If there are fewer than 8 zeroes

This topic has expert replies
Master | Next Rank: 500 Posts
Posts: 186
Joined: Sat Dec 24, 2016 12:38 am
Thanked: 5 times
Followed by:3 members

If there are fewer than 8 zeroes

by rsarashi » Thu Mar 09, 2017 9:24 am

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

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

OAA

User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Thu Mar 09, 2017 9:37 am
rsarashi 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 smallest value that has 7 zeroes to the right of the decimal point is 0.00000001.
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)� cannot have more than 7 zeroes to the right of the decimal point, (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)� ≥ 0.00000001
(t/10³)� ≥ 1/10�
t�/10¹² ≥ 1/10�
10�t� ≥ 10¹²
t� ≥ 10�.

None of the three options for t satisfies the constraint that t� ≥ 10�.

The correct answer is A.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Master | Next Rank: 500 Posts
Posts: 415
Joined: Thu Oct 15, 2009 11:52 am
Thanked: 27 times

by regor60 » Thu Mar 09, 2017 9:46 am
(t/1000)^4 is the same thing as t^4/1000^4 = t^4/(10^3)^4 = t^4/10^12

In general, 1/10^X has X-1 zeros after the decimal and before first nonzero digit, for example, 1/10^3 = .001

So, 1/10^8 has 7 zeroes, which is less than 8, as the question requires

So, the goal is to make t^4/10^12 = 1/10^8.

Cross multiplying to solve for t^4 = 10^12/10^8 = 10^4

Therefore, t has to be equal to or greater than 10 to satisfy question, therefore answer is A

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Thu Mar 09, 2017 9:48 pm
rsarashi 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

OAA
Hi rsarashi,

A number that has seven 0s after the decimal and before a non-zero digit can be represented by 0.0000000x; where x is a non-zero digit.

We can write 0.0000000x as x/(10^8)

We have a number (t/1000)^4 that has fewer than eight 0s after the decimal and before the first non-zero digit.

Thus, (t/1000)^4 ≥ x/(10^8); where x = 1; 1/(10^8) is the smallest number that has seven 0s after the decimal and before a non-zero digit.

Thus, (t/1000)^4 ≥ 1/(10^8)

=> t^4/10^12 ≥ 1/10^8

=> (t^4/10^4)*(1/10^8) ≥ 1/10^8

=> t^4/10^4 ≥ 1

=> t ≥ 10 or -10 ≥ t

No option qualifies.

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Tokyo | Manchester | Geneva | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Thu Mar 09, 2017 10:05 pm
rsarashi 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

OAA
Another approach...

Since there are only three values: 3, 5, and 9, we can plug-in and test.

Let's test t = 9. We chose the largest of the option values since if t = 9 fails, others need not be tested as they would return relatively less value than the one that is at t = 9.

@ t = 9,

(t/1000)^4 = t^4/10^12 = 9^4/10^12 = 6561 / 10^12.

Since the numerator 6541 is a four-digit number and the exponent of 10 is 12, the decimal number would have eight 0s [12 - 4 = 8] after the decimal and before 6541.

=> 6561 / 10^12 = 0.000000006541.

To have (t/1000)^4 with seven or less than seven 0s after the decimal and before the first non-zero digit, t^4 must be at least a five digit number. The smallest five digit number is 10000.

=> t^4 ≥ 10000

=> t^4 ≥ 10^4

=> t ≥ 10 or -10 ≥ t

No option qualifies!

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Math Essentials Guide

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Tokyo | Manchester | Geneva | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

Master | Next Rank: 500 Posts
Posts: 186
Joined: Sat Dec 24, 2016 12:38 am
Thanked: 5 times
Followed by:3 members

by rsarashi » Sat Mar 11, 2017 5:41 pm
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)� cannot have more than 7 zeroes to the right of the decimal point, (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)� ≥ 0.00000001
Hi GMATGuruNY ,

Thank you so much for your reply.

Just a quick question. Can you please explain that why (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001?

Please explain.

Thanks..

