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# Which of the following fractions will lead to a terminating decimal?

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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19 Jul 2019, 22:05
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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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19 Jul 2019, 22:40
1
Soln:

Concept to apply: A fraction will always be terminating if the denominator of fraction in the least form has NO PRIME FACTOR other than 2,5, or both 2 and 5

Now lets check each option :

I) 83/80
80 has prime factors - 2,2,2,2,5 (2^4 * 5^1)
only 2 and 5 in denominator.

So 83/80 is terminating decimal

II) 23/120
120 has prime factors - 2,2,2,5,3 (2^3*5^1*3^1)
- since prime factors includes 3 , other than 2 and 5 .

23/120 is non terminating decimal.

III) 33/480 - We can reduce 33/480 to 11/160 ( by dividing by 3). This is final reduced form, now check denominator.
- Prime Factors of 160 are 2,2,2,2,2,5 (2^5*5^1)
- Applying concept mentioned above since only 2 and 5 in denominator
33/480 is terminating decimal

Ans choice D is CORRECT[
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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19 Jul 2019, 23:36
1
I. $$\frac{83}{80}$$
$$\frac{83}{8}$$*$$\frac{1}{10}$$
1.0375*$$\frac{1}{10}$$ = 0.10375 (yes this is a terminating decimal since this has finite number of digits after decimal point)

II. $$\frac{23}{120}$$
$$\frac{23}{40}$$*$$\frac{1}{3}$$
= 0.575 *$$\frac{1}{3}$$
=0.19166666666......(this will not end in a finite number of digits)
so this is not a terminating decimal

III. $$\frac{33}{480}$$
= $$\frac{33}{3}$$*$$\frac{1}{160}$$
= 11*$$\frac{1}{160}$$
= 11*$$\frac{1}{2}$$*$$\frac{1}{2}$$*$$\frac{1}{2}$$*$$\frac{1}{2}$$*$$\frac{1}{10}$$

=0.6875 (yes this is a terminating decimal)

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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19 Jul 2019, 23:39
Which of the following fractions will lead to a terminating decimal?

I. 83/80

II. 23/120

III. 33/480

A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III

A simple condition that can be used to check for terminating decimals is to check for the presence of 2*5 as factors in the denominator.
In this case, I II and III all of them have multiples of 10 in the denominator, so all of them will lead to a terminating decimal.

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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19 Jul 2019, 23:56
Which of the following fractions will lead to a terminating decimal?

I. $$\frac{83}{80}$$

II. $$\frac{23}{120}$$

III. $$\frac{33}{480}$$

Now, there is theorem (rarely published in GMAT materials) on which this questions is based: A fraction has terminating decimal if its denominator has prime factors of only 2 and 5 in any power.

Now, need to prime factorize denominators in the answers:
1. $$80 = 5*2*2*2*2$$ - only 2 and 5, therefore has terminating decimal
1. $$120 = 3*2*2*2*5$$ - there is 3 in it , therefore out
1. $$480 = 120*4$$ - as we seen, 120 has 3 in it, therefore out

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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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Updated on: 20 Jul 2019, 05:49
We are to determine which of the following will lead to a terminating decimal.

I. 83/80
83/80=8.3/8
but we know 1/8=0.125. Hence 8.3/8 = 8.3*0.125 = 83*125*10^-4
Since 83*125 leads to an integer, which is 10375, then 83*125*10^-4 = 10375/10000 = 1.0375.
therefore I leads to a terminating decimal.

II. 23/120.
23/120 = 23/(10*4*3) = 23*[(1/10*1/4*1/3)]. We know that 1/4 = 0.25 and 1/10 = 0.1.
hence 1/4*1/10=0.025.
However, 1/3 leads to a non-terminating decimal, and multiplying 0.025 by 1/3 leads to a non-terminating decimal.
Therefore II does not lead to a terminating decimal

III. 33/480
33/480 = 11/160 = 11*1/160 = 11/[(1/10*1/16)]. We know 1/10 = 0.1.
and 1/16 = 1/2*1/8 0.5*0.125 = 0.0625
hence 11/60 = 11*0.1*0.0625 = 0.06875, which is a terminating decimal.

