Hello,
Can you please assist with this? This question is from MGMAT. Thanks for your help.
If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.
OA: C
is it possible to write x as a terminating decimal?
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Hi gmattesttaker2,
Terminating decimal means that there are finite number of digits following the decimal.
e.g. 1/2=0.5 is a terminating decimal, whereas 1/3 = 0.33333... is not a terminating decimal. Now 1 divided by 3 or 7 or their multiples would always result in a non terminating decimal. Whereas 1 divided by 2 or 5 or their multiples would result in a terminating decimal.
We know 0<x<1
Now let us look at the options.
(1) 24x is an integer. 24's factors are 2,3,4,6,12 & 24. Therefore 24x would be an integer only if x = 1/2, 1/3, 1/4, 1/6 , 1/12 or 1/24. Out of these 1/2 is terminating ( 0.5) , so is 1/4 (0.25) or 1/8(0.125).... whereas 1/3 (0.3333...) or 1/12 or 1/24 are not. So Insufficient.
(2) 28x is an integer. Following the same reason as above, this results in x = 1/2 or 1/4 or 1/7 or 1/14 or 1/28. Clearly, in this case also , 1/2 and 1/4 are terminating , whereas 1/7 or 1/14 or 1/28 are not. Insufficient.
Combining, we can say that 1/2 and 1/4 are the only two values of x , which enable writing x as a terminating decimal. Hence Sufficient. So OA is C
Hope this makes it clear.
Terminating decimal means that there are finite number of digits following the decimal.
e.g. 1/2=0.5 is a terminating decimal, whereas 1/3 = 0.33333... is not a terminating decimal. Now 1 divided by 3 or 7 or their multiples would always result in a non terminating decimal. Whereas 1 divided by 2 or 5 or their multiples would result in a terminating decimal.
We know 0<x<1
Now let us look at the options.
(1) 24x is an integer. 24's factors are 2,3,4,6,12 & 24. Therefore 24x would be an integer only if x = 1/2, 1/3, 1/4, 1/6 , 1/12 or 1/24. Out of these 1/2 is terminating ( 0.5) , so is 1/4 (0.25) or 1/8(0.125).... whereas 1/3 (0.3333...) or 1/12 or 1/24 are not. So Insufficient.
(2) 28x is an integer. Following the same reason as above, this results in x = 1/2 or 1/4 or 1/7 or 1/14 or 1/28. Clearly, in this case also , 1/2 and 1/4 are terminating , whereas 1/7 or 1/14 or 1/28 are not. Insufficient.
Combining, we can say that 1/2 and 1/4 are the only two values of x , which enable writing x as a terminating decimal. Hence Sufficient. So OA is C
Hope this makes it clear.
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gmattesttaker2 wrote:Hello,
Can you please assist with this? This question is from MGMAT. Thanks for your help.
If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.
OA: C
Numbers That can produce non-terminating decimals : 3 & 7.
X can be number which can contain 3 or 7 or both. So, inorder to be sure that X is terminating we need both "3"and "7
(1) 24x = 3 x 2 x 2 x 2 --> "7" is missing.. Insufficent
(2) 28x = 7 x 2 x 2 --> "3" is missing.. Insufficient
Combining (1) & (2) ..... both 3 and 7 are present... So,.. [C]
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gmattesttaker2 wrote: If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.
Target question: Is it possible to write x as a terminating decimal?
This is a great candidate for rephrasing the target question. Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100
Given: 0 < x < 1
Let's say that x = a/b where the fraction a/b is written in simplest terms.
There's a nice rule that says something like,
If a/b results in a terminating decimal, then the denominator, b, MUST be the product of 2's and 5's only!
So, for example, if b = 20, the fraction a/b will result in a terminating decimal. The same holds true for other values of b such as 4, 5, 25, 40, 2, 8, and so on.
REPHRASED target question: Is b the product of 2's and 5's only?
Statement 1: 24x is an integer.
x = a/b. So, if 24x is an integer, b must be a divisor of 24.
So, b could equal 2, 3, 4, 6, 8, 12, or 24 [aside: I omitted 1 as a possibility, since we're told that 0 < x < 1]
So, for example, b could equal 8, in which case b IS the product of 2's and 5's only
Or b could equal 3, in which case b is NOT the product of 2's and 5's only
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: 28x is an integer.
x = a/b. So, if 28x is an integer, b must be a divisor of 28.
So, b could equal 2, 4, 7, 14, or 28
So, for example, b could equal 4, in which case b IS the product of 2's and 5's only
Or b could equal 7, in which case b is NOT the product of 2's and 5's only
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 says that b could equal 2, 3, 4, 6, 8, 12, or 24
Statement 2 says that b could equal 2, 4, 7, 14, or 28
So, we can conclude that b = 2 or 4
Both of these possible b values ARE the product of 2's and 5's only.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
Answer = C
Cheers,
Brent
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Yes it is, and EVERY fraction with 8 in the denominator (e.g., 1/8, 2/8, 3/8, etc.) can be written as a terminating decimal.mikepamlyla wrote:Brent,
Great explanation. But isn't 8 a factor of only 2s and 5s (2x2x2)?
1/8 = 0.125
2/8 = 0.25
3/8 = 0.375
etc
Cheers,
Brent
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To Solve the question we should know about an observation about the fractions which is any fraction, in it's lowest possible form, will be NON-terminating if the denominator has any prime factor other than 2 and 5If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.
Statement 1) It only mentions that denominator of x is a a factor of 24 because 24x is an integer.
Therefore x can be reciprocal of 2,3,4,6,12,24 but if x is 1/2 then it's terminating and if x is 1/3 then it's non-terminating therefore INSUFFICIENT
Statement 2) It only mentions that denominator of x is a a factor of 28 because 28x is an integer.
Therefore x can be reciprocal of 2,4,7,14,24 but if x is 1/2 then it's terminating and if x is 1/7 then it's non-terminating therefore INSUFFICIENT
COMBINING the two statement
Denominator of x has to be factor of 24 and 28 both so possible values of x are reciprocal of 2 or 4 and in both cases x is a TERMINATING decimal therefore SUFFICIENT
Hence Answer option C
"GMATinsight"Bhoopendra Singh & Sushma Jha
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