User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Sun Mar 12, 2017 3:52 am
rsarashi wrote:
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)� cannot have more than 7 zeroes to the right of the decimal point, (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)� ≥ 0.00000001
Hi GMATGuruNY ,

Thank you so much for your reply.

Just a quick question. Can you please explain that why (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001?

Please explain.

Thanks..
If (t/1000)� = 0.00000001, then (t/1000)� has EXACTLY 7 ZEROES to the right of the decimal point.
Since 0.00000001 is the smallest number with exactly 7 zeroes to the right of the decimal point, any value less than 0.00000001 must have 8 OR MORE ZEROES to the right of the decimal point.
For example, the next smallest value than 0.00000001 -- 0.00000000999, where the 9's repeat forever -- has 8 zeroes to the right of the decimal point.
Thus, for (t/1000)� to have fewer than 8 zeroes to the right of the decimal point, it cannot be less than 0.00000001.
In the words, (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)� ≥ 0.00000001.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Wed Mar 15, 2017 3:29 pm
rsarashi 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 of these numbers:

I. 3

If t = 3, then (t/1000)^4 = (3/1000)^4 = (.003)^4 has twelve decimal places to the right of the decimal point with the digits 81 as the 2 rightmost digits (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 to the right of the decimal point with the digits 625 as the 3 rightmost digits (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 to the right of the decimal point with the digits 6561 as the 4 rightmost digits (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

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Thu Mar 16, 2017 8:15 pm
rsarashi wrote: Just a quick question. Can you please explain that why (t/1000)� must be GREATER THAN OR EQUAL TO 0.00000001?
Because "the next smallest number"* would be .000000009999...(whatever), and that'd have at least 8 zeros. So the smallest such number that satisfies the constraints of our problem would be .00000001.

*In reality, .000000099999.... (9 forever) actually is the same number! But it gives a good enough idea, any decimal smaller than but arbitrarily close to .00000001 would be .0000000099999...(some numbers).

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Sun Jul 22, 2018 7:21 am
rsarashi 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

OAA
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
Brent Hanneson - Creator of GMATPrepNow.com
Image

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Wed Jul 25, 2018 4:34 pm
rsarashi 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
If t = 1, then (1/1000)^4 = (1/10^3)^4 = 1/10^12. Therefore, the decimal expansion would have 12 decimal places with the last (rightmost) digit being a 1. That is, there are 11 zeros between the decimal point and the last digit 1. For any of the given t values, 3, 5 and 9, it will not change the number of decimal places; however, it may change the number of zeros between the decimal point and the first nonzero digit.

If t = 3, then t^4 = 3^4 = 81. So 81 will occupy the last two of the 12 decimal places, that means there are 10 zeros between the decimal point and the first nonzero digit 8.

If t = 5, then t^5 = 5^4 = 625. So 625 will occupy the last three of the 12 decimal places, that means there are 9 zeros between the decimal point and the first nonzero digit 6.

If t = 9, then t^5 = 9^4 = 6561. So 6561 will occupy the last four of the 12 decimal places, that means there are 8 zeros between the decimal point and the first nonzero digit 6.

Since we are looking for fewer than 8 zeroes between the decimal point and the first nonzero digit, none of the given t values will make this happen.

Answer: A

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Junior | Next Rank: 30 Posts
Posts: 10
Joined: Sat Oct 27, 2018 10:35 am

My Method

by ca7ch22 » Wed Mar 06, 2019 4:10 pm
The simplest method for me was to just write out the numbers and count the zeros.

$$(\frac{t}{1,000})^4 = \frac{t^4}{10^{12}}$$
If t = 3:
$$\frac{81}{10^{12}}$$
0.000 000 000 081 = 10 zeroes

If t = 5:
$$\frac{625}{10^{12}}$$
0.000 000 000 625 = 9 zeroes

If t = 9:
$$\frac{6,561}{10^{12}}$$
0.000 000 006 561 = 8 zeroes

Answer (A): NONE

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800
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
Contact Rich at [email protected]
Image