Find-the-number-of-rational-numbers-r-0-lt-r-lt-1-such-that-when-r-is-written-as-a-fraction-in-lowest-term-The-numerator-and-the-denominator-have-a-sum-of-1000-

Question Number 111155 by Aina Samuel Temidayo last updated on 02/Sep/20

Find the number of rational numbers

r, 0 < r < 1, such that when r is written

as a fraction in lowest term . The

numerator and the denominator

have a sum of 1000 .

Answered by Her_Majesty last updated on 02/Sep/20

r = \frac{n}{1000 - n}; 0 < r < 1 \Rightarrow 0 < n < 500 \land n \neq 2 k \land n \neq 5 k

b e t w e e n 0 a n d 500 : 249 n u m b e r s n = 2 k a n d

99 n u m b e r s n = 5 k b u t w e h a v e 49 n u m b e r s

n = 2 \times 5 k \Rightarrow 50 n u m b e r s l e f t

s u m o f n o t a l l o w e d n u m b e r s = 299

\Rightarrow a l l o w e d n u m b e r s = 200 = a n s w e r

Commented by Aina Samuel Temidayo last updated on 02/Sep/20

Thanks