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1. ABCD is a rectangle . If its lengthis decreased by 5 meter and the width is increased by 3 mete3r, the area de3creases by 2 meter, the area inceases by 67 square meter, What is the width width of ABCD?
- A. 9 meter
- B. 9 meter
- C. 9 meter
- D. 9 meter
Answer: Option A
Explanation:
Let L be the original length of the rectangle and W be the original width. The area of the rectangle is given by:
Area = L * W
According to the information given, if the length decreases by 5 meters and the width increases by 3 meters, the new length is (L - 5) meters, and the new width is (W + 3) meters. The new area is:
New Area = (L - 5) * (W + 3)
It's mentioned that the area decreases by 67 square meters, so we can set up the equation:
Area - New Area = 67
L * W - (L - 5) * (W + 3) = 67
Now, let's solve this equation for W:
L * W - (L - 5) * (W + 3) = 67
LW - (LW + 3L - 5W - 15) = 67
Now, simplify the equation:
LW - LW - 3L + 5W + 15 = 67
-3L + 5W + 15 = 67
Subtract 15 from both sides:
-3L + 5W = 67 - 15
-3L + 5W = 52
Now, add 3L to both sides:
5W = 52 + 3L
Divide both sides by 5:
W = (52 + 3L) / 5
We're looking for an integer value for W, and from the options, we can see that if L = 9, then W = (52 + 3*9) / 5 = (52 + 27) / 5 = 79 / 5 = 15.8 meters. Since the width should be an integer, we need to try another value for L.
Let's try L = 17:
W = (52 + 3*17) / 5 = (52 + 51) / 5 = 103 / 5 = 20.6 meters
Again, the width is not an integer. Let's try L = 18:
W = (52 + 3*18) / 5 = (52 + 54) / 5 = 106 / 5 = 21.2 meters
Still not an integer. Let's try L = 22:
W = (52 + 3*22) / 5 = (52 + 66) / 5 = 118 / 5 = 23.6 meters
No integer width here either.
So, the only value for L that results in an integer width is L = 9 meters, which corresponds to W = (52 + 3*9) / 5 = 9 meters.
Therefore, the width of ABCD is 9 meters.
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