Q 1 - Two congruent squares overlap to form 3 congruent, non-overlapping rectangles, as shown. If the perimeter of each of these rectangles is 18, what is the area of each?
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A 1 - We can look at this picture like this:
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(This is one of the two squares that overlap)
And this:
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Set the long side of the rectangle as X, and the short side as Y.
Now, we know that:
1. There are 2 long sides and 2 short sides in the rectangle
2. The perimeter is 18
3. The perimeter consists of all the sides of a shape
So, we can set:
18 = 2X + 2Y
Now, we look at the square.
We know that:
1. The square consists of 2 rectangles
2. Squares have 4 equal sides
3. The square we are looking at right now has two kinds of sides:
a. A side consisting of 1 long side. (X)
b. A side consisting of 2 short sides. (2Y)
Since we know that the sides of squares are equal, we can infer that X = 2Y
Then, we substitute it back in the first equation we got:
18 = 2X + 2Y
18 = 2X + X
18 = 3X
X = 6
Substitute X back in the equation again to find Y:
18 = 2(6) + 2Y
18 = 12 + 2Y
6 = 2Y
Y = 3
We want to know what the area of the rectangle is. The equation for that is multiplying the two different sides of the rectangle together. So:
X Y = Area
6 3 = 18
Answer: 18
Q 2 - What is the greatest possible sum of two multiples of 12, each less than 100, whose greatest common factor is 24?
A 2 - If the question says that the greatest common factor of the two numbers is 24, then both of the numbers must be a multiple of 24.
The two numbers that are less than 100 can only have these possibilities:
1. 24 4 = 96
2. 24 3 = 72
3. 24 2 = 48
4. 24 1 = 24
Since the question wants the greatest possible sum, we should choose the two biggest numbers and add them together.
96 + 72 = 168
Answer: 168
Q 3 - The right side of the equation 3(ABC) = BBB represents a 3-digit number with 3 identical digits. If different letters...