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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   85% (hard)

Question Stats: 56% (02:07) correct 44% (02:07) wrong based on 2579 sessions

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If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

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Math Expert V
Joined: 02 Aug 2009
Posts: 8792
If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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19
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

Hi,

since we have ONLY 10s in denominator, the # of zeroes between the decimal point and the first nonzero digit in the decimal expansion of $$(\frac{t}{1000})^4$$ will depend on 't'...

ONE 10 will put the decimal point and the REMAINING 10s will result in " zeroes between the decimal point and the first nonzero digit in the decimal expansion of $$(\frac{t}{1000})^4$$"..

so we have $$10^{12}$$, so if we have t as a single digit, there will be 11 zeroes, that is 12 - number of digits in t.....
BUT 0s are <8, so t should have digits>12-8 or >4......

lets see the choices -

I. 3 -----$$3^4 = 81$$... two digits ..# of 0s $$= 12 - 2 = 10 > 8.$$...NO
II. 5 ------$$5^4 = 625$$... three digits ..# of 0s = 12 - 3 = 9 > 8....NO
III. 9------$$9^4 = 6561$$... four digits ..# of 0s = 12 - 4 = 8 = 8....NO, we are looking for <8

none
A
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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Top Contributor
3
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

First: (t/1000)^4 = (t^4)/(1000^4)
Now recognize that 1000^4 = (10^3)^4 = 10^12
So, (t/1000)^4 = (t^4)/(1000^4) = (t^4)/(10^12)

IMPORTANT: When we divide a number by 10^12, we must move the decimal point 12 spaces to the left
So, for example, 1234567/10^12 = 0.000001234567
Likewise, 8888/10^12 = 0.000000008888
And, 66666666666666/10^12 = 66.666666666666

Now let's check each option

I. 3
It t = 3, then (t^4)/(10^12) = (3^4)/(10^12)
= 81/(10^12)
= 0.000000000081
There are 10 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement I

II. 5
It t = 5, then (t^4)/(10^12) = (5^4)/(10^12)
= 625/(10^12)
= 0.000000000625
There are 9 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement II

III. 9
It t = 9, then (t^4)/(10^12) = (9^4)/(10^12)
= 6561/(10^12)
= 0.000000006561
There are 8 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement III

Cheers,
Brent
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If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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9
Hi GMATters,

Here's my video explanation of this question:

Enjoy!

Rowan
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##### General Discussion
Intern  Joined: 11 Aug 2015
Posts: 2
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2
1
hey chetan thanks for the explanation but i don't get this one at all. is there another way possible and if so can u please post it ? thanks

chetan2u wrote:
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

Hi,

since we have ONLY 10s in denominator, the # of zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4 will depend on 't'...

ONE 10 will put the decimal point and the REMAINING 10s will result in " zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4"..

so we have $$10^{12}$$, so if we have t as a single digit, there will be 11 zeroes, that is 12 - number of digits in t.....
BUT 0s are <8, so t should have digits>12-8 or >4......

lets see the choices -

I. 3 -----$$3^4 = 81$$... two digits ..# of 0s $$= 12 - 2 = 10 > 8.$$...NO
II. 5 ------$$5^4 = 625$$... three digits ..# of 0s = 12 - 3 = 9 > 8....NO
III. 9------$$9^4 = 6561$$... four digits ..# of 0s = 12 - 4 = 8 = 8....NO, we are looking for <8

none
A
Math Expert V
Joined: 02 Aug 2009
Posts: 8792
If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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2
shivmalhotra10 wrote:
hey chetan thanks for the explanation but i don't get this one at all. is there another way possible and if so can u please post it ? thanks

chetan2u wrote:
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

Hi,

since we have ONLY 10s in denominator, the # of zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4 will depend on 't'...

ONE 10 will put the decimal point and the REMAINING 10s will result in " zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4"..

so we have $$10^{12}$$, so if we have t as a single digit, there will be 11 zeroes, that is 12 - number of digits in t.....
BUT 0s are <8, so t should have digits>12-8 or >4......

lets see the choices -

I. 3 -----$$3^4 = 81$$... two digits ..# of 0s $$= 12 - 2 = 10 > 8.$$...NO
II. 5 ------$$5^4 = 625$$... three digits ..# of 0s = 12 - 3 = 9 > 8....NO
III. 9------$$9^4 = 6561$$... four digits ..# of 0s = 12 - 4 = 8 = 8....NO, we are looking for <8

none
A

Hi
I'll try another way..

First just logic
What is $$\frac{1}{100}=0.01$$...
$$\frac{20}{100}=0.2$$ and so on...
So we have number of zeroes the way asked is the 0s in denominator - number of digits in numerator...

