The sum of a number and its reciprocal is 13/6. Find the number.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Let the number we have to find be N.

The sum of the number and its reciprocal is 13/6

=> N + 1/N = 13/6

=> N^2 + 1 = (13/6)N

=> 6N^2 + 6 = 13N

=> 6N^2 - 13N + 6 = 0

=> 6N^2 - 9N - 4N + 6 = 0

=> 3N(2N - 3) - 2(2N - 3) = 0

=> (3N - 2)(2N - 3) = 0

=> N = 2/3 and 3/2

The required number is 2/3 or 3/2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The reciprocal of a number is the number that multiplied to original number, gives 1.

We'll take the x number and it's reciprocal, 1/x.

We'll verify the constraint:

x*(1/x)=1

Now, we'll impose the constraint from enunciation:

x + 1/x = 13/6

We'll mutliply by 6x all over:

6x^2 + 6 = 13x

We'll move all terms to one side:

6x^2 - 13x + 6 = 0

We'll apply quadratic formula:

x1= [13+sqrt(169 - 144)]/12

x1 = (13+5)/12

x1 = 18/12

x1 = 3/2

x2 = (13-5)/12

x2 = 8/12

x2 = 2/3

The number could be:x= 3/2 or x = 2/3.

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