SOLUTION: Hi kim had 580 red and green beads. after she gave 1/3 of the red beads, she bought another 80 green beads. now the number of green beads was half the number of red beads rema

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Question 1038491: Hi
kim had 580 red and green beads.
after she gave 1/3 of the red beads, she bought another 80 green beads.
now the number of green beads was half the number of red beads remaining.
How many more red beads than green beads did she have at first.
Thanks
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
r Red beads
g Green beads

r%2Bg=580


New combination of beads:
system%28%282%2F3%29r%2C+and%2C+g%2B80%29
and further with the description,
g%2B80=%281%2F2%29%282%2F3%29r
g%2B80=%281%2F3%29r


Solve the system of equations system%28r%2Bg=580%2Cg%2B80=r%2F3%29.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
kim had 580 red and green beads.
after she gave 1/3 of the red beads, she bought another 80 green beads.
now the number of green beads was half the number of red beads remaining.
How many more red beads than green beads did she have at first.

let g equal the number of green beads.
let r equal the number of red beads.

start with r + g = 580.

she gave away 1/3 of the red beads and added 80 to the green beads.
this resulted in the green beads being equal to 1/2 of the remaining red beads.

the remaining red beads equal r - 1/3 * r = 2/3 * r

you get g + 80 = 1/2 * 2/3 * r

you have 2 equations that needs to be solved simultaneously.
they are:

r + g = 580
g + 80 = 1/2 * 2/3 * r

simplify the second equation to get g + 80 = 1/3 * r.
solve for g to get g = 1/3 * r - 80.

replace g in the first equation of r + g = 580 with its equivalent value of 1/3 * r - 80 to get r + 1/3 * r - 80 = 580.
simplify to get 4/3 * r - 80 = 580.
add 80 to both sides to get 4/3 * r = 660
multiply both sides by 3/4 to get r = 3/4 * 660 = 495.

since r + g = 580 and r = 495, then g must be equal to 85, because 495 + 85 = 580.

you have:

r = 495
g = 85.

r + g = 580 becomes 495 + 85 = 580 which becomes 580 = 580.
this confirms the solution satisfies this part of the problem statement.

g + 80 = 1/2 * 2/3 * r becomes 85 + 80 = 1/2 * 2/3 * 495 which becomes 165 = 165.
this confirms the solution satisfies this part of the problem statement.

the solution is.....

number of red beads was originally 495 and number of green beads was originally 85.