The answer therefore is D. since only I and III leads to a terminating decimal.

Originally posted by eakabuah on 20 Jul 2019, 00:40.
Last edited by eakabuah on 20 Jul 2019, 05:49, edited 1 time in total.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 00:52
1
I. 83/80

83/80 = 1 + (3/80)
1/80 = 0.0125 , So (3/80) is also terminating.
No need to do the further calculation, from the above result we can directly say the (83/80) will lead to the terminating decimal.

II. 23/120

1/120 = 0.0083333..... (non terminating)
So, 23/120 is also non-terminating decimal.
Note: (a/120) will be terminating decimal if 'a' is multiple of '3'. Here, 23 is not a multiple of 3.

III. 33/480

33/480= 11/160
1/160 = 0.00625
1/160 is leading to terminating decimal. So, 11/160 will also lead to terminating decimal.

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 00:53
1
Task is to find denominators with either 2 or 5 or both. Only I and III have 2 and 5. II has 2, 5, and 3 in its denominator (infinite number of digits).
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 01:01
1
Rule for Terminating or Non-Terminating Recurring decimals,

=> For fractions in their simplest form, if the denominator has only powers of 2 and 5 then the fraction will be terminating.

=> For fractions in their simplest form, if the denominator has powers other than for 2 and 5 then the fraction will be non terminating recurring.

Let us now check the 3 options given,

Option 1: 83/80

This fraction is in it's lowert form.

Also, 80 = 2^4 * 5

=> 83/80 will be terminating.

Option 2: 23/120

This fraction is in it's lowest form.

Also, 120 = 2^3 * 3 * 5

=> 23/120 will be non-terminating recurring.

Option 3: 33/480

33/480
=> 11/160

This fraction is in it's lowet form now.

Also, 160 = 2^5 * 5

=> 33/480 will be terminating.

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 02:57
1
Which of the following fractions will lead to a terminating decimal?

I.
83
80

8380

II.
23
120

23120

III.
33
480

33480

A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III

Solution:

The concept in this question is that a fraction will lead to a terminating decimal if and only if the reduced fraction has a prime factorization that contains only 2's or5's or both.If the prime factorization of the reduced fraction contains denominatior anything other than 2s or 5s,the decimal equivalent will not terminate.
So considering the various options:

2 ) 23/120 = 23/($$2^3$$*3*5)

3)$$\frac{33}{480}$$ = $$\frac{11}{(2^5*5)}$$

1) $$\frac{83}{80}$$ = 83 /$$(2^4$$*5)

As we see that only 1 and 3 option have 5 and 2 in their denominators and hence they will lead to a terminating decimal. Optionn 2 have 3 also in its denominator hence it will not be a terminating decimal.Hence D is the answer IMO.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 03:35
1
(i) 83/80 --> 83/(8*10) --> 83/(2^4*5), 1/(2^4*5) gives terminating decimal
(ii) 23/120 --> 23/(12*10) --> 23/(2^2*3*2*5) --> 23/(2^3*3*5) doesn't give a terminating decimal because of 3
(iii) 33/480 --> 3*11/(2^5*3*5) --> 11/(2^5*5), 1/(2^5*5) gives terminating decimal

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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 03:48
A fraction leads to a terminating decimal only if its denominator has 2 or 5 as its prime factors. Any other prime factor in the denominator will make the decimal non-terminating.

I. $$\frac{83}{80} = \frac{83}{2^4*5^1}$$
As the denominator has only 2 and 5 as its factors, this is a terminating decimal.

II. $$\frac{23}{120} = \frac{23}{2^3*3^1*5^1}$$
As the denominator has 3 as one of its prime factors, this is a non-terminating decimal.

III. $$\frac{33}{480} = \frac{33}{2^5*3^1*5^1}$$
As the denominator has 3 as one of its prime factors, this is a non-terminating decimal.

Hence, Option (B)
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 04:14
Question: Which of the following fractions will lead to a terminating decimal?

If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates.

