If there are fewer than 8 zeroes between the decimal point and the

Question

If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of  (\frac{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

chetan2u · Accepted Answer

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

PGTLrowanhand · Answer

Hi GMATters,

Here's my video explanation of this question:

VnLRYs2jd5w

Enjoy!

Rowan

BrentGMATPrepNow · Answer

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

Answer: A

Cheers,
Brent

JeffTargetTestPrep · Answer

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.

Answer: A

shivmalhotra10 · Answer

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 · Answer

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

manlog · Answer

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.

vfcordeiro · Answer

\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?

EMPOWERMathExpert · Answer

(\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 	imes 0.0003 = 0.000009  (6 digits:  5 zeroes ,  1 nonzero digit )
 0.0005 	imes 0.0005 = 0.000025  (4 digits:  4 zeroes ,  2 nonzero digits )
 0.0009 	imes 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  A .

unraveled · Answer

In  t^4 * 10^{-12}  for it to have 0's < 8 the number  t^4  must be a 5-digit number. And the least 5-digit number is 10000 i.e.  10^4 .
Hence  t^4 = 10^4  or greater.
So, the question stem becomes t ≥ 10?

As we can see none of the options among I, II and III satisfy the condition, the answer is NONE.

Answer A.

MHIKER · Answer

(\frac{t}{1000})^4, \ which \ follows \ that \frac{t^4}{(1000)^4}=\frac{t^4}{(10^3)^4} =\frac{t^4}{10^{12}}

So,  t^4  is being divided by  10^{12} , which will leave 12 zeros behind  t , if  t=1,

We need  0<8,  Let's check the options:

I.  t=3, 3^4=\frac{81}{10^{12}}; 12-2=10 , zeros behind. Eliminated

II.  t=5, 5^4=\frac{625}{10^{12}}; 12-3=9 , zeros behind. Eliminated

III.  t=9, 9^4=\frac{6541}{10^{12}}=12-4=8 , zeros behind. Eliminated, as we fewer than 8 zeros.

The answer is A.

EMPOWERgmatRichC · Answer

Hi All,

We’re told that there are FEWER than 8 zeroes between the decimal point and the first NON-ZERO digit when converting (T/1000)^4 to a decimal. We’re asked which of the following 3 numbers could be the value of T.

While this question looks a bit complicated, it’s based on some standard multiplication rules that you probably already know. As such, we can beat it with a bit of brute force and some Arithmetic.

I. 3…. If T = 3, then we have (3/1000)^4. The numerator of this fraction is 3^4 = (3)(3)(3)(3) = (9)(9) = 81. The number 1000 has 3 zeroes, so when we multiply (1000)(1000)(1000)(1000), we’ll have 12 zeroes… which means that this denominator will create 12 decimal places.

If you don’t immediately realize that last pattern, then you can use a few simple examples to prove it.

3/10 = 0.3… which is 1 decimal place
3/100 = 0.03… which is 2 decimal places
3/1000 = 0.003… which is 3 decimal places
Etc.

The ‘81’ in the numerator would take up the last 2 ‘spots’ in the 12 decimal places, which means that there would be 10 zeroes before that ’81.’ This does NOT match what we were told though (there are supposed to be FEWER than 8 zeroes), so 3 CANNOT be the value of T. Eliminate Answer B.

Using this same approach, we can now work through the other two Roman Numerals….

II. 5… If T = 5…. Then (5)(5)(5)(5) = (25)(25) = 625. The ‘625’ would take up the last 3 ‘spots’ in the 12 decimal places, which means that there would be 9 zeroes before that ‘625.’ This also does NOT match what we were told (again, there are supposed to be FEWER than 8 zeroes), so 5 CANNOT be the value of T. Eliminate Answers C and E.

III. 9 … If T = 9…. Then (9)(9)(9)(9) = (81)(81) = 6561. The ‘6561’ would take up the last 4 ‘spots’ in the 12 decimal places, which means that there would be 8 zeroes before that ‘6561.’ This also does almost what we’re looking for, but it does NOT match what we were told (again, there are supposed to be FEWER than 8 zeroes; this would give us EXACTLY 8 zeroes), so 59CANNOT be the value of T. Eliminate Answer D

Final Answer:  A

GMAT Assassins aren’t born, they’re made,
Rich

GMATinsight · Answer

Answer: Option A

Video solution by  GMATinsight

https://www.youtube.com/watch?v=8Ojv_pev_F8

TobbeSwe · Answer

This is how I solved it:

1) Find the least possible value that would fit the requirement --> 10^(-8) = 0,00000001

2) Simplify the original equation --> (t/1000)^4 = (t*10^(-3))^4 = (t^4)*(10^(-12))

3) Set the equation equal to each other --> (t^4)*(10^(-12)) = 10^(-8)

4) Multiply in 10^8 on both sides of the equation --> (t^4)*(10^(-12))*(10^8) = 10^(-8)*(10^8) --> (t^4)*(10^(-4)) = 1

5) Solve equation: From the above equation we see that the least possible value of t that would fit the requirement is 10: (10^4)*(10^(-4)=10^0=1

6) Compare with numerals: Since all numeral values are <10 our answer is A), None

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=r-fPnPi5ubE

joe123x · Answer

I think another way is to acknowledge that the least value of t that satisfy this restriction is actually 10, since 10^4/ 10^12 gives us 10^-8

KarishmaB · Answer

Responding to a pm:

(\frac{t}{1000})^4 
To understand what the question is saying, put t = 1. We will get
 (\frac{1}{1000})^4 = \frac{1}{100..(12 zeroes)} = .000.. (11 zeroes)1

If we want 7 or fewer zeroes, we must have 5 non-zero digits in the numerator. e.g.
 \frac{12345}{100...(12 zeroes)} = .000.. (7 zeroes)12345

I. 3
3^4 = 81 (just 2 digits)
Doesn't work

II. 5
5^4 = 625 (just 3 digits)
Doesn't work

III. 9
9^4 = 729*9 which will be 4 digits because 729*10 = 7290
Doesn't work.

Answer (A)

Note that 10^4 = 10000 is the smallest 5 digit number so we will need the value of t to be at least 10. If we are able to identify this up front, we won't need to do any calculations.

​​​​​​​

pc343 · Answer

not sure if this thread is still active, but I've got a maybe dumb question:

from reading through all the explanations here, I have a brief understanding of it, but can someone tell me theoretically why we can't simplify t/1000 first?

for the record I got this question wrong because of 5/1000

jaykayes · Answer

Fewer than 8 zeros means that the minimum value of the expression is 0.00000001
Write this in exponential form as 10^(-8)
So the minimum value of part inside the brackets will be the fourth root of this, which is 10^(-2)
This can be written as 1/10^2
Rearrange this to make it match the part inside the brackets.
1/10^2 = 10/10^3 = 10/1000 = t/1000
That means t must be a minimum of 10.
Therefore t cannot be = 3, 5, or 9.
Answer choice A.

If there are fewer than 8 zeroes between the decimal point and the

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