BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem
TTP OnDemand GMAT Video Course
Featured Course

Target Test Prep GMAT OnDemand

Scott Woodbury-Stewart’s private virtual classroom — 400 hours of master-class video lessons for the GMAT Focus Edition.

  • 715+ score guarantee — highest in the industry (99th percentile)
  • 52 chapters · 1,500+ lessons · 4,000+ practice questions
  • 400 hours of video · 1,500+ instructor-led HD smartboard lessons
  • 300,000+ students accepted to Harvard, Stanford, Wharton, Booth & Sloan & more
  • 24/7 live support + weekly Zoom office hours with GMAT instructors
  • TTP AI Assist — 24/7 AI-powered virtual tutor for instant help
  • 1,200+ flashcards + AI-powered study assistant & daily calendar
  • OnDemand, LiveTeach & GMAT Bootcamp formats available
  • Also: GRE, SAT Math & Executive Assessment courses
  • MBA Admissions Consulting now available
  • 🏆 2025 EdTech Breakthrough Award: Test Prep Solution Provider of the Year
  • 200,000+ students served
  • 5-day free trial — $0 to start, no auto-billing, cancel anytime
★★★★★ 5.0 (559 reviews)
130-pt guarantee
$0 to start then $127/mo
Start Free 5-Day Trial Read 559 Reviews →

Terminating Decimals

Expert replies
Post new topic Post Reply
Previous Topic Next Topic
Newbie | Next Rank: 10 Posts
Posts: 8
Joined: Wed Oct 06, 2010 11:35 am
Thanked: 1 times
by chrisjim5 » Thu Oct 21, 2010 11:02 am
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?
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Source: — Data Sufficiency |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 2623
Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members
Location: Montreal
Thanked: 1090 times
GMAT Score:780

by Ian Stewart » 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.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members
Thanked: 59 times

by fskilnik@GMATH » 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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Newbie | Next Rank: 10 Posts
Posts: 6
Joined: Fri Sep 24, 2010 2:09 amFollowed by: 2 members
Thanked: 10 times

by g.manukrishna » 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.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Legendary Member
Posts: 1077
Joined: Mon Dec 13, 2010 1:44 amFollowed by: 33 members
Thanked: 118 times
GMAT Score:710

by bblast » 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 :)

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 :
https://www.beatthegmat.com/710-bblast-s ... 90735.html
My take on the GMAT RC :
https://www.beatthegmat.com/ways-to-bbla ... 90808.html
How to prepare before your MBA:
https://www.youtube.com/watch?v=upz46D7 ... TWBZF14TKW_
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Legendary Member
Posts: 608
Joined: Sun Jun 19, 2011 11:16 amFollowed by: 8 members
Thanked: 37 times

by saketk » Sun Oct 02, 2011 11:09 am
Nothing to add here. +1 for B
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Legendary Member
Posts: 2789
Joined: Tue Jul 26, 2011 12:19 amFollowed by: 43 members
Location: Chennai, India
Thanked: 206 times
GMAT Score:640

by GmatKiss » Mon Oct 03, 2011 9:48 am
IMO:B
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Master | Next Rank: 500 Posts
Posts: 183
Joined: Wed Sep 21, 2011 7:06 amFollowed by: 2 members
Location: Washington, DC
Thanked: 6 times
GMAT Score:500

by Rastis » Fri Nov 18, 2011 11:14 am
Waaaaaaaaaaaaaay to hard
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 382
Joined: Thu Mar 31, 2011 5:47 pm
Thanked: 15 times

by ArunangsuSahu » 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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Legendary Member
Posts: 626
Joined: Fri Dec 23, 2011 2:50 amFollowed by: 10 members
Location: Ahmedabad
Thanked: 31 times

by ronnie1985 » 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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 72
Joined: Mon Apr 11, 2011 6:45 pm
Thanked: 2 times

by shubhamkumar » 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.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Master | Next Rank: 500 Posts
Posts: 183
Joined: Wed Sep 21, 2011 7:06 amFollowed by: 2 members
Location: Washington, DC
Thanked: 6 times
GMAT Score:500

by Rastis » Thu Apr 12, 2012 6:27 am
Can someone provide and easier to understand explanation please?
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
Master | Next Rank: 500 Posts
Posts: 134
Joined: Fri Apr 06, 2012 3:11 amFollowed by: 5 members
Thanked: 35 times

by Shalabh's Quants » 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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members
Location: New York, NY
Thanked: 13060 times
GMAT Score:790

by GMATGuruNY » 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:

https://www.beatthegmat.com/terminating- ... 92476.html
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
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Master | Next Rank: 500 Posts
Posts: 171
Joined: Tue Jan 08, 2013 7:24 am
Thanked: 1 times

by rajeshsinghgmat » Thu Feb 28, 2013 2:02 am
B in answer.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Post new topic Post Reply