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The difference of two number is 4. If the difference of their reciprocals is $\frac{4}{21}$, find the numbers.
Given:
The difference of two number is 4. The difference of their reciprocals is $\frac{4}{21}$.
To do:
We have to find the numbers.
Solution:
Let the two numbers be $x$ and $x-4$.
According to the question,
Difference of the reciprocals $=\frac{4}{21}$.
This implies,
$\frac{1}{x-4} -\frac{1}{x} =\frac{4}{21}$
$\frac{1(x)-1(x-4)}{(x-4)x} =\frac{4}{21}$
$\frac{x-x+4}{x^2-4x} =\frac{4}{21}$
$\frac{4}{x^2-4x}=\frac{4}{21}$
$4(21)=4(x^2-4x)$ (By cross multiplication)
$21=x^2-4x$
$x^2-4x-21=0$
Solving for $x$ by factorization method, we get,
$x^2-7x+3x-21=0$
$x(x-7)+3(x-7)=0$
$(x-7)(x+3)=0$
$x-7=0$ or $x+3=0$
$x=7$ or $x=-3$
If $x=7$, $x-4=7-4=3$
If $x=-3$, $x-4=-3-4=-7$
The required numbers are $3$ and $7$ or $-3$ and $-7$.
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