<PRE><B>Question 263308</B>
r = number of red birds
b = number of blue birds
x = number of worms each red bird got
y = number of worms each blue bird got.


30 birds in all means:


r + b = 30


red birds got 108 worms in total means:


x*r = 108


blue birds got 108 worms in total means:


y*b = 108


each blue bird got 3 fewer worms than each red bird means:


y = x-3


I searched for a common variable that I could substitute for each of the other variables and I came up with the variable x.


from the equation x*r = 108, I got:


r = 108/x


from the equation y*b = 108 and the equation y = x-3, I got (x-3) * b = 108 which led to:


b = 108/(x-3)


substituting these values in the equation of r + b 30, I got:


108/x + 108/(x-3) = 30


now I have one equation in one unknown that can be solved.


multiply both sides of this equation by x*(x-3) to get:


108*(x-3) + 108*x = 30*x*(x-3)


simplify by removing parentheses to get:


108*x - 3*108 +  108*x = 30*x^2 - 90*x


combine like terms and simplify further to get:


216*x - 324 = 30*x^2 - 90*x


subtract 216*x from both sides of the equation and add 324 to both sides of the equation to get:


0 = 30*x^2 - 90*x - 216*x + 324


combine like terms to get:


0 = 30*x^2 - 306*x + 324 which is the same as:


30*x^2 - 306*x + 324 = 0


divide both sides of this equation by 6 to get:


5*x^2 - 51*x + 54 = 0


factor this quadratic equation to get:


(5*x-6) * (x-9) = 0


solve for x to get:


x = 9 or x = 6/5


if x = 9, then:


x*r = 108 becomes 9*r = 108 which means that r = 108/9 = 12.


(x-3)*b = 108 becomes 6*b = 108 which means that b = 18.


r + b = 30 becomes 12 + 18 = 30 which is true.


12*x = 108 means that x = 9 which is true.


18*y = 108 means that y = 6 which is 3 less than 9 which is true.


all equations are satisfied when x = 9


when x = 6/5, then:


x*r = 108 becomes 6/5*r = 108 which means that r = 5*108/6 = 5*18 = 90.


since r + b = 30, this means that b would have to be negative which is not possible so the value of x = 6/5 is not good and is rejected.


the question was how many blue birds were there and the answer is there were 18 blue birds.


last confirmation:


18 blue birds
12 red birds.
each red bird caught 9 worms.
each blue bird caught 6 worms.


12 * 9 = 108
18 * 6 = 108


each blue bird caught 3 less worms than each red bird.


answer is confirmed.






</PRE>