SOLUTION: Hi joanne and dave each plucked a certain number of oranges in an orchard. Joanne gave 1/5 of her orange s to dave. Dave then gzve 1/6 of what he had back to her. Next joanne gave

Question 1122788: Hi
joanne and dave each plucked a certain number of oranges in an orchard. Joanne gave 1/5 of her orange s to dave. Dave then gzve 1/6 of what he had back to her. Next joanne gave 1/3 back to dave. At the end joanne had 36 orange s and dave 88.
how many did easch of yhem pluck initially.
Thanks
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
joanne had 50 to start.
dave had 74 to start.

joanne gave 1/5 of what she had to dave.
1/5 of 50 = 10

joanne now had 40
dave now had 84.

dave gave 1/6 of what he had back to joanne.
1/6 of 84 = 14

joanne now had 54
dave now had 70

joanne gave 1/3 of what she had back to dave.
1/3 of 54 = 18.

joanne now had 36
dave now had 88.

solution looks good.
solution is that joanne plucked 50 originally and dave plucked 74 originally.

you can work forwards or you can work back.
either way is a chore.

i worked forward as follows:

let A = the amount of oranges that joanne had originally and let B = the amount of oranges that dave had originally.

step 1:

joanne gave 1/5 of her oranges to dave.

joanne now had 4/5 * A

dave now had B + 1/5 * A

step 2:

dave gave 1/6 of what he had back to joanne.

joanne now had 4/5 * A + 1/6 * (B + 1/5 * A)
simplify this to get:
joanne now had 4/5 * A + 1/6 * B + 1/6 * 1/5 * A
simlify further to get:
joanne now had 4/5 * A + 1/6 * B + 1/30 * A
4/5 * A is equivalent to 24/30 * A
equation becomes:
joanne now had 24/30 * A + 1/6 * B + 1/30 * A
combine like terms to get:
joanne now had 25/30 * A + 1/6 * B

dave now had B + 1/5 * A - 1/6 * (B + 1/5 * A)
simplify to get:
dave now had B + 1/5 * A - 1/6 * B - 1/6 * 1/5 * A
simplify further to get:
dave now had 5/6 * B + 1/5 * A - 1/30 * A
1/5 * A is equivalent to 6/30 * A
equation becomes:
dave now had 5/6 * B + 6/30 * A - 1/30 * A
combine like terms to get:
dave now had 5/6 * B + 5/30 * A

step 3:

joanne gave 1/3 of what she had back to dave.

joanne now had 25/30 * A + 1/6 * B - 1/3 * (25/30 * A + 1/6 * B)
simplify this to get:
joanne now had 25/30 * A + 1/6 * B - 1/3 * 25/30 * A - 1/3 * 1/6 * B
simplify further to get:
joanne now had 25/30 * A + 1/6 * B - 25/90 * A - 1/18 * B
25/30 * A is equivalent to 75/90 * A
1/6 * B is equivalent to 3/18 * B
equation becomes:
joanne now had 75/90 * A + 3/18 * B - 25/90 * A - 1/18 * B
combine like terms to get:
joanne now had 50/90 * A + 2/18 * B

dave now had 5/6 * B + 5/30 * A + 1/3 * (25/30 * A + 1/6 * B)
simplify to get:
dave now had 5/6 * B + 5/30 * A + 25/90 * A + 1/18 * B
5/6 * B is equivalent to 15/18 * B
5/30 * A is equivalent to 15/90 * A
equation becomes:
dave now had 15/18 * B + 15/90 * A + 25/90 * A + 1/18 * B
combine like terms to get:
dave now had 16/18 * B + 40/90 * A

step 4.

joanne ended up with 36.
this means that:
50/90 * A + 2/18 * B = 36

dave ended up with 88.
this means that:
16/18 * B + 40/90 * A = 88

you have 2 equations that need to be solved simultaneously.

they are:

50/90 * A + 2/18 * B = 36
16/18 * B + 40/90 * A = 88

rearrange the terms so that like terms are underneath each other.

equations become:

50/90 * A + 2/18 * B = 36
40/90 * A + 16/18 * B = 88

multiply both sides of both equations by 90 to get:

50 * A + 10 * B = 3240
40 * A + 80 * B = 7920

multiply both sides of the first equation by 8 and leave the second equation as is to get:

400 * A + 80 * B = 25920
40 * A + 80 * B = 7920

subtract the second equation from the first to get:

360 * A = 18000

solve for A to get:

A = 18000 / 360 = 50

in the second equation of 40 * A + 80 * B = 7920, replace A with 50 to get:
40 * 50 + 80 * B = 7920
simplify to get:
2000 + 80 * B = 7920
subtract 2000 from both sides of this equation to get:
80 * B = 5920
solve for B to get:
B = 5920 / 80 = 74.

your solution is that A = 50 and B = 74 which means that:

joanne started with 50 and dave started with 74.

this was already confirmed to be a good solution.

as i said before, it's a chore.
you have to be very diligent about keeping track of what you're doing.
you could also work your way back from the answer, but that's also a chore.
which one is easier is up for grabs.
try it both ways and see which one you like better.

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