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

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

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

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

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

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.

\(\frac{t^4}{1000^4}\) 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?

\((\frac{t}{1000})^4\) 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 .

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.

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

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.

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:

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

Something divided by 10^12 must give at most 7 zeros between the decimal point and the first non-zero digit.

1/(10^12) = a total of 12 zeros, where 11 comes after the decimal point. So we have to add 4 zeros to the numerator.

10000/(10^12) = a total of 12-4=8 zeros, where 7 comes after the decimal point.

The least value of t^4 is 10000, or 10^4.

\((t/1000)^4\) ------- > t\(^4/((10)^3)^4\)-----> \(t^4 /(10)^12 \)

plug 3,5,9 ------ > Number of zeros after the decimals has to be FEWER THAN 8 ( can't be = 8)

\(3^4/10^12\) ----------> 9 zeros before a non-zero value.

\(5^4/10^12 \)------------>9 zeros before a non-zero value.

\(9^4/10^12\) ------------> 8 zeros before a non-zero value.

hence, none of the values has fewer than 8 zeros before the first non-zero number. A.

Answer: Option A

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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

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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

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