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The difference of two natural number is 5 and the difference of their reciprocals is $\frac {1}{10}$. Find the numbers.
Given: Two difference of two natural numbers$=5\ cm$ and the the difference of their reciprocals is $\frac{1}{10}$.
To do: To find the numbers.
Solution:
Let us assume the natural numbers are $a$ and $b$.
As per given condition,
Difference of the number is 5.
$a-b=5$
$\Rightarrow a=b+5............( 1)$
Difference of the reciprocals is $\frac{1}{10}$.
$\frac{1}{a} -\frac{1}{b} =\frac{1}{10}$
$\Rightarrow \frac{b-a}{ab} =\frac{1}{10}$
$\Rightarrow \frac{5}{ab} =\frac{1}{10}$
$\Rightarrow ab=50$
$\Rightarrow ( b+5) b=50$
$\Rightarrow b^{2} +5b-50=0$
$\Rightarrow b^{2} +10b-5b-50=0$
$\Rightarrow b( b+10) -5( b+10) =0$
$\Rightarrow ( b-5)( b+10) =0$
If$b-5=0$
$\Rightarrow b=5$
If $b+10=0$
$b=-10$
$\because$ b is a natural number, it can't be negative.
$\therefore$ We reject $b=-10$,
Thus we accept only $b=5$, on putting this value in $( 1)$ ,
$a=5+5=10$
Therefore, the natural numbers are 5 and 10.
Updated on: 10-Oct-2022
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