I. $$\frac{83}{80}$$
$\text{Prime Factorization Calculation }\\ \text{Example: 80} \\ \begin{pmatrix*} &&80&& \\ &\swarrow&|&\searrow&\\ 2&&\downarrow&&2^3\\ &&5&&\\ &&&&\\ &&&& \end{pmatrix*} \rightarrow \text{Prime Factors of 80 are } 2,2,2,2,5\\ \text{Terminating!}$
II. $$\frac{23}{120}$$
$\text{Prime Factorization Calculation }\\ \text{Example: N = 120} \\ \begin{pmatrix*} &&120&& \\ &\swarrow&|&\searrow&\\ 2^3&&\downarrow&&5\\ &&3&&\\ &&&&\\ &&&& \end{pmatrix*} \rightarrow \text{Prime Factors of 120 are } 2,2,2,3,5\\ \text{NOT Terminating!}$
III. $$\frac{33}{480}$$
$\text{Prime Factorization Calculation }\\ \text{Example: N = 480} \\ \begin{pmatrix*} &&480&& \\ &\swarrow&|&\searrow&\\ 2^5&&\downarrow&&5\\ &&3&&\\ &&&&\\ &&&& \end{pmatrix*} \rightarrow \text{Prime Factors of 480 are } 2,2,2,2,2,3,5\\ \text{NOT Terminating!}$
Correct Answer: B. $$\implies\text{I only}$$
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 04:15
1
Solution:

General property:
A fraction in its lowest term can be expressed as a terminating decimal if and only if the denominator has powers of only 2 and/or 5.

Lets prime factorize the division.

I. $$\frac{83}{2^4 X 5}$$ , Hence we see that the division only contains 2 & 5. So this fraction will be terminating.

II.$$\frac{23}{2^3 X 3 X 5}$$We observe that there is 3 in the prime factorization , Hence this fraction is non terminating

III.$$\frac{33}{480}$$

=$$\frac{{11 X 3}}{2^5 X 3 X 5}$$, we observe that 3 gets cancelled with the 3 in the numerator leaving only 2 & 5 as factors in the denominator. Hence this fraction will be terminating.

So the answer here must be D
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 05:05
1
Which of the following fractions will lead to a terminating decimal?

I. 83/80 = 1.0375

II. 23/120 = 0.19166(6)

III. 33/480 = 0.06875

A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 05:13
1
Which of the following fractions will lead to a terminating decimal?

I. 83/80

II. 23/120

III. 33/480

For a decimal to be terminating, the denominator must be in power of 2's or 5's or both.

i) 80= 5*2^4
ii) 120= 5*3*2^3
iii) 160 = 5*2^5

so, the correct answer choice is (D)
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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 05:33
1
As we know, we call the Terminating decimal if and only if the denominator has powers of only 2 and/or 5.

I. 83/80=83/(16*5)=83/(2^4*5) (yes, denominator consists of only powers of 2 and 5)

II. 23/120=23/(2^3*3*5) (No, it is not terminating decimal, denominator has power of 3.)

III. 33/480 (we can divide both parts of fraction by 3) --> 11/160
--> 11/160=11/(16*10)=11/(2^5*5) (yes, denominator consists of only powers of 2 and 5)

We got only I and III right
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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 06:31
1
For a fraction to be a terminating decimal, the denominator must be in the form 2^m*5^n in reduced form.

1) 80 = 2^4*5. Hence it will be lead to a terminating decimal.
2) 120= 2^3 * 3 * 5. Non terminating
3) Reducing fraction further it can be written as $$\frac{11}{160}$$.
160=2^5*5. Terminating

Hence option D
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 06:38
1
Only options 1 and 3 have denominators in forms of 2^a*5^b. Option 2 has 2^a*3^b*5^c.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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20 Jul 2019, 06:38
1
I. 83/80 =83/8 *10 = 10.3875/10 = 1.03875 --> Terminating decimal

II. 23/120 = 23/12*10 = 1.916666666.. /10 = 0.19166666666... ---> is not a terminating decimal

III. 33/480 = 11*3/4*3*4*10 = 11/16*10 =0.9=6875/10 = 0.06875--> terminating decimal

D. I and III only--- Is the correct answer
Re: Which of the following fractions will lead to a terminating decimal?   [#permalink] 20 Jul 2019, 06:38

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