Let's see the Q now ...
Denominator has $$1000^4$$, so 12 Zeroes....
We are looking for 0s <8.... so numerator should have digits >12-8...

Now the best way to solve this as it does not really require to be made complicated
A simple way which will work here is take t as smallest 2 digit number 10... so (10/1000)^4= (.01)^4= .00000001 so 7 zeroes...
This should tell us that any 't' <10 will give us MORE than 7 Zeroes and 10 or more will give7 or less Zeroes...

Now the choices given are all less than 10, so all of them will give 8 or more Zeroes

Ans None
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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got it now. thank u chetan
Manager  B
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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Another way is to understand that there will 12 zeros and some other number of other digits. Then just write down 12 zeros and substitute at the end 81, 625 and 6561. The only thing is left - to calculate the number of zeros left.
Intern  Joined: 13 Jun 2015
Posts: 2
Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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1
1
$$\frac{t^4}{1000^4}$$ has to have 7 or less zeros between the decimal point and the first non zero digit decimal

$$\frac{t^4}{1000^4}$$ = $$\frac{t^4}{1000^{3*4}}$$ = $$\frac{t^4}{10^{12}}$$

Isolating $$10^{-7}$$ that will generate 7 digits $$\frac{t^4}{10^5}*10^{-7}$$

Thus to have 7 or less digits $$\frac{t^4}{10^5}\geq{1}$$. Thus is clear that neither 3, 5 or 9 will satisfy the equation

make sense?
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Location: Italy
Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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I got it wrong because it says " fewer than 8 zeroes between the decimal point and the first nonzero digit ....". So I did not count the first one. Shouldn't they be more precise ?
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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14
4
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

We are given that the decimal expansion of (t/1000)^4 has fewer than 8 zeroes between the decimal point and the first nonzero digit. We are also given that 3, 5 and 9 are possible values of t. Let’s test each Roman numeral:

I. 3

If t = 3, then (t/1000)^4 = (3/1000)^4 = (.003)^4 has twelve decimal places with the digits 81 (notice that 3^4 = 81). So there must be 10 zeros between the decimal point and the first nonzero digit 8 in the decimal expansion. This is not a possible value of t.

II. 5

If t = 5, then (t/1000)^4 = (5/1000)^4 = (.005)^4 has twelve decimal places with the digits 625 (notice that 5^4 = 625). So there must be 9 zeros between the decimal point and the first nonzero digit 6 in the decimal expansion. This is not a possible value of t.

III. 9

If t = 9, then (9/1000)^4 = (9/1000)^4 = (.009)^4 has twelve decimal places with the digits 6561 (notice that 9^4 = 6561). So there must be 8 zeros between the decimal point and the first nonzero digit 6 in the decimal expansion. This is not a possible value of t.

Recall that we are looking for fewer than 8 zeros between the decimal point and the first nonzero digit in the decimal expansion. So none of the given numbers are possible values of t.

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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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(t/1000)^4=(t/(10^3)^4=t^4/10^12
10^12 will move decimel point to the left 12 times
if we test 9=> 9^4= 6561 or 0,6*10^4
So 4-12=8.
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If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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1
chetan2u wrote:
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

Hi,

since we have ONLY 10s in denominator, the # of zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4 will depend on 't'...

ONE 10 will put the decimal point and the REMAINING 10s will result in " zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4"..

so we have $$10^{12}$$, so if we have t as a single digit, there will be 11 zeroes, that is 12 - number of digits in t.....
BUT 0s are <8, so t should have digits>12-8 or >4......

lets see the choices -

I. 3 -----$$3^4 = 81$$... two digits ..# of 0s $$= 12 - 2 = 10 > 8.$$...NO
II. 5 ------$$5^4 = 625$$... three digits ..# of 0s = 12 - 3 = 9 > 8....NO
III. 9------$$9^4 = 6561$$... four digits ..# of 0s = 12 - 4 = 8 = 8....NO, we are looking for <8

none
A

Hi, do you agree with this rationale?

According to the stem, [(t/1000)^4]>1x10^-8
1x10^-8 has 7 0s between the decimal and first non-zero digit. 1x10^-8 is the smallest number with 7 0s between the decimal and first non-zero digit.

Therefore, t^4>=(1x10^-8)*(1x10^12)
t^4>=(1x10^4)
t^4>=10000

None of the choices for t give t^4 greater than 10000 and therefore the answer is none.
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

The number of zeroes between 't' and the decimal point has to be <8. Then (t/10^(7 or lesser)) matches.
Now we need to take into account the size of 't'. So (t/10^(7 or lesser+size of 't'))

The expression in question is t^4/10^12; to match this size of 't' should be anything above or equal to 5.
t^4 with no option has the size>5, so option A
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

(t^4)/(10^12) should have less then 8 zero between first non zero number

option 1 81/( 10^12)
option 2 625/(10^12)
option 3 729 /(10^12)

none of them satisfies given requirement
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

My 2 cents.

