Question 1208344: Hi
Alan Ben and Caleb shared some marbles. Alan took 40% of the marbles and was given 1 more . Ben took 25% of the remaining and was given 3 more. Caleb took the remaining 12 marbles. How many were there at first.
Found 4 solutions by josgarithmetic, ikleyn, greenestamps, MathTherapy:
Answer by josgarithmetic(39616)   (Show Source):
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
Alan Ben and Caleb shared some marbles. Alan took 40% of the marbles and was given 1 more .
Ben took 25% of the remaining and was given 3 more.
Caleb took the remaining 12 marbles. How many were there at first.
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Let T be the total number of marbles.
Alan has 0.4T +1 marbles; the remaining is T - (0.4T+1) = 0.6T-1.
Ben has 0.25(0.6T-1) + 3 marbles;
Caleb has 12 marbles.
Write an equation for the total
(0.4T + 1) + (0.25(0.6T-1) + 3) + 12 = T.
Simplify and find T
0.4T + 1 + 0.15T - 0.25 + 3 + 12 = T
0.55T + 15.75 = T
15.75 = T - 0.55T
15.75 = 0.45T
T = 15.75/0.45 = 35.
ANSWER. The total number of marbles is/was 35.
This total number of marbles was not changed before and after: it remained THE SAME.
Solved.
The question in the problem is posed INCORRECTLY.
The correct question is: "How many marbles were there in total ?"
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Let me explain you, WHY I think that the question is posed incorrectly
and WHY a TRUE question should be posed differently.
In this problem, the marbles were distributed inside some group of three people.
No marbles went outside. Therefore, the total number of marbles at the beginning was
the same as at the end. So, there is a hidden symmetry between the future and the past
in this problem. The symmetry is that the number of the marbles at the origin
and at the end is the same.
If the problem's creator formulates the problem according to my instruction, he (or she)
demonstrates that he/she recognizes this symmetry and does not violate it.
If he/she formulates differently, he/she demonstrates that he/she does not care
about his/her language and leaves to the reader to figure out these details on his/her own.
My point is that the problem should sound harmonically, should be posed clearly
and should not distract the reader from the essence of the task.
That's all.
Behind all this is respect / (or disrespect) for the symmetry, for the harmony and for the reader.
Nothing more and nothing else.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Note there is no need to change the question that is asked. If Alan took some marbles, then Ben took some, and then Caleb took the ones that were left, then there were no marbles left. So asking how many marbles there were at first is the right question.
The algebra involved if you work this kind of problem in the "forwards" direction can get a bit ugly. Often a problem like this can be solved more easily by working backwards.
Caleb took the last 12 marbles. The last thing that happened before that was Ben took 3 marbles, so before Ben took those 3 marbles the number of marbles was 12+3 = 15.
The last thing that happened before Ben took those 3 marbles was he took 25% (1/4) of the marbles. So the 15 marbles before Ben took his last 3 are 75% of the number there were before Ben took any. So 100% of the number before Ben took any was 20.
Before Ben took any, the last thing that happened was Alan took 1 additional marble; so before he took his last one the number of marbles was 20+1 = 21.
And before Alan took his last one, he took 40% of the original number of marbles. So the 21 marbles after he took 40% of the original number are 60% of the original number, which means 100% of the original number of marbles was 35.
With all the words of explanation, that sounds like a very long solution. But if your mental math is good, it doesn't take much time.
Answer by MathTherapy(10551)  (Show Source):
You can put this solution on YOUR website!
Hi
Alan Ben and Caleb shared some marbles. Alan took 40% of the marbles and was given 1 more . Ben took 25% of the remaining and was given 3 more. Caleb took the remaining 12 marbles. How many were there at first.
Let original number of marbles, be M
Then Alan took 40%(M) = .4M, and given 1 more, ended up with .4M + 1
After Alan's .4M + 1, number remaining = (M - .4M) - 1 = .6M - 1
With Ben taking 25% of remainder, given 3 more, he ended up .25(.6M - 1) + 3
Number remaining after Ben's take: (1 - .25)(.6M - 1) - 3
.75(.6M - 1) - 3
.45M - .75 - 3 = .45M - 3.75
Since Caleb took the remainder (.45M - 3.75), which was 12, we get: .45M - 3.75 = 12
.45M = 15.75
Original number of marbles, or 
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