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(D) Let the two natural numbers be $x$ and $y$.Given that the sum of the numbers is $x + y = 5$,which implies $y = 5 - x$.The sum of their reciprocals

Vedclass · Accepted Answer

$2, 3$

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(D) Let the two natural numbers be $x$ and $y$.Given that the sum of the numbers is $x + y = 5$,which implies $y = 5 - x$.The sum of their reciprocals is $\frac{1}{x} + \frac{1}{y} = \frac{5}{6}$.Substituting $y = 5 - x$ into the equation: $\frac{1}{x} + \frac{1}{5 - x} = \frac{5}{6}$.Taking the common denominator: $\frac{(5 - x) + x}{x(5 - x)} = \frac{5}{6}$.This simplifies to $\frac{5}{5x - x^2} = \frac{5}{6}$.Dividing both sides by $5$,we get $\frac{1}{5x - x^2} = \frac{1}{6}$.Therefore,$5x - x^2 = 6$,which rearranges to the quadratic equation $x^2 - 5x + 6 = 0$.Factoring the quadratic equation: $(x - 2)(x - 3) = 0$.Thus,$x = 2$ or $x = 3$.If $x = 2$,then $y = 5 - 2 = 3$. If $x = 3$,then $y = 5 - 3 = 2$.The two numbers are $2$ and $3$.

The sum of two natural numbers is $5$ and the sum of their reciprocals is $\frac{5}{6}$. Find those numbers.

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