Question 1193163
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I can't imagine how the response from the other tutor would be of any use to the student, unless the student just wanted to get the answer without learning anything....<br>
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Let's solve the problem in the "forward" direction, building an ugly-looking equation directly from the given information, and then solving that equation to find the answer to the problem.<br>
Then we will look at how we solved that ugly equation to see how we could solve the problem faster and more easily by working "backwards".<br>
And finally we will look at solving the problem backwards using the concept of an inverse function, which will be faster yet.<br>
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Let x be the number of beads Jane started with.<br>
Note that when she gives away 1/3 of the beads she has, she has 2/3 of that number of beads left; the number of beads remaining gets multiplied by 2/3.<br>
She gave 1/3 of what she had to Sally, plus 4 more; the number she had left was {{{(2/3)x-4}}}<br>
She then gave 1/3 of what she had left to Mary, plus 3 more; the number she had left was {{{(2/3)((2/3)x-4)-3}}}<br>
She then gave 1/3 of what she had left to Betty, plus 3 more; the number she had left was {{{(2/3)((2/3)((2/3)x-4)-3)-3}}}<br>
That number she had left was 3:<br>
{{{(2/3)((2/3)((2/3)x-4)-3)-3=3}}}<br>
Solve the equation....<br>
add 3: {{{(2/3)((2/3)((2/3)x-4)-3)=6}}}
multiply by 3/2: {{{((2/3)((2/3)x-4)-3)=9}}}
add 3: {{{(2/3)((2/3)x-4)=12}}}
multiply by 3/2: {{{(2/3)x-4=18}}}
add 4: {{{(2/3)x=22}}}
multiply by 3/2: {{{x=33}}}<br>
ANSWER: Jane started with 33 beads<br>
Now let's look at solving the problem backwards, starting with the 3 beads Jane finished with.<br>
There are two kinds of things that happened to the number of beads Jane started with:
(1) She gave away 1/3 of what she had; leaving her with 2/3 of what she had previously.  That means the number of beads she had left got multiplied by 2/3.  To work backwards, we need to divide by 2/3, which means multiplying by 3/2.
(2) She gave away some number of beads; to work backwards, we need to add that number of beads.<br>
Now work backwards, using those steps.<br>
She finished with 3 beads.
Before that 3 was subtracted from her number of beads, so add 3 beads: 3+3=6
Before that her number of beads was multiplied by 2/3, so multiply by 3/2: 6(3/2)=9
Before that 3 was subtracted from her number of beads, so add 3: 9+3=12
Before that her number of beads was multiplied by 2/3, so multiply by 3/2: 12(3/2)=18
Before that 4 was subtracted from her number of beads, so add 4: 18+4=22
Before that her number of beads was multiplied by 2/3, so multiply by 3/2: 22(3/2)=33<br>
Observe that the operations we performed to work backwards to get the answer are exactly the operations we performed to solve our ugly equation.<br>
With a little practice with this kind of problem, working backwards is faster and easier for most students than writing and solving an ugly equation.<br>
The part of the process that is hardest to understand is that (for example in this problem) when 1/3 of the beads are given away, the number remaining is multiplied by 2/3, so when working backwards at this step we need to multiply by 3/2.<br>
Finally, for the more interested student, here is a way to do this "working backward" much more quickly.<br>
Working backwards in a problem is like finding an inverse function.  The problem tells us what happened to the original number of marbles; the final number of marbles is some function of the original number f(x).  Given the final number of beads, we can find the original number using the inverse of f(x).<br>
The function that changes the original number of marbles does the following operations:
multiply by 2/3;
subtract 4;
multiply by 2/3;
subtract 3;
multiply by 2/3,
subtract 3<br>
The inverse function needs to perform the opposite operations, in the opposite order:
add 3;
multiply by 3/2;
add 3;
multiply by 3/2;
add 4;
multiply by 3/2<br>
Again that sequence of operations is the same as we used in both of the previous methods for solving the problem.<br>