Found 3 solutions by Edwin McCravy, greenestamps, MathTherapy:
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
Juliet gave her 1/5 of her marbles to Ken. After that, Ken gave 2/3 of what
he had and an additional 4 more marbles to Juliet. Then, Juliet lost 29
marbles on the way home. In the end, Juliet had 97 marbles and Ken had 21
marbles left. How many marbles did each of them have at first?
It's easier to work it backwards.
>>In the end, Juliet had 97 marbles and Ken had 21 marbles left.
>>Juliet lost 29 marbles on the way home.
97+29=126. So before she lost those, Juliet had 126 marbles and Ken had 21.
>>Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet.
Suppose Ken had N marbles before that.
2/3 of N is 2/3N. 4 more is 2/3N+4. So Ken had 21+2/3N+4 before he gave
that to Juliet. But that must be equal to what Ken had before he gave that
to her, which was N. So we have the equation
21+2/3N+4 = N
25+2/3N = N
75+2N = 3N
75 = N
So Ken had 75 and then gave Juliet 2/3 of 75, which is 50, plus 4 more,
which makes it 54 that he gave to Juliet. So before she got those from Ken,
Juliet had 54 fewer marbles, or 126-54=72. And of course, Ken had 75.
Ken had 75 and Juliet had 72 back then.
>>>Juliet gave 1/5 of her marbles to Ken.
Suppose Juliet had M before giving those to Ken.
1/5 of M is 1/5M. So she had 1/5M more before she gave that to Ken. That
is, she had 72+1/5M. But that must be equal to what she had before she gave
that to Ken, which was M. So we have the equation
72+1/5M = M
360+M = 5M
360 = 4M
90 = M
1/5 of 90 is 18. So before she gave that to Ken, he must have had 18 fewer
than 75. 75-18 = 57.
So the answer is that at first, Juliet had 90 and Ken had 57.
Let's check:
Juliet began with 90 and Ken began with 57.
>>>Juliet gave her 1/5 of her marbles to Ken.
1/5 of 90 is 18. So Juliet then had 90-18=72 and Ken had 57+18=75
>>>After that, Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet.
2/3 of 75 is 50. An additional 4 makes it 54. So Juliet then had 72+54=126
and Ken had 75-54=21.
>>>Then, Juliet lost 29 marbles on the way home.
So she ended up with 126-29=97 and Ken still had his 21.
It checks!
Edwin
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
For SOME students, working the problem backwards might be easier....
But solving it "forwards" is a good exercise in algebra, which some students might find more to their liking that solving it backwards.
Juliet Ken
----------------------------------------------------------------------
x y
(4/5)x y+(1/5)x Juliet gave 1/5 of hers to Ken
(4/5)x+(2/3)(y+(1/5)x)+4 (1/3)(y+(1/5)x))-4 Ken gave 2/3 of his marbles plus 4 more to Juliet
(4/5)x+(2/3)(y+(1/5)x)+4-29 (1/3)(y+(1/5)x))-4 Juliet lost 29 marbles
Juliet ended with 97 marbles:
(4/5)x+(2/3)(y+(1/5)x)+4-29=97
(4/5)x+(2/3)y+(2/15)x=122
(14/15)x+(2/3)y=122
14x+10y=1830 [1]
Ken ended with 21 marbles:
(1/3)(y+(1/5)x))-4=21
(1/3)y+(1/15)x=25
5y+x=375 [2]
Solve [1] and [2] by elimination, dividing [1] by 2:
5y+7x=915
5y+x=375
6x=540
x=90
5y+90=375
5y=285
y=57
ANSWERS:
Juliet started with x=90 marbles
Ken started with y=57 marbles
CHECK:
Start: (J,K)=(90,57)
Juliet gives 1/5 of hers (18) to Ken: (J,K)=(72,75)
Ken gives 2/3 of his plus 4 more (54) to Juliet: (J,K)=(126,21)
Juliet loses 29: (97,21)
Answer by MathTherapy(10553) (Show Source): You can put this solution on YOUR website!
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Juliet gave her 1/5 of her marbles to Ken. After that, Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet. Then, Juliet lost 29 marbles on the way home. In the end, Juliet had 97 marbles and Ken had 21 marbles left. How many marbles did each of them have at first?
Let original amounts Juliet and Ken had, be J, and K, respectively
After giving Ken of her marbles, Juliet had remaining
After receiving of Juliet's marbles, Ken then had:
After giving of what he had and an additional 4 more marbles to Juliet, Ken then had: remaining
Since Ken had a final count of 21 marbles, we then get:
After receiving and an additional 4 more marbles from Ken, Juliet then had: remaining
Since Juliet lost 29 marbles and ended up with 97 marbles, we then get:
J + 5K = 375 ------ eq (i)
14J + 10K = 1,830 ---- eq (ii)
2J + 10K = 750 ------ Multiplying eq (i) by 2 ------- eq (iii)
12J = 1,080 ---- Subtracting eq (iii) from eq (ii)
Original number of marbles Juliet had, or
90 + 5K = 375 ------- Substituting 90 for J in eq (i)
5K = 375 - 90
5K = 285
Original number of marbles Ken had, or
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