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

Bunuel: Good approach!

nice approach Bunuel. It shortens the long process and makes it less error prone.

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

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

Hope this helps...!!!

1st method is really awasome to follow...thanks Bunuel

Wow Bunuel, that was really good!

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

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.

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

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.

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

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

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

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

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

\(10^4\) remains in denominator

Answer = 4

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

Awesome approach Bunuel...

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

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.

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.

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