The difference of two numbers is 5 and the difference of their reciprocals is 5/14 find the numbers

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The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.

Solution

Let the two natural numbers be x and y A/q

$y-x=5$.....(1)

$\frac{1}{x}-\frac{1}{y}=\frac{5}{14}$....(2) From( 1.)

$y=x+5$

Putting the value of y in eqn. (2)

$\frac{1}{x}-\frac{1}{x+5}=\frac{5}{14}=>\frac{x+5-x}{x(x+5)}=\frac{5}{14}=>\frac{5}{x^2+5x}=\frac{5}{14}=>x^2+5x-14=0=>x^2+(7-2)x-14=0=>x^2+7x-2x-14=0=>x(x+7)-2(x+7)=0=>(x+7)(x-2)=0$

Either $x+7=0$
$=>x=-7$Not acceptable because -7 is not natural number Or,

$x-2=0=>x=2$So, the first number is 2

Other number $=x+5=2+5=7$

so the numbers are 7 and 2

Mathematics

Secondary School Mathematics X

Standard X