Welcome! Check out our free B-School Guides to learn how you compare with other applicants.
Login or Register

Terminating Decimals

This topic has 3 expert replies and 13 member replies
Goto page
  • 1,
  • 2
Next
chrisjim5 Just gettin' started! Default Avatar
Joined
06 Oct 2010
Posted:
8 messages
Terminating Decimals Post Thu Oct 21, 2010 11:02 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Here is a question:

    Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 36, 0.72, and 3.005 are terminating decimals.
    If a, b, c, d and e are non-negative integers and p = 2^a*3^b and q = 2^c*3^d*5^e, is p/q a terminating decimal?

    (1) a > c
    (2) b > d

    Can you please provide an answer to this?

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!

    GMAT/MBA Expert

    Ian Stewart GMAT Instructor
    Joined
    02 Jun 2008
    Posted:
    2178 messages
    Followed by:
    289 members
    Thanked:
    994 times
    GMAT Score:
    780
    Post Thu Oct 21, 2010 11:38 am
    chrisjim5 wrote:
    Here is a question:

    Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 36, 0.72, and 3.005 are terminating decimals.
    If a, b, c, d and e are non-negative integers and p = 2^a*3^b and q = 2^c*3^d*5^e, is p/q a terminating decimal?

    (1) a > c
    (2) b > d

    Can you please provide an answer to this?
    You can recognize whether a fraction will produce a terminating decimal by:

    1. Reducing your fraction completely

    2. Then looking only at the prime factors of the denominator. If the denominator has any prime factor besides 2 or 5, the fraction will give a *repeating* (infinite) decimal. If the only prime factors of the denominator are 2 and/or 5, the fraction will give a terminating decimal.

    So fractions like 3/16 (only prime factor of denominator is 2), 9/125 (only prime factor of denominator is 5) and 3/40 (only prime factors of denominator are 2 and 5) will all produce terminating decimals. Fractions like 1/13, 9/35, and 11/120 will all produce non-terminating decimals since each is completely reduced, and has a factor different from 2 or 5 in the denominator. The first step above is critical; while a fraction like 7/35 might appear to have a factor of 7 in the denominator, that 7 actually cancels with the 7 in the numerator to give us 1/5, a terminating decimal.

    So in this question, we have the fraction:

    (2^a*3^b) / (2^c*3^d*5^e)

    The only reason this might not terminate is because of the 3's; if our 3^d in the denominator does not cancel out completely, we will get a repeating decimal. If it does cancel, we will get a terminating decimal. Statement 2 tells us that it will cancel completely, so is sufficient. Statement 1 doesn't help. So the answer is B.

    _________________
    Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

    private GMAT tutor in Toronto

    GMAT/MBA Expert

    fskilnik GMAT Instructor
    Joined
    09 Oct 2010
    Posted:
    306 messages
    Followed by:
    21 members
    Thanked:
    54 times
    Post Thu Oct 21, 2010 11:59 am
    The solution provided by Ian is perfect and very informative, but I guess many test takers go to the exam without this knowledge... that´s why I believe the solution I present below is also nice:

    From the "a, b, c, d and e are non-negative integers" hypothesis, I believe the question should be "seen" as:

    2^(a-c) * 3^(b-d) over 5^e is a terminating decimal ?

    Important: this is just a visually helpful thing, because (for instance) the value of (a-c) may be negative, and that means that 2^(a-c) can be at the denominator, "in reality"...

    (1) This sttm tells us that 2 is really in numerator, but what about the 3(´s) ?

    > Take a = 2, c =1 (to be only 2^1) and b = 2 and d = 1 (to let 3 be 3^1, so numerator...) and e = 1 , then we have: (2 * 3) over 5, and if you multiply both numerator and denumerator by 2 , you get (2^2 * 3) over... 10, that is, certainly terminating because it is an integer divided by 10, so you just move decimal point, you do not "alter terminallity"...

    > Take a =2 , c = 1 (to be only 2^1 again) and b = 1 and d = 2 (to let 3 be 3^(-1), so denumerator) and e =1 , then we have: 2 over (3 times 5) and now we know we are with a non-terminating decimal because of this (for instance):

    2/(15) = 2*2 / (15*2) = 4/30 = (1/10) * (4/3) and divide by 10 does not alter the "terminallity of a certain decimal" (as mentioned above) and we know that 4/3 is not terminating, because it is equal to 1+ 1/3 and (1 is an integer and) 1/3 is non-terminating, for sure (0.333333...)

    Obs.: this is not stupid calculations, I believe. This is the "insight" that is behind Ian´s statements...

    (2) Now we know that 3´s are on the numerator, therefore we may have only 2´s , only 5´s or both in the denumerator. From all shown, you should be able to recognize that 2´s and 5´s are no problem, because multiplying by 10´s in enough quantity you turn the fractions into integers, therefore terminating decimals.

    This one DECIDES affirmatively on the question asked, that is, (2) is sufficient.

    Regards,
    Fabio.

