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

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

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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]

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New post Updated on: 19 Jul 2019, 08:28
1
2
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]

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New post Updated on: 20 Jul 2019, 02:07
1
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]

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New post 19 Jul 2019, 08:17
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Which of the following fractions will lead to a terminating decimal?

I. 83/80

II. 23/120

III. 33/480

To find out answer to this question, we need to find out if any fraction simplifies into multiple of (1/3).
If we see the first fraction denominator is 80. Powers of 2 and 5. Hence terminating decimal.
Second fraction 23/120. Simplifying it we will get a fraction which is a multiple of 1/3. hence non terminating.
Third fraction 33/480. In first look at the denominator we feel it is a multiple of 3 hence non-terminating. But numerator and denominator both are multiple of 3. And highest power of 3 in both is 3^1. So they will cancel each other out. So simplified denominator will only be a multiple of 2 and 5. Hence terminating.

Hence Answer is 1 and 3. Answer is D.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:19
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IMO answer is D:

First number will lead to 83/(8*10), as denominator has all even numbers, definitely terminate
second number: 23/120 = 23/3*40. 23/3 is a non terminating decimal. so non terminating
third number: 33/48*10 =11/16*10, all even numbers at the denominator, so terminate


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

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New post 19 Jul 2019, 08:21
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Which of the following fractions will lead to a terminating decimal?

I. 83/80

II. 23/120

III. 33/480

The key to answering this question is knowing the theory behind terminating decimals. Terminating decimals are 1.09 and non-terminating decimals are 1.33333.
In order to check for terminating decimals, you need to check if the denominator is 2^x or 5^y or both.

Now let's take each of the options check for the same.

80 = 8*10 = 2^4 & 5^1
Hence this is correct.

120 = 12*10 = 2^3 , 3^1 , 5^1, Since it has 3 it is incorrect.

480 again only has 2 and 5, so it is correct.

which means 1st and 3rd are correct.

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


D is the correct answer.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:29
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We have terminating fractions if and only if the denominator contains only 2s and/or 5s

I: 83/80 = \(\frac{83}{2^{4} × 5}\) (terminates)

II: 23/120 = \(\frac{23}{2^{3} × 3 × 5}\) (does not terminate since denominator contains a 3)

III: 33/480 = \(\frac{33}{2^{5} × 3 × 5}\) = \(\frac{11}{2^{5} × 5}\) (terminates)

Therefore, only I and III are terminating decimals (answer is D)
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:30
To get a terminating decimal we need to have multiples of 2 or 5 or both (2 and 5).

Other than 2 or 5, the fraction will be non terminating decimal.
Lets consider prime factorisation of various denominators:
I 80 - 2^4 * 5 will have terminating decimal
II 120 - 2^3 * 3 * 5 will have non terminating decimal
III 480 - 2^5 * 3 * 5 will have non terminating decimal

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

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New post 19 Jul 2019, 08:31
1. 83/80 gives a terminating decimal
While 2&3 doesn't terminate. So the answer is B

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

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New post 19 Jul 2019, 08:31
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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

important points
Numbers in 4 or 5 in fraction are always terminating whereas with fractions as 3 are non terminating

I. 83/80 it will of form \(83 * (\frac{1}{4}*\frac{1}{2} * \frac{1}{10})\)
= 83 * 0.125*0.5*0.1
It does not go indefinite hence acceptable

II. 23/120

this if form \(23 * (\frac{1}{4}*\frac{1}{3} * \frac{1}{10})\)
hence 0.3 is indefinite 0.333333333333 ....
hence non terminating
III. 33/480
it can further be simplified to
11/160
which is again of the form \(11 *(\frac{1}{2}*\frac{1}{2}*\frac{1}{2}*\frac{1}{2} \frac{*1}{10})\)
all are terminating in fraction
Hence is III is acceptable
thus answer is D. I and III only
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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post Updated on: 19 Jul 2019, 10:19
1
Which of the following fractions will lead to a terminating decimal?

I. 83/80 = 1 + 3/80

II. 23/120

III. 33/480 = 11/160


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

I. 83/80 = 1 + 3/80
3/80 = 3 * .0125 = .0375
83/80 = 1. 0375
Terminating decimal

II. 23/120
1/120 = .008333
23/120 = .1916666
Non terminating decimal

III. 33/480 = 11/160
11/160 = .06875
Terminating decimal

Alternatively,
If a simplified fraction, has only multiples of either 2 or 5 in denominator, then it is a terminating decimal.

