If 15 times the reciprocal of a number is added to the ratio of 9 times the number minus 7 and the number plus 2, the result is 9. What is the number?

Question

If 15 times the reciprocal of a number is added to the ratio of 9 times the number minus 7 and the number plus 2, the result is 9. What is the number?

Answer 1

let x = the number

 

15(1/x) + (9x-7)/(x+2) = 9

 

Multiply the equation by x(x+2) to obtain:  

 

15(x+2) +x(9x-7) = 9x(x+2)

 

15x + 30 + 9x² - 7x = 9x² + 18x

 

9x² + 8x + 30 = 9x² + 18x

 

        8x + 30 = 18x

 

              10x = 30

 

                 x = 3

 

 

Answer 2

We have to convert the words into math:

 

15 times the reciprocal of the number means 15*(1/x)

the ratio of 9 times the number minus 7 and the number plus 2 means (9x-7)/(x+2)

 

Putting it together:

 

(15/x) + (9x-7)/(x+2) = 9

 

multiply the whole equation by x*(x+2) to get rid of the fractions:

 

15*(x-2) + x*(9x-7) = 9x*(x+2)

 

Distribute all the terms:

15x - 30 + 9x² - 7x = 9x^{2 +} 18x

 

Subtract 9x² from both sides:

 

15x - 30 - 7x = 18x

 

Group the 15x and the -7x on the left side:

 

8x - 30 = 18x

 

Subtract 8x from both sides:

 

10x = 30

 

Divide both sides by 10:

 

x = 3