Question 263308: Red birds and blue birds, 30 in all, feasted on worms in my yard. The red birds got 108 worms in total, as did the blue birds. Each blue bird got 3 fewer worms than each red bird. How many blue birds were there?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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