## Tie the Knots

Math brain teasers require computations to solve.Joe needed a rope 75 feet long. He went to a local hardware store, but found that they did not have ropes nearly that long. They had only 3', 5', 12', and 20' segments of rope. Joe would have bought only 20' segments, as they were the cheapest per foot; unfortunately, they had such a limited quantity of segments (though none were out of stock) that he had to buy all of the 20', 12', and 5' segments available, and 3' segments for the remainder. In fact, he bought a greater quantity of each length than the next longer length (i.e., more 3' segments than 5' segments, more 5' segments than 12' segments, and more 12' segments than 20' segments). Just to be sure he had some margin for error, he bought a total of 90 feet, even though he needed only 75 feet. When he got home, he was glad he had decided to buy a little extra, because only then did he realize that he was going to lose some length as he tied the segments together to make the full rope. He tied two pieces together to check how much length he would lose, and found it to be 6 inches lost from each of the two segments that were tied together. Does Joe have enough to make his 75-foot rope?

### Answer

Yes. He will have a 76-foot rope.The only combination that fits the statements in the puzzle is:

1 20' segment

2 12' segments

5 5' segments

7 3' segments

This is a total of 15 segments, which will require 14 knots to tie together. Each knot takes 6 inches off of two segments, or 1 foot total; therefore, of his 90 feet of rope, he will lose 14 feet in the knots, making a 76-foot rope.

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