If r and s are positive integers, can the fraction r/s be expressed as

Question

If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?

(1) s is a factor of 100
(2) r is a factor of 100

Professor · Answer

B. yes. factors of 100 = 1, 2, 4, 5, 10, 20, 25, 50, 100. it doesnot matter the value of r, r/s terminates in decimal.

laxieqv · Answer

But we don't know anything about r, what's if r itself is an unterminated decimal?!
For example: r= 33.333333333333333 and s = 100 ---> r/s = 0.33333333333


I go for C coz:
for every r/s, in case s <>100 , we can multiply s with a factor k ( k=100/s) to make the denominator become 100, then r/s = rk/ 100 
Because r is a factor of 100 ---> r is integer ---> rk is an integer ( coz k is also an integer) ---> rk/100 will be terminated in decimal.

For example: 
r= 1, s = 5 ---> r/s= 20/100= 0.2

Professor · Answer

hmmmmmm........

your view is definitly appealing but i believe r should be terminated decimal otherwise how we write r?

remgeo · Answer

Go with B. 
Anything divided by 100 is a terminating decimal

joemama142000 · Answer

no oa but i believe this is E. can anyone explain this better?

I. dont know s
100/3= non term
100/5=yes term
insuff

II. dont know r
(1/3)/100=non term
100/100 =yes term

both.
200/100= yes
100/300=no term

laxieqv · Answer

Buddy, s must be a factor of 100, 300 is not a factor of 100 :wink:

BTW, I'm still stuck to C coz we have to consider the case in which r itself is a non-terminated decimal!

giddi77 · Answer

Agree with Laxie,
If it is specified that r is an integer, then B would be fine.

Since r is not necessarily an integer it could be a fraction too. 
Say r =2/3 --> r/s non-terminated decimal
if r = any integer --> r/s terminated decimal

So it cannot be B.

It has to be C as both r & s are factors of 100. 
If it were an ETS question, (I hope) r would be specified as an integer and would make C a nice trap!

joemama142000 · Answer

ill go with C thanks lax

utkgogia2003 · Answer

ans is E 
S1- s can be irrational
S2-r can be irratonal
S1+s2- 200/300-.66
None is sufficent

Bunuel · Answer

THEORY: Reduced fraction \frac{a}{b} (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only b (denominator) is of the form 2^n5^m , where m and n are non-negative integers. For example: \frac{7}{250} is a terminating decimal 0.028 , as 250 (denominator) equals to 2*5^3 . Fraction \frac{3}{30} is also a terminating decimal, as \frac{3}{30}=\frac{1}{10} and denominator 10=2*5 . Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not. For example \frac{x}{2^n5^m} , (where x, n and m are integers) will always be terminating decimal. (We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction \frac{6}{15} has 3 as prime in denominator and we need to know if it can be reduced.) Questions testing this concept: does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.html any-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.html if-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html 700-question-94641.html is-r-s2-is-a-terminating-decimal-91360.html pl-explain-89566.html which-of-the-following-fractions-88937.html BACK TO THE ORIGINAL QUESTION: If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits? (1) s is a factor of 100. Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50 and 100. All these numbers are of the form 2^n5^m (for example 1=2^0*5^0, 2=2^1*5^0, ...), therefore no matter what is the value of r, r/s will always will be terminating decimal. Sufficient. (2) r is a factor of 100. We need to know about the denominator. Not sufficient. Answer: A. OPEN DISCUSSION OF THIS QUESTION IS HERE: if-r-and-s-are-positive-integers-141000.html

If r and s are positive integers, can the fraction r/s be expressed as

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