    _________________
    GMATH high-level Quant Prep
    www.GMATH.NET

    Thanked by: chrisjim5, elenaelena, IngaLav
    g.manukrishna Just gettin' started!
    Joined
    24 Sep 2010
    Posted:
    6 messages
    Followed by:
    1 members
    Thanked:
    8 times
    Post Thu Jan 06, 2011 1:51 am
    The ans is B

    when we simplify the question we get 2^(a-c) * 3^(b-d) / 5^e

    when we have prime factors other than 2 and 5 in the denominator we get non-terminating decimal.

    1) a>c is in sufficient. If a>c or a<c it doesnt matter since we don't know about b and d.
    2) if b>d there is no 3 in the denominator. So, we have terminating number.

    Hence 2 alone is sufficient to ans the question.

    bblast GMAT Titan
    Joined
    13 Dec 2010
    Posted:
    1079 messages
    Followed by:
    33 members
    Thanked:
    117 times
    Test Date:
    9th Sept 2011
    Target GMAT Score:
    730+
    GMAT Score:
    710
    Post Thu Jan 06, 2011 4:55 am
    mgmat cat question hmmmmmmmmm

    already enough on this by the experts, my say in short.

    remember 2,4,6,8 and 10(because it has 2 and 5) are terminators(the bad guys)(schwarzenegger in terminator 1)


    any fraction with additional 3,6(because it has 3),7 and 9(because it has 3) as denominators is not a terminator and the good guy(schwarzenegger in terminator 2)

    hope this helps, u'll remember this now Smile

    this question can simply be rephrased as

    is the no of 3's in p greater than no of 3's in q ???
    i'e
    is b>d

    the answer is right in one of the statements 8)

    _________________
    Cheers !!

    Quant 47-Striving for 50
    Verbal 34-Striving for 40

    My gmat journey :
    http://www.beatthegmat.com/710-bblast-signing-off-thank-you-all-t90735.html
    My take on the GMAT RC :
    http://www.beatthegmat.com/ways-to-bblast-the-gmat-rc-t90808.html
    How to prepare before your MBA:
    https://www.youtube.com/watch?v=upz46D7l8fA&list=PLUmBNvYMnppJRMpR9fwfcsTWBZF14TKW_

    saketk GMAT Destroyer! Default Avatar
    Joined
    19 Jun 2011
    Posted:
    608 messages
    Followed by:
    7 members
    Thanked:
    36 times
    Target GMAT Score:
    700+
    Post Sun Oct 02, 2011 11:09 am
    Nothing to add here. +1 for B

    GmatKiss GMAT Titan Default Avatar
    Joined
    26 Jul 2011
    Posted:
    2789 messages
    Followed by:
    40 members
    Thanked:
    205 times
    Target GMAT Score:
    700+
    GMAT Score:
    640
    Post Mon Oct 03, 2011 9:48 am
    IMO:B

    Rastis Really wants to Beat The GMAT!
    Joined
    21 Sep 2011
    Posted:
    102 messages
    Followed by:
    1 members
    Thanked:
    2 times
    Test Date:
    April 2012
    Target GMAT Score:
    700
    Post Fri Nov 18, 2011 11:14 am
    Waaaaaaaaaaaaaay to hard

    Post Sun Jan 01, 2012 10:34 am
    p/q = 2^(a-c)*3^(b-d)/5^e

    Except b<d where 3 comes in the denominator in all other cases the fraction is a terminating decimal

    Statement 1: INSUFFICIENT

    Statement 2: SUFFICIENT

    ronnie1985 GMAT Destroyer!
    Joined
    23 Dec 2011
    Posted:
    626 messages
    Followed by:
    9 members
    Thanked:
    29 times
    Test Date:
    June
    Target GMAT Score:
    750
    Post Fri Mar 30, 2012 9:10 am
    if the fraction has powers of 2 or 5 only hen the fraction is terminating decimal
    S1: a>c does not tell about power of 3
    S2: b-d>0 hence the fraction is terminating decimal
    (B) is ans

    _________________
    Follow your passion, Success as perceived by others shall follow you

    shubhamkumar Rising GMAT Star Default Avatar
    Joined
    11 Apr 2011
    Posted:
    72 messages
    Thanked:
    2 times
    Post Mon Apr 02, 2012 4:06 am
    Ian Stewart wrote:
    chrisjim5 wrote:
    Here is a question:

    Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 36, 0.72, and 3.005 are terminating decimals.
    If a, b, c, d and e are non-negative integers and p = 2^a*3^b and q = 2^c*3^d*5^e, is p/q a terminating decimal?

    (1) a > c
    (2) b > d

    Can you please provide an answer to this?
    You can recognize whether a fraction will produce a terminating decimal by:

    1. Reducing your fraction completely

    2. Then looking only at the prime factors of the denominator. If the denominator has any prime factor besides 2 or 5, the fraction will give a *repeating* (infinite) decimal. If the only prime factors of the denominator are 2 and/or 5, the fraction will give a terminating decimal.

