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1. Double cover
A sheet of paper is exactly the same size as a rectangular table top. The paper is cut in half and the two halves are placed on the table as shown.
What is the ratio of the area of table left uncovered to the area that is covered twice?
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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Let the area of the table top be $A$. Since the paper is cut in half, each piece of paper has an area of $\frac{A}{2}$

step 2

When the two pieces of paper are placed on the table as shown, they overlap to cover a square section in the middle. The area of this overlap is $\frac{A}{4}$ because each piece of paper covers half of the table, and the overlap is half of one piece of paper

step 3

The area covered twice is $\frac{A}{4}$

step 4

The area left uncovered is the total area of the table minus the area covered by the two pieces of paper. Since each piece of paper covers $\frac{A}{2}$, the total area covered by the two pieces of paper is $A$. Therefore, the area left uncovered is $A - A = 0$

step 5

However, since the pieces of paper overlap, the area covered twice is $\frac{A}{4}$, and the area left uncovered is $A - \left(\frac{A}{2} + \frac{A}{2} - \frac{A}{4}\right) = A - \left(A - \frac{A}{4}\right) = \frac{A}{4}$

step 6

The ratio of the area left uncovered to the area covered twice is $\frac{\frac{A}{4}}{\frac{A}{4}} = 1$

Answer

The ratio of the area left uncovered to the area covered twice is 1.

Key Concept

Ratio of areas

Explanation

The key concept here is understanding how to calculate the areas of the table that are left uncovered and covered twice, and then finding the ratio of these areas.

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