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

x, y are different positive integers with (1/x)+(1/y)=(1/5). find x+y=?

$${x},\:{y}\:{are}\:{different}\:{positive}\:{integers} \\ $$$${with}\:\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{1}}{\mathrm{5}}.\:{find}\:{x}+{y}=? \\ $$

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

xy=5(x+y) x or y must be a multiple of 5, say x=5k ky=5k+y (k−1)y=5k 1) k−1=5, y=k ⇒k=6, y=6, x=5×6=30 2) k−1=1, y=5k ⇒k=2, y=10, x=10 for x≠y: (x, y)=(30, 6)=(6, 30) ⇒x+y=36

$${xy}=\mathrm{5}\left({x}+{y}\right) \\ $$$${x}\:{or}\:{y}\:{must}\:{be}\:{a}\:{multiple}\:{of}\:\mathrm{5},\: \\ $$$${say}\:{x}=\mathrm{5}{k} \\ $$$${ky}=\mathrm{5}{k}+{y} \\ $$$$\left({k}−\mathrm{1}\right){y}=\mathrm{5}{k} \\ $$$$\left.\mathrm{1}\right)\:{k}−\mathrm{1}=\mathrm{5},\:{y}={k}\: \\ $$$$\Rightarrow{k}=\mathrm{6},\:{y}=\mathrm{6},\:{x}=\mathrm{5}×\mathrm{6}=\mathrm{30} \\ $$$$\left.\mathrm{2}\right)\:{k}−\mathrm{1}=\mathrm{1},\:{y}=\mathrm{5}{k} \\ $$$$\Rightarrow{k}=\mathrm{2},\:{y}=\mathrm{10},\:{x}=\mathrm{10} \\ $$$${for}\:{x}\neq{y}:\:\left({x},\:{y}\right)=\left(\mathrm{30},\:\mathrm{6}\right)=\left(\mathrm{6},\:\mathrm{30}\right) \\ $$$$\Rightarrow{x}+{y}=\mathrm{36} \\ $$

Answered by kapoorshah last updated on 01/Oct/22

(1/y) = (1/5) − (1/x) (1/y) = ((x − 5)/(5x)) y = ((5x )/(x − 5)) y = ((5(x − 5) + 25 )/(x − 5)) y = 5 + (( 25 )/(x − 5)) x − 5 = 1 ⇒ x = 6 y = 30 x − 5 = 5 ⇒ x = 10 y = 10 rejected x − 5 = 25 ⇒ x = 30 y = 6

$$\frac{\mathrm{1}}{{y}}\:=\:\frac{\mathrm{1}}{\mathrm{5}}\:−\:\frac{\mathrm{1}}{{x}} \\ $$$$\frac{\mathrm{1}}{{y}}\:=\:\frac{{x}\:−\:\mathrm{5}}{\mathrm{5}{x}} \\ $$$${y}\:=\:\frac{\mathrm{5}{x}\:}{{x}\:−\:\mathrm{5}} \\ $$$${y}\:=\:\frac{\mathrm{5}\left({x}\:−\:\mathrm{5}\right)\:+\:\mathrm{25}\:}{{x}\:−\:\mathrm{5}} \\ $$$${y}\:=\:\mathrm{5}\:+\:\frac{\:\mathrm{25}\:}{{x}\:−\:\mathrm{5}} \\ $$$${x}\:−\:\mathrm{5}\:=\:\mathrm{1}\:\Rightarrow\:{x}\:=\:\mathrm{6}\:\:\:{y}\:=\:\mathrm{30} \\ $$$${x}\:−\:\mathrm{5}\:=\:\mathrm{5}\:\Rightarrow\:{x}\:=\:\mathrm{10}\:\:\:{y}\:=\:\mathrm{10}\:\:{rejected} \\ $$$${x}\:−\:\mathrm{5}\:=\:\mathrm{25}\:\Rightarrow\:{x}\:=\:\mathrm{30}\:\:\:{y}\:=\:\mathrm{6} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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

very nice sir! thanks!

$${very}\:{nice}\:{sir}!\:{thanks}! \\ $$

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

5x+5y−xy=0 (x−5)(y−5)=25 x−5=1, y−5=25 ⇒x=6, y=30 ✓ x−5=5, y−5=5 ⇒ x=10, y=10

$$\mathrm{5}{x}+\mathrm{5}{y}−{xy}=\mathrm{0} \\ $$$$\left({x}−\mathrm{5}\right)\left({y}−\mathrm{5}\right)=\mathrm{25} \\ $$$${x}−\mathrm{5}=\mathrm{1},\:{y}−\mathrm{5}=\mathrm{25}\:\Rightarrow{x}=\mathrm{6},\:{y}=\mathrm{30}\:\checkmark \\ $$$${x}−\mathrm{5}=\mathrm{5},\:{y}−\mathrm{5}=\mathrm{5}\:\Rightarrow\:{x}=\mathrm{10},\:{y}=\mathrm{10} \\ $$

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