# Crab's Weight

## Outline Mathematics

Word Problems

In the absence of competition, even salesmen not minding their customer's mood may do good business

"What's the weight of this one?" asked Mrs Pothersniff, pointing to a large crab there on the wet slab.

Fred considered the question a moment. He had to have his little joke, and Fred was no respecter of persons. "It weighs exactly three quarters of its weight, and three quarters of three quarters of a pound," he told the pompous customer. "I guess you'll want it dressed."

The lady wasn't amused! But what did the crab weigh?

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

"What's the weight of this one?" asked Mrs Pothersniff, pointing to a large crab there on the wet slab.

Fred considered the question a moment. He had to have his little joke, and Fred was no respecter of persons. "It weighs exactly three quarters of its weight, and three quarters of three quarters of a pound," he told the pompous customer. "I guess you'll want it dressed."

The lady wasn't amused! But what did the crab weigh?

Let w denote the weight of the crab. Three quarters of three quarters is 3/4 times 3/4,3/4 times 3/4,4/3 times 4/3 which is 9/16,16/9,9/16,10/16. This leads to the equation

w = 3/4 w + 9/16,w = 9/16,w = 3/4 w,3/4 w = 9/16,w = 3/4 w + 9/16.

This we can simplify to

w/4 = 9/16,w/4 = 9/16,w/9 = 9/16,w/3 = 9/16.

To solve this, mutiply both sides by 4,2,3,4,6,16: w = 9/4,w = 9/8,w = 9/4,w = 16/9,w = 4/9.

Do not forget to check your solution.

### References

- J. A. H. Hunter,
*Mathematical Brain-Teasers*, Dover Publications, 1976

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Copyright © 1996-2018 Alexander Bogomolny

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