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If there are fewer than 8 zeroes

This topic has 6 expert replies and 2 member replies

Top Member

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
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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 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)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
    (t/10³)⁴ ≥ 1/10⁸
    t⁴/10¹² ≥ 1/10⁸
    10⁸t⁴ ≥ 10¹²
    t⁴ ≥ 10⁴.

    None of the three options for t satisfies the constraint that t⁴ ≥ 10⁴.

    The correct answer is A.

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    Post 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

    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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    Thanked by: rsarashi
    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
    _________________
    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.

    Thanked by: rsarashi

    Top Member

    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..

    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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    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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