The sum of two fractions is 11/12 and their product is 1/8. What is the lesser of the two fractions? Express your answer as a common fraction.
+129852 a + b = 11/12 ⇒ b = 11/12 - a (1)
ab = 1/8 (2) sub (1) into 2
a (11/12 - a) = 1/8
-a^2 + (11/12)a - 1/8 = 0 multiply through by -1
a^2 - (11/12)a + 1/8 = 0 mulyiply through by 24
24a^2 - 22a + 3 = 0 factor as
(6a -1) ( 4a - 3) = 0
Setting each factor to 0 and solving for a gives us
a = 1/6 or a = 3/4
The smaller fraction is 1 / 6
+876 Given a quadratic $ax^2 + bx + c,$
$ - \frac{b}{a} = \frac{11}{12}$
$\frac{c}{a} = \frac{1}{8}$
$a = \text{LCM}(12, 8) = 24$
$- \frac{b}{24} = \frac{11}{12}$
$-b = 22$
$b = -22$
$\frac{c}{24} = \frac{1}{8}$
$c = 3$
$24x^2 - 22x + 3 = 0$
$x = \frac{-(-22) \pm \sqrt{(-22)^2 - 4 \cdot 24 \cdot 3}}{24 \cdot 2} = \frac{22 \pm \sqrt{484 - 288}}{48} = \frac{22 \pm \sqrt{196}}{48} = \frac{22 \pm 14}{48} = \frac{11 \pm 7}{24}.$
$x = \frac{18}{24} = \frac{3}{4}$
$x = \frac{4}{24} = \frac{1}{6}$
$\boxed{\frac{1}{6}}$ is the answer
