Count the numbers in numerator as equivalent to zero's in denominator

= \frac{15625}{100000 . 10^4}

10^4  remains in denominator

Answer = 4

luckyme17187 · Answer

t = 1/(2^9*5^3)

2^9 = 8^3

8^3 *5^3 = 40^3

1/64000 = there will be at least 3 zeros

1/64=0.0something ,,, adding another 3 zeros then 4 zeros

kakal0t29 · Answer

We got
T=1/2^9+5^3 
2^9=(2*2*2)^3
5^3
T=1/8^3*5^3=1/40^3=1/4*10^4=0.25*10^4=0.000025

So the solution will be  B

GMATinsight · Answer

Please mark your mistake in Highlighted part \frac{1}{40^3} = \frac{1}{(4^3 * 10^3)} = (\frac{1}{64})*10^{-3} = 0.015625 *10^{-3} = 0.000015625 Your answer however is correct but that seems to just by chance. :wink:

JeffTargetTestPrep · Answer

Solution:

We use the term "leading zeros" to describe the zeros between the decimal point and the first nonzero decimal digit. To complete this problem we can use the following rule to determine the number of leading zeros in a fraction when it is converted to a decimal:

If X is an integer with k digits, then 1/X will have k – 1 leading zeros unless X is a perfect power of 10, in which case there will be k – 2 leading zeros.

We see that t is in the form 1/X. Because the denominator X has more twos than fives, we know X is not a perfect power of 10. Before considering the fraction as a whole, we first must determine the number of digits in the denominator.

Rewriting the denominator, we get 2^9 x 5^3 = (2^6 x 2^3) x 5^3 = 2^6 x (2^3 x 5^3) = 64 x (1,000) = 64,000, which is a 5-digit integer. Thus, k = 5.

Using our rule, we see that the fraction t has 5 - 1 = 4 leading zeros.

Answer B.

GoodnessShallPrevail · Answer

Between decimal point and 1st non-zero digit to the right of decimal point:
1 divided by a number greater than 10 will yield 1 Zero
1 divided by a number greater than 100 will yield 2 Zeros
1 divided by a number greater than 1000 will yield 3 Zeros

Hence, answer is 3 Zeros + 1 Zero = 4 Zeros

Kinshook · Answer

Asked: If  t = \frac{1}{(2^9*5^3)}  is expressed as a terminating decimal, how many zeros will t have between the decimal point and the fist nonzero digit to the right of the decimal point?

t = \frac{1}{(2^9*5^3)} = \frac{2^9*5^9}{2^9*5^3} * 10^{-9} = 5^6 * 10^{-9} = 15625 * 10^{-9} = .000015625

IMO B

GMATGuruNY · Answer

Determine how many powers of 10 are in the denominator.

\frac{1}{2^95^3} = \frac{1}{2^6(2^35^3)} = \frac{1}{64*10^3} = \frac{1}{64,000}

https://s10.postimage.org/tstmzq4lh/64000_division.jpg

There are four 0's to the right of the decimal.

B .

Abolnasr · Answer

WhatsApp Image 2023-08-30 at 12.30.58 PM.jpeg

KarishmaB · Answer

t = \frac{1}{(2^9*5^3)} = \frac{1}{64000} (Three 5s multiply with three 2s to make three 0s and we are left with six 2s which make 64) Now, \frac{1}{64} = .01 something (We don't care what something is) When we divide this by 1000, the decimal moves 3 places to the left and we get .00001 Answer (B) Check these post on terminating decimals for theory and practice questions: https://anaprep.com/number-properties-t ... -decimals/ https://anaprep.com/number-properties-t ... plication/

findingmyself · Answer

Expression can be written us: The game is to make everything possible as a square of 10 in the denominator: 1/2^6*2^3*5^3

1/2^6*10^3--->10^-3/2^6--->10^-3/64--->(100*10^-5)/64

100/64 is some 1.5 approx

1.5*10^-5--->0.15*10^-4 Which means FOUR ZEROS between decimal point and first non zero digit (Number of zero is equal to the power of 10)

If t = 1/(2^9*5^3) is expressed as a terminating decimal, ho

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