    So fractions like 3/16 (only prime factor of denominator is 2), 9/125 (only prime factor of denominator is 5) and 3/40 (only prime factors of denominator are 2 and 5) will all produce terminating decimals. Fractions like 1/13, 9/35, and 11/120 will all produce non-terminating decimals since each is completely reduced, and has a factor different from 2 or 5 in the denominator. The first step above is critical; while a fraction like 7/35 might appear to have a factor of 7 in the denominator, that 7 actually cancels with the 7 in the numerator to give us 1/5, a terminating decimal.

    So in this question, we have the fraction:

    (2^a*3^b) / (2^c*3^d*5^e)

    The only reason this might not terminate is because of the 3's; if our 3^d in the denominator does not cancel out completely, we will get a repeating decimal. If it does cancel, we will get a terminating decimal. Statement 2 tells us that it will cancel completely, so is sufficient. Statement 1 doesn't help. So the answer is B.
    Nice Logic Ian.Thanks!!
    Arrow If the denominator has any prime factor besides 2 or 5, the fraction will give a *repeating* (infinite) decimal. If the only prime factors of the denominator are 2 and/or 5, the fraction will give a terminating decimal.

    Rastis Really wants to Beat The GMAT!
    Joined
    21 Sep 2011
    Posted:
    102 messages
    Followed by:
    1 members
    Thanked:
    2 times
    Test Date:
    April 2012
    Target GMAT Score:
    700
    Post Thu Apr 12, 2012 6:27 am
    Can someone provide and easier to understand explanation please?

    Shalabh's Quants Really wants to Beat The GMAT!
    Joined
    06 Apr 2012
    Posted:
    134 messages
    Followed by:
    5 members
    Thanked:
    35 times
    Post Thu Apr 12, 2012 7:24 am
    Rastis wrote:
    Can someone provide and easier to understand explanation please?
    Relatively easier approach for you.

    Lets put this question as 2^(a-c).3^(b-d)/5^e.

    Its for sure that this question wishes us to predict nature of 2^(a-c).3^(b-d)/5^e by deducing nature of 2^(a-c), 3^(b-d), & 5^e.

    => lets put a=c=b=d=0. This will help us find the nature of 5^e.

    => so 2^(a-c).3^(b-d)/5^e = 2^0.3^0/5^e = 1/5^e.

    => Now you may try few values of e such as 1, 2, 3... to see if 1/5^e is terminating or non-terminating. We can conclude 1/5^e is terminating.So 5^e has no role to decide if expression is non-terminating.


    Now lets put b=d=e=0. This will help us find the nature of 2^(a-c).

    => so 2^(a-c).3^(b-d)/5^e = 2^(a-c).3^0/5^0 = 2^(a-c).

    => Now you may try few values of (a-c) such as ...., -3, -2, -1 to see if 2^(a-c) is terminating or non-terminating. We can conclude 2^(a-c) is terminating.So 2^(a-c) has no role to decide if expression is non-terminating. Whether a > < c. So stat. 1 is not necessary. Redundant.


    Now lets put a=c=e=0. This will help us find the nature of 3^(b-d).

    => so 2^(a-c).3^(b-d)/5^e = 2^0.3^(b-d)/5^0 = 3^(b-d).

    => Now you may try few values of(b-d) such as ...., -3, -2, -1 to see if 3^(b-d) is terminating or non-terminating. We can conclude 3^(b-d) is non-terminating if (b-d) is negative.So to make it terminating (b-d) should be positive or b>d. So stat. 2 is not necessary & sufficient.


    Ans B.

    _________________
    Shalabh Jain,
    e-GMAT Instructor

    GMAT/MBA Expert

    Post Thu Apr 12, 2012 7:45 am
    Rastis wrote:
    Can someone provide and easier to understand explanation please?
    My explanation is similar to the ones above, but perhaps you will find it helpful:

    http://www.beatthegmat.com/terminating-decimal-t92476.html

    _________________
    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.
    Contact me about long distance tutoring!

    Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
    rajeshsinghgmat Really wants to Beat The GMAT! Default Avatar
    Joined
    08 Jan 2013
    Posted:
    171 messages
    Thanked:
    1 times
    Post Thu Feb 28, 2013 2:02 am
    B in answer.

    Best Conversation Starters

    1 j_shreyans 82 topics
    2 aditya8062 24 topics
    3 abhasjha 21 topics
    4 RiyaR 19 topics
    5 tanvis1120 14 topics
    See More Top Beat The GMAT Members...

    Most Active Experts

    1 image description GMATGuruNY

    The Princeton Review Teacher

    155 posts
    2 image description Brent@GMATPrepNow

    GMAT Prep Now Teacher

    141 posts
    3 image description David@GMATPrepNow

    GMAT Prep Now Teacher

    67 posts
    4 image description David@VeritasPrep

    Veritas Prep

    51 posts
    5 image description Jim@StratusPrep

    Stratus Prep

    43 posts
    See More Top Beat The GMAT Experts