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If there are fewer than 8 zeroes

This topic has 6 expert replies and 2 member replies
rsarashi Master | Next Rank: 500 Posts Default Avatar
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If there are fewer than 8 zeroes

Post Thu Mar 09, 2017 9:24 am
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

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Post Sun Mar 12, 2017 3:52 am
rsarashi wrote:
Quote:
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)⁴ cannot have more than 7 zeroes to the right of the decimal point, (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)⁴ ≥ 0.00000001
Hi GMATGuruNY ,

Thank you so much for your reply.

Just a quick question. Can you please explain that why (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001?

Please explain.

Thanks..
If (t/1000)⁴ = 0.00000001, then (t/1000)⁴ 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)⁴ 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)⁴ must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)⁴ ≥ 0.00000001.

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If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
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Top Reply
Post 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

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Jeffrey Miller Head of GMAT Instruction

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Post Sun Mar 12, 2017 3:52 am
rsarashi wrote:
Quote:
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)⁴ cannot have more than 7 zeroes to the right of the decimal point, (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)⁴ ≥ 0.00000001
Hi GMATGuruNY ,

Thank you so much for your reply.

Just a quick question. Can you please explain that why (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001?

Please explain.

Thanks..
If (t/1000)⁴ = 0.00000001, then (t/1000)⁴ 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)⁴ 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)⁴ must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)⁴ ≥ 0.00000001.

_________________
Mitch Hunt
GMAT Private Tutor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.

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Post 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

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Post Thu Mar 16, 2017 8:15 pm
rsarashi wrote:
Just a quick question. Can you please explain that why (t/1000)⁴ 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).

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rsarashi Master | Next Rank: 500 Posts Default Avatar
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Post Sat Mar 11, 2017 5:41 pm
Quote:
Any value less than 0.00000001 will have MORE than 7 zeroes to the right of the decimal point.
Since (t/1000)⁴ cannot have more than 7 zeroes to the right of the decimal point, (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001:
(t/1000)⁴ ≥ 0.00000001
Hi GMATGuruNY ,

Thank you so much for your reply.

Just a quick question. Can you please explain that why (t/1000)⁴ must be GREATER THAN OR EQUAL TO 0.00000001?

Please explain.

Thanks..

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Post 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
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Thanked by: rsarashi
Post 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
_________________
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Locations: New York | Tokyo | Manchester | Geneva | and many more...

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