I. 83/80
80=2^4.5 has only multiples of 2 and 5 in denominator => Terminating decimal

II. 23/120
120 = 2^3.3.5 has 3 in denominator => Non-terminating decimal

III. 33/480 = 11/160
160=2^5.5 has only multiples of 2 and 5 in denominator => Terminating decimal

IMO D
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Originally posted by Kinshook on 19 Jul 2019, 08:31.
Last edited by Kinshook on 19 Jul 2019, 10:19, 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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New post 19 Jul 2019, 08:33
only 1,

terminating decimals will occur only when a 2 and or 5 are the only prime factors in the denominator
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Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post Updated on: 19 Jul 2019, 09:01
1
IMO : D

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


Sol:

if the denominator has 2's and 5's as prime factors then it will be a terminating decimal,
and if the denominator has other primes like 3,7,11 then the decimal will be a non terminating decimal.

120 = 2 * 2 * 2 * 3 * 5
80 = 2 * 2 * 2 * 2 * 5
480=2* 2* 2* 2* 2* 3* 5

I)83/80 =83*1/2 * 2 * 2* 2 * 5 this means only 2's and 5's in the denominator and thus terminating decimal.
II)23/120=23/2 * 2 * 2 * 3 * 5 this means only 2's ,3's and 5's in the denominator and so this is a non terminating decimal because 3 exist in decimal.
III) 33/480= 3*11/2*2* 2* 2* 2* 3* 5 this gives 11/160 and 160 only have 2's and 5's so this will be a terminating decimal


SO D is correct.

Originally posted by abhishekdadarwal2009 on 19 Jul 2019, 08:33.
Last edited by abhishekdadarwal2009 on 19 Jul 2019, 09:01, 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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New post 19 Jul 2019, 08:36
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fractions will lead to a terminating decimal if the denominator multiplied 2 or 5 or both and nothing else.

I. 80=8*10=2^4*5. Ok.
II.120=12*10=3*2^3*5. It has 3 thus not terminating
III. 480=2^5*3*5. But here we can reduce 3 in the numerator and denominator and thus this fraction is terminating.

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

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New post 19 Jul 2019, 08:39
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A fraction will be a terminating fraction when at its simplest form the denominator must be only of the form 2^n*5^m (were n,m can also be 0) without any other prime factors.

I)83/(2^4*5^1) - Terminating fraction (as denominator when prime factorized contains only 2 and 5 raised to different powers)
II) 23/(3^1*2^3*5^1) - Non terminating fraction (as denominator contains a 3^1 apart from 2^3*5^1)
III) Can be simplified to 11/(2^5*5^1) - Terminating fraction (as denominator when prime factorized contains only 2 and 5 raised to different powers)

IMO D - I and III Only
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:39
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A terminating decimal is a number which has finite digits after the decimal point.

1. 83/80 = 1.0375
2. 23/120 = 0.19166666666666666....
3. 33/480 = 0.06875

Thus only 1 and 3 have finite numbers after the decimal point.

IMO the answer is D.

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

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New post 19 Jul 2019, 08:41
IMO the correct answer is option B- I only
Explanation given as attachment -
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:42
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IMO D.

By just finding out the values, we can know if the fraction is terminating-

I. \frac{83}{80} = 1.0375

II. \frac{23}{120} = 0.19166666...

III. \frac{33}{480} = 0.06875

Hence, I and III give us terminating decimal points making option D the answer.
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Re: Which of the following fractions will lead to a terminating decimal?  [#permalink]

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New post 19 Jul 2019, 08:44
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D

I think this kind of problem should not be posted for competition because everyone has access to calculator, and can quickly check the answers without knowing the logic properly.

Anyways, here is my reasoning. In order to have a fraction that is fully terminable, the denominator of the fraction needs to be in the form of 2^n*5^m, after the fraction has been reduced to its simplest form.

I. 83/80, 80 can be factored as 2^4*5, so fraction is terminable

II. 23/120, 120 can be factored as 2^3*3*5, so fraction cant be terminable because of 3 in the denominator

III. 33/480 = 11/160, 160 can be factored as 2^5*5, so fraction is terminable

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

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New post 19 Jul 2019, 08:47
Which of the following fractions will lead to a terminating decimal?

I. 83808380

II. 2312023120

III. 3348033480

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

B is the ans
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Re: Which of the following fractions will lead to a terminating decimal?   [#permalink] 19 Jul 2019, 08:47

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