Golden Ratio

You see here a rectangle with a special shape. The shape has the property that if you make a square at the bottom and cut it off, the remaining rectangle is similar to the original rectangle. That is . Find the ratio of to .

This is a famous ratio, known as the golden ratio. Rectangles of this shape are frequently used in architecture.

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Pickanumba, Give Digits

Ola says to Argyle: Pick your favorite number with no more than three digits. Multiply it by 7, add 2, multiply the result by 13, subtract 25, multiply the result by 11, and subtract 10. Argyle did it. When Ola asked for the 10’s digit, the 100’s digit, and the 1000’s digit of his answer, Argyle replied 7, 3, and 4 respectively. Ola then told him his original number. What was it?

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Sandwiches

Al and Bob are just sitting down to lunch when they are joined by their friend Carol. Al has brought 5 sandwiches to eat, Bob has brought 3, but Carol has not brought any. She does have $4, however. The three of them divide the food evenly, and then Carol gives Al and Bob her $4 to pay for her share. How should Al and Bob share the money?

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