x-y-are-different-positive-integers-with-1-x-1-y-1-5-find-x-y-

Question Number 177083 by mr W last updated on 30/Sep/22

x, y a r e d i f f e r e n t p o s i t i v e i n t e g e r s

w i t h \frac{1}{x} + \frac{1}{y} = \frac{1}{5} . f i n d x + y = ?

Answered by mr W last updated on 30/Sep/22

x y = 5 (x + y)

x o r y m u s t b e a m u l t i p l e o f 5,

s a y x = 5 k

k y = 5 k + y

(k - 1) y = 5 k

1) k - 1 = 5, y = k

\Rightarrow k = 6, y = 6, x = 5 \times 6 = 30

2) k - 1 = 1, y = 5 k

\Rightarrow k = 2, y = 10, x = 10

f o r x \neq y : (x, y) = (30, 6) = (6, 30)

\Rightarrow x + y = 36

Answered by kapoorshah last updated on 01/Oct/22

\frac{1}{y} = \frac{1}{5} - \frac{1}{x}

\frac{1}{y} = \frac{x - 5}{5 x}

y = \frac{5 x}{x - 5}

y = \frac{5 (x - 5) + 25}{x - 5}

y = 5 + \frac{25}{x - 5}

x - 5 = 1 \Rightarrow x = 6 y = 30

x - 5 = 5 \Rightarrow x = 10 y = 10 r e j e c t e d

x - 5 = 25 \Rightarrow x = 30 y = 6

Commented by mr W last updated on 01/Oct/22

v e r y n i c e s i r! t h a n k s!

Answered by mr W last updated on 01/Oct/22

5 x + 5 y - x y = 0

(x - 5) (y - 5) = 25

x - 5 = 1, y - 5 = 25 \Rightarrow x = 6, y = 30 ✓

x - 5 = 5, y - 5 = 5 \Rightarrow x = 10, y = 10

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