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Re: Which of the following fractions has a decimal equivalent [#permalink]

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07 Sep 2013, 06:48

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akankshasoneja wrote:

salsal wrote:

Which of the following fractions has a decimal equivalent that is a terminating decimal?

a) 10/189 b) 15/196 c) 16/225 d) 25/144 e) 39/128

is there any simple method to find it?

Denominators of options a,c,d contains powers of 3...numerators of these options when divided by 3 will have non-terminating decimals Denominator of option b contains power of 7...numerator 15 when divided by 7 will give non terminating decimal Option E has denominator in powers of 2...so when 39 divided by 2 will give a terminating decimal

Which of the following fractions has a decimal equivalent that is a terminating decimal?

a) 10/189 b) 15/196 c) 16/225 d) 25/144 e) 39/128

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 the 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.

BACK TO THE QUESTION:

Only option E (when reduced to its lowest form) has the denominator of the form \(2^n5^m\): 39/128=39/2^7.

Re: Which of the following fractions has a decimal equivalent [#permalink]

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07 Sep 2013, 10:07

You can solve this question in less than 30 seconds if you understand the concept of terminating decimal. The denominator must have only power's of 2 or 5 in the denominator no other powers ( if it has any other prime factors like 3,7, etc it won't be terminating). 2's and 5's can be in any possible combination but it must only have 2's and 5's

a) 10/189

denominator sum of digits is 18 so its divisible by 3 eliminate

b) 15/196

This has a prime factor of 7 when do the prime factorization of the denominator.. Eliminate

c) 16/225

denominator sum of digits is 9 so its divisible by 3 eliminate

d) 25/144

denominator sum of digits is 9 so its divisible by 3 eliminate

You can solve this question in less than 30 seconds if you understand the concept of terminating decimal. The denominator must have only power's of 2 or 5 in the denominator no other powers ( if it has any other prime factors like 3,7, etc it won't be terminating). 2's and 5's can be in any possible combination but it must only have 2's and 5's

This is true if a fraction is reduced to its lowest term.

Consider this: the denominator of 3/30 has other primes than 2 or 5, but 3/30 IS a terminating decimal because 3 in the denominator gets reduced: 3/30=1/10=0.1.

Re: Which of the following fractions has a decimal equivalent [#permalink]

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01 Oct 2013, 19:02

1

This post received KUDOS

Bunuel wrote:

salsal wrote:

Which of the following fractions has a decimal equivalent that is a terminating decimal?

a) 10/189 b) 15/196 c) 16/225 d) 25/144 e) 39/128

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 \(2*5^2\). 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 the 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.

BACK TO THE QUESTION:

Only option E (when reduced to its lowest form) has the denominator of the form \(2^n5^m\): 39/128=39/2^7.

Which of the following fractions has a decimal equivalent that is a terminating decimal?

a) 10/189 b) 15/196 c) 16/225 d) 25/144 e) 39/128

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 \(2*5^2\). 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 the 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.

BACK TO THE QUESTION:

Only option E (when reduced to its lowest form) has the denominator of the form \(2^n5^m\): 39/128=39/2^7.

Re: Which of the following fractions has a decimal equivalent [#permalink]

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02 Oct 2013, 07:12

Bunuel wrote:

AccipiterQ wrote:

I'm confused, 128 is 2^7, you said it had to be 2^n*5^m....there is no 5^m in 128

Yes, it is 128 = 2^7*5^0.

So ANY number with a 2^x or 5^x (where x is greater than or equal to 1) will fall into this then?

I'm confused though, so 6/15 a terminating decimal, because 15 is 5^1*3^1*2^0, but then why is 16/225 is not terminating? It follows the same pattern; 225 is 5^2*3^2*2^0

I'm confused, 128 is 2^7, you said it had to be 2^n*5^m....there is no 5^m in 128

Yes, it is 128 = 2^7*5^0.

So ANY number with a 2^x or 5^x (where x is greater than or equal to 1) will fall into this then?

I'm confused though, so 6/15 a terminating decimal, because 15 is 5^1*3^1*2^0, but then why is 16/225 is not terminating? It follows the same pattern; 225 is 5^2*3^2*2^0

Re: Which of the following fractions has a decimal equivalent [#permalink]

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13 Feb 2016, 05:16

every terminating decimal has 10,100,1000,10000 etc. in denominator when written as fraction. So, all fractions should have only 2 and/or 5 as a factors in their reduced state

a) 10/189, it is reduced, and 189 does not have any 2 or 5 b) 15/196, reduced, and 196=(2*7)^2, so does not fit c) 16/225, reduced, and 225=(3*5)^2, so does not fit d) 25/144, reduced, and 144=(3*4)^2, so does not fit e) 39/128, reduced, and 128=2^7, so can have 10^7 in denominator and fits

Which of the following fractions has a decimal equivalent that is a terminating decimal?

a) 10/189 b) 15/196 c) 16/225 d) 25/144 e) 39/128

NOTE: this is one of those questions that require us to check/test each answer choice. In these situations, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top.

Re: Which of the following fractions has a decimal equivalent [#permalink]

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01 Mar 2017, 00:59

If numerator and denominator in fraction doesn’t have any common factor and the denominator has a prime factor except 2 and 5, then it is going to be non terminating decimal. Out of every option only Option E has 2 as the prime factor. Rest all have 3also. Hence option E s the answer