Question 1175013
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After reading your post, I turn on my mind.


     {{{10/3}}} = 3 {{{1/3}}} = 3 + {{{1/3}}}.


So, the number is  3  and its reciprocal is  {{{1/3}}}.     <U>ANSWER</U>
</pre>

Solved.


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Wait a minute &nbsp;(!)


There is &nbsp;&nbsp;A N O T H E R &nbsp;&nbsp;solution, &nbsp;too,  &nbsp;and you should know it:


<pre>
    Another answer is THIS:  the number is  {{{1/3}}}  and its reciprocal is 3.
</pre>


Now all possible solutions are in place (!)



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Let me add something for your education.


This problem can be solved formally.

This way goes through solving quadratic equation.



Other people &nbsp;(and other teachers) &nbsp;consider such problems as &nbsp;"joke" problems,

that can be solved mentally in 3-4-5 seconds.



For advanced young students of &nbsp;5-th grade level, &nbsp;who are able freely manipulate fractions, 

it is a &nbsp;"joke" problem, &nbsp;which &nbsp;SHOULD &nbsp;be solved mentally, &nbsp;and it gives them fan to do it.



For other students, &nbsp;the formal way is often very boring and torturing way to solve it formally.


The formal way is THIS :


<pre>
Let x be the number; then reciprocal is  {{{1/x}}}.

The equation is THIS:

    x + {{{1/x}}} = {{{10/3}}}.


Multiply both sides by 3x


    3x^2 + 3 = 10x

    3x^2 - 10x + 3 = 0

    {{{x[1,2]}}} = {{{(10 +- sqrt(10^2 - 4*3*3))/(2*3)}}} = {{{(10 +- sqrt(64))/6}}} = {{{(10 +- 8)/6}}}.


One root is  {{{x[1]}}} = {{{(10+8)/6}}} = {{{18/6}}} = 3.


The other root is  {{{x[2]}}} = {{{(10-8)/6}}} = {{{2/6}}} = {{{1/3}}}.



Thus we solved the problem formally and obtained &nbsp;BOTH &nbsp;solutions (!)
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 . . . . . . . 



Good student should know both ways.


Worst or better, &nbsp;the average student will, &nbsp;probably, &nbsp;learn the hard way in the school.


But I am &nbsp;129% &nbsp;sure that &nbsp;NOBODY &nbsp;and &nbsp;NEVER &nbsp;will show him &nbsp;(or her) &nbsp;the easy way.


But without knowing this easy way, &nbsp;Math education can not be considered as complete.


Therefore, &nbsp;I started with this easy &nbsp;"joking" &nbsp;mode.



Happy learning (!)