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Solve Word Problems Involving Fractional Equations

The numerator of a fraction is 12 more than the denominator. If the value of the fraction is 3 less than the value of the denominator, what is the value of the fraction?

denominator = x
numerator = x+12
value = x-3

$fraction = value$

$\frac{x+12}{x} = x-3$

$x+12=(x-3)(x)$

$x+12=x^{2}-3x$

$x+12-(x+12)=x^{2}-3x-(x+12)$

$0=x^{2}-3x-x-12$

$0=x^{2}-4x-12$

factored form: (x+a)(x+b)

(a)(b)=-12

a+b=-4

factors of -12:

±~~1, 12~~ → do not add up to -4
±~~3, 4~~ → do not add up to -4
±6, 2 → -6, +2 → add up to -4

$0=x^{2}-4x-12$

$0=(x-6)(x+2)$

$x-6=0$

$x-6+6=0+6$

$x=6$

$x+2=0$

$x+2-2=0-2$

$x=-2$

Value: x-3

x=6, Value: 6-3 = 3

x=-2, Value: -2-3 = 5