Numerator of fraction is 2 less than denominator. If we lower the numerator by this fraction by one and we increase denominator by 3, the fraction shall be equal 1/4. Determine the fraction.
Answers
Answer:
fraction F is so that :
F = n/d
with :
n = d - 2
(n+3)/(d+3) = F + 3/20 ==> (n+3)/(d+3) = n/d + 3/20
==> (d - 2 + 3)/(d+3) = (d - 2)/d + 3/20 (replace, using : n = d - 2 )
==> (d + 1)/(d+3) = (d - 2)/d + 3/20
==> [(d + 1)*d*(d+3)]/(d+3) = [(d - 2)*d*(d+3)]/d + [3*d*(d+3)]/20 (multiply by d*(d+3) )
==> (d + 1)*d = (d - 2)(d+3) + 3/20 * d*(d+3)
==> 20(d + 1)*d = 20(d - 2)(d+3) + 3d*(d+3) ( (multiply by 2à : to get rid of 20 at denominator !)
==> 20d^2 + 20d = 20(d^2 + d - 6) + 3d^2 + 9d
==> 20d^2 + 20d = 20d^2 + 20d - 120 + 3d^2 + 9d
==> ................0 = ................0 - 120 + 3d^2 + 9d
==> 3d^2 + 9d - 120 = 0
==> d^2 + 3d - 40 = 0
discrminant is :
∆ = 3^2 + 4*40 = 13^2
candidate solutions for d are :
d1 and d2 = (-3 +/- 13)/2
d1 = -8 ==> n1 = d1 - 2 = -10 ==> Fraction is : F1 = n1/d1 = 5/4
d2 = 5 ==> n2 = d2 - 2 = 3 ==> F2 = 3/5
verification : let's verify the increase
for F1 : (5+3)/(4+3) - 5/4 = 8/7 - 5/4 = (32 - 35)/28 is NOT 3/20
for F2 : (3+3)/(5+3) - 3/5 = 3/4 - 3/5 = (15 - 12)/20 = 3/20 IP