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

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Bunuel
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Which of the following fractions will lead to a terminating decimal? [#permalink] New post   19 Jul 2019, 08:00
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Which of the following fractions will lead to a terminating decimal?

I. \(\frac{83}{80}\)

II. \(\frac{23}{120}\)

III. \(\frac{33}{480}\)

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


 

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Which of the following fractions will lead to a terminating decimal? [#permalink] New post   Updated on: 19 Jul 2019, 08:28
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A terminating decimal will have denominator in the 2^a*5^b

\(80 = 2^4*5^1\)

\(120 = 2^3*3^1*5^1\)

\(480 = 2^5*3^1*5^2\) but in this case the numerator is a multiple of 3 so the 3 cancels out and we are left \(2^5*5^2\) in the denominator so its a terminating decimal


I & III only

Answer is (D)

Originally posted by firas92 on 19 Jul 2019, 08:12.
Last edited by firas92 on 19 Jul 2019, 08:28, edited 1 time in total.
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Which of the following fractions will lead to a terminating decimal? [#permalink] New post   Updated on: 20 Jul 2019, 02:07
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For a fraction to be terminating, the fraction in its lowest form should have a denominator that should have prime factors consisting of only 2 OR only 5 OR a combination of 2 and 5.

I. \(\frac{83}{80}\)
There is no common factor in Numerator and denominator to reduce the fraction.
Denominator 80 = 2 * 2 * 2 * 2 * 5 -> Only Prime factors of 2 and 5.

Hence, it will lead to a terminating decimal.

II. \(\frac{23}{120}\)
There is no common factor in Numerator and denominator to reduce the fraction.
Denominator 120 = 2 * 2 * 3 * 5 -> There are other factor besides 2 and 5.

Hence, it will NOT lead to a terminating decimal.

III. \(\frac{33}{480}\)
The fraction can be reduced to \(\frac{11}{160}\)
Denominator 160 = 2 * 2 * 2 * 2 * 2 * 5 -> Only Prime factors of 2 and 5.

Hence, it will lead to a terminating decimal.

Answer D

Originally posted by Sayon on 19 Jul 2019, 08:14.
Last edited by Sayon on 20 Jul 2019, 02:07, edited 1 time in total.
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Re: Which of the following fractions will lead to a terminating decimal? [#permalink] New post   19 Jul 2019, 10:33
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If the denominator of the fraction, in its reduced form, contains no other prime number facors other than 2 and 5, then the fraction is terminating.

I) \(\frac{83}{80}\)= \(\frac{83}{2^4*5}\)
Denominator contains only 2 and 5 as prime factors, hence \(\frac{83}{80}\) leads to terminating decimal.

II) \(\frac{23}{120}\)= \(\frac{23}{2^3*3*5}\)
Denominator contains 2, 3 and 5 as prime factors, hence \(\frac{23}{120}\) does not lead to terminating decimal.

III)\(\frac{33}{480}\)= \(\frac{3*11}{2^5*3*5}\)=\(\frac{11}{2^5*5}\)
Denominator contains only 2 and 5 as prime factors, hence \(\frac{33}{480}\) leads terminating decimal.

IMO D
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Which of the following fractions will lead to a terminating decimal? [#permalink] New post   Updated on: 19 Jul 2019, 23:41
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Which of the following fractions will lead to a terminating decimal?

We just simply need to perform division calculations, and see do we have a terminating decimal or not:

I. 83/80
Attachment:
83.PNG
83.PNG [ 5.51 KiB | Viewed 5875 times ]


II. 23/120
Attachment:
230.PNG
230.PNG [ 8.25 KiB | Viewed 5832 times ]


III. 33/480
Attachment:
33.PNG
33.PNG [ 6.63 KiB | Viewed 5811 times ]


If we will test by concept then:

Number will have finite number of integers after decimal of denominator consists of only 2, 5 or both.
I. Has both 2 and 5, so it will terminate
II. Has 2 and 5, but also 3, will not terminate
III. Has only 2 and 5, thus will terminate

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

D is the answer. :heart

Originally posted by GKomoku on 19 Jul 2019, 12:16.
Last edited by GKomoku on 19 Jul 2019, 23:41, edited 1 time in total.
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Re: Which of the following fractions will lead to a terminating decimal? [#permalink] New post   20 Jul 2019, 04:15
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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] New post   11 Feb 2024, 18:02
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Re: Which of the following fractions will lead to a terminating decimal? [#permalink]
11 Feb 2024, 18:02
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