Question 1195818
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I will show two different methods for solving the problem, starting from different starting places, to demonstrate how taking a few moments to look for different ways of setting up a problem can make a big difference in how complicated the solution is.<br>
First method....
Probably the most obvious starting point is to let the variable be what the question asks for -- the number of apples in the basket.  The solution then might go like this....<br>
(1) x = original number of apples<br>
(2) kado divided the apples into three equal portions and took one of them.  So the number of apples he took was (1/3)x, and the number remaining was (2/3)x.<br>
(3) chado then came and saw the two piles of apples with a total of (2/3)x apples.  He mixed them and divided them into three equal piles, with one apple left over.
Let y be the number of apples in each of the three piles; then, with one apple left over, the number of apples remaining, (2/3)x, is equal to the number in the three piles, plus the one extra apple:<br>
3y+1 = (2/3)x
3y = (2/3)x-1
y = (2/9)x-1/3<br>
He then took one pile, containing (2/9)x-1/3 apples, plus the extra apple; so the number he took was (2/9)x+2/3.
But the number he took was the same as the number kado took, so<br>
(2/9)x+2/3 = (1/3)x
2/3 = (1/9)x
x = 9(2/3) = 6<br>
ANSWER: The original number of apples was 6.<br>
CHECK:
original number: x = 6
number in each pile kado made: 6/3 = 2
number kado took: (1/3)x = 2
number remaining: 6-2 = 4
number in each pile chado made: 1
number chado took (one pile plus the extra apple): 1+1 = 2
number left for nado: 4-2 = 2<br>
That solution method worked; but it involved some ugly fractions.  So let's start from a different place and see if the solution is easier.<br>
Let x be the number of apples in each pile chado made.
Then the number of apples chado found was 3x+1.
chado took one pile (x) plus the extra apple, so he took x+1 apples.
That left (3x+1)-(x+1) = 2x apples for nado.
But chado and nado got equal numbers of apples:
x+1 = 2x
x = 1
So chado and nado each got 2 apples.
But all three of them got the same number of apples; so the original number of apples was 3*2 = 6.<br>
ANSWER: 6<br>
Observe that the solution by this method was far easier than by the other method....<br>