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a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.

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Director
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a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.  [#permalink]

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New post 16 Sep 2019, 09:08
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\(a=\frac{x}{y}\), such that "a' is a fraction written in lowest terms and \(0<a<1\). Also, x+y=100. How many values of 'a' are possible.

A. 15
B. 18
C. 20
D. 25
E. 30
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Re: a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.  [#permalink]

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New post 16 Sep 2019, 10:15
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since 0<a<1 we learn that x<y ==> combined with x+y=100 we get x=1,...,49 and y=99,...,50
there are 50 fractions we can create.
there are 25 pairs of (x,y) which are both even, therefore not at their lowest term form
there are 5 more pairs where x=10*k+5, such that we count all multiples of 5 but not the multiple of 10.
multiples of 3 are out of sync, so does the rest of the multiples of primes.
so we get 50-25-5=20.

please let me know if you agree with my solution.
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Re: a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.  [#permalink]

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New post 16 Sep 2019, 14:41
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a=x/y where 0<a<1 and a is a fraction written in lowest terms. Since x+y=100 and logically x<y, there are max. 50 combinations of x and y:
x=1,2,...,50
y=51,52,...,100

If both x and y are even (there are 25 combinations of x and y that are both even), x/y can NOT be written in lowest term, e.g. x/y=2/98, 4/94,..., 48/52, 50/50.

If both x and y are multiples of 5 (there are 5 combinations of x and y that are multiples of 5), x/y can NOT be written in lowest term, e.g. x/y=5/95, 15/85,..., 45/55.

Rest of the (x, y) combinations can be written in the lowest term. No of possible values of 'a' = 50 - 25 - 5 = 20.

Answer is (C)

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a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.  [#permalink]

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New post 16 Sep 2019, 19:02
almogsr Your solution is perfect.

There is a short-cut to find the co-primes of 100 that are less than 50

Totient(x)= \(x*(1-\frac{1}{a})(1-\frac{1}{b})\)....., where a,b are prime factors of x
totient(50)=\(50*(1-\frac{1}{2})(1-\frac{1}{5})\)=\(50*\frac{1}{2}*\frac{4}{5}\)=20


almogsr wrote:
since 0<a<1 we learn that x<y ==> combined with x+y=100 we get x=1,...,49 and y=99,...,50
there are 50 fractions we can create.
there are 25 pairs of (x,y) which are both even, therefore not at their lowest term form
there are 5 more pairs where x=10*k+5, such that we count all multiples of 5 but not the multiple of 10.
multiples of 3 are out of sync, so does the rest of the multiples of primes.
so we get 50-25-5=20.

please let me know if you agree with my solution.
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a=x/y, such that "a' is a fraction written in lowest terms and 0<a<1.   [#permalink] 16 Sep 2019, 19:02
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