Max has a strip of paper that is 3 1/3 (mixed number) inches in length. He needs to cut the strips into pieces that are 1/6 of an inch in length. How many pieces can she cut from her original strip?
Answers
Answer:
So the next strip will begin at what was the 4-inch mark on the
original board, and take up the next 3 inches:
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
<---------> <-> <---------> <->
strip 1 saw strip 2 saw
Repeat again, and we have
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
<---------> <-> <---------> <-> <---------> <->
strip 1 saw strip 2 saw strip 3 saw
So we have a 2-inch strip left that is too small to make another 3-
inch strip from.
How could we have calculated this? Well, each strip uses up not just
3 inches, but 4 inches of wood; so we can divide 14 by 4 to find how
many times we can repeat the process--that is, in cutting three times,
we used up 3 times 4 inches of wood, counting both the strips we made
and the sawdust that was produced.
There is just one little detail to worry about. What if we had had
a 15-inch board to start with, so that the last strip was exactly 3
inches wide, and we didn't have to make the last cut?
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
<---------> <-> <---------> <-> <---------> <-> <--------->
strip 1 saw strip 2 saw strip 3 saw strip 4
Dividing 15 by 4 gives 3 with a remainder of 3, and we'd have to
recognize that the remainder is enough to leave one more strip.
There's a neat trick I can see that would allow us always to find the
number of strips we can make by dividing; it involves pretending that
the board started out 1 inch wider than it is, so that we would need
that one last cut and have nothing left. But you don't need that
trick in order to solve the proble