Question

The difference between of two natural numbers is 5 and the difference of reciprocals is 1/10 . Find the numbers​

Answer 1

Answer:

Refer to the attachment above

Answer 2

✏$\sf{Let\: the \:two \:numbers\: be \:a \:and\: b\: with\: a>b.}$$\sf{This\: means \:that\: a-b = 5 }$

✏$\sf\underline{And}$$\sf{ since,\: a > b \:means,\: \dfrac{1}{a} < \dfrac{1}{b}}$

✏$\sf{\underline{So,}}$$\sf{\implies \dfrac{1}{b} - \dfrac{1}{a} = \dfrac{1}{10}}$$\sf{or,\: \dfrac{(a - b)}{ab} = \dfrac{1}{10} }$

$\sf{ \: \: }$

✒$\bold{\fbox{Since, \:a - b = 5}}$

$\sf{Means,\: \dfrac{5}{ab} = \dfrac{1}{10} means\: ab = 50}$

$\sf{ \: \: }$

✒$\sf{\underline{Now,}}$

$\sf{{(a + b)}^{2} = {(a - b)}^{2} + 4ab}$

✒$\sf{\underline{\underline{Hence}}}$

$\sf{{(a + b)}^{2} = {(5)}^{2} + 4 \times 50 }$

$\sf{a + b = +- 15}$

✏

$\sf{\underline{Let\: us\: take\: a + b =15,\: first }}$

$\sf{now,\:a - b = 5 \:adding \:these\: 2 \:equations}$

$\sf{\underline{Now\: take \:a+b = -15 }}$

$\sf{and \:a-b =5 }$

$\sf{add\: these\: 2\: equations\: to \:get}$

$\sf{2a = -10 \:means \:a = -5 }$

$\sf{ \: \: }$

✒$\sf{\underline{\underline{Hence\: b = 10}}}$✏$\bold{So \:the\: numbers\: are \:either \:10\: or \:5}$$\bold{Or \:-5 \:and \:-10}$