I think it is easier if we change as this :

= (t x 10^-3)^4
= t^4 x 10^-12

So, in here, if t =1, we have 12 zeroes.
So what we need to find is how many digits do t^4 contains.

I. 3
3^4 is two digit number, meaning, there will be 10 zeroes...so this is no good

II. 5
5^4 is three digit number, meaning, there will be 9 zeroes...so no good

III. 9
9^4 is four digit number, meaning, there will 8 zeroes...so no good.

Hence, A
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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JeffTargetTestPrep wrote:
Bunuel wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

We are given that the decimal expansion of (t/1000)^4 has fewer than 8 zeroes between the decimal point and the first nonzero digit. We are also given that 3, 5 and 9 are possible values of t. Let’s test each Roman numeral:

I. 3

If t = 3, then (t/1000)^4 = (3/1000)^4 = (.003)^4 has twelve decimal places with the digits 81 (notice that 3^4 = 81). So there must be 10 zeros between the decimal point and the first nonzero digit 8 in the decimal expansion. This is not a possible value of t.

II. 5

If t = 5, then (t/1000)^4 = (5/1000)^4 = (.005)^4 has twelve decimal places with the digits 625 (notice that 5^4 = 625). So there must be 9 zeros between the decimal point and the first nonzero digit 6 in the decimal expansion. This is not a possible value of t.

III. 9

If t = 9, then (9/1000)^4 = (9/1000)^4 = (.009)^4 has twelve decimal places with the digits 6561 (notice that 9^4 = 6561). So there must be 8 zeros between the decimal point and the first nonzero digit 6 in the decimal expansion. This is not a possible value of t.

Recall that we are looking for fewer than 8 zeros between the decimal point and the first nonzero digit in the decimal expansion. So none of the given numbers are possible values of t.

Perfect !!!!! Kudos
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If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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1
$$(\frac{t}{1000})^4$$ in decimal form is $$(0.000t)^4$$. This means that $$0.000t$$ is multiplied to itself 4 times. It will be important to observe that:

$$0.0003 \times 0.0003 = 0.000009$$ (6 digits: 5 zeroes, 1 nonzero digit)
$$0.0005 \times 0.0005 = 0.000025$$ (4 digits: 4 zeroes, 2 nonzero digits)
$$0.0009 \times 0.0009= 0.000081$$ (4 digits: 4 zeroes, 2 nonzero digits)

These examples show that when a decimal has 3 places containing zeroes is raised to its second power, we’re expecting 3 x 2 = 6 digits after the decimal point.
Similarly, we’re expecting 4 x 3 = 12 numbers after the decimal place when 0.0003, 0.0005, and 0.0009 are raised to the fourth power.

First, let’s check the values of$$3^4$$,$$5^4$$, and $$9^4$$:

$$3^4 = 81$$, $$5^4 = 625$$, and $$9^4 = 6561$$

This means that for$$(0.0003)^4$$ or $$(3/1000)^4$$, there will be 12-2 = 10 zeroes and 2 nonzero digits after the decimal point.

Similarly, $$(5/1000)^4$$ will have 9 zeroes and 3 nonzero digits while $$(9/1000)^4$$ will have 8 zeroes and 4 nonzero digits after the decimal point.

We can see that for the three options, none of them had fewer than 8 zeroes. Thus, the final answer is .
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Re: If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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(t/1000)^4 = t^4/10^12 = 0.000..x (with maximum 7 digit 0 after . and before x and x>=1) so t^4/10^12 >= 1/10^8
==> t^4 >= 10^4 ==> t>=10
None of the answers is true => Option A
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If there are fewer than 8 zeroes between the decimal point and the  [#permalink]

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Consider simple fractions
$$\frac{1}{1000}$$=0.001 (1 digit give n-1 0s)
$$\frac{11}{1000}$$=0.011 (1 digit give n-2 0s)
$$\frac{111}{1000}$$=0.111 (1 digit give n-3 0s)

We clearly know that question has 12 0s in the denominator ($$10^3$$= 3 0s, $$10^12$$=12 0s)

So it must have less than 8 0s, which means it has at max 7 0s. Let us consider the case with 7 0s after the decimal, thus numerator must have n-7 digits i.e 12-7 = 5 digits

Not let us see the number of digits for 1,2,3
I. 3 => 3^4 =81 NO
II. 5 => 5^4 =625 NO
III. 9 => 9^4 =81*81=729*9=4 digits NO

Thus the correct answer is A. None If there are fewer than 8 zeroes between the decimal point and the   [#permalink] 14 Sep 2019, 06:17

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