SOLUTION: Two adjacent sides of a rectangle have the ratio of 3:4. If the longer side is increased by 6, and the shorter side is decreased by 2, the area of the rectangle increases by 38. Ho

Question 28401: Two adjacent sides of a rectangle have the ratio of 3:4. If the longer side is increased by 6, and the shorter side is decreased by 2, the area of the rectangle increases by 38. How long are the sides of the original rectangle?

Answer by sdmmadam@yahoo.com(530) (Show Source): You can put this solution on YOUR website!
Two adjacent sides of a rectangle have the ratio of 3:4. If the longer side is increased by 6, and the shorter side is decreased by 2, the area of the rectangle increases by 38. How long are the sides of the original rectangle?

Two adjacent sides of a rectangle are its length and its width.
Let the length be a units and let the width be b units.
In the given ratio, observe that the nr is smaller than the denominator.
Therefore the ratio given is between the width and the length.
(as by convention the length of a rectangle is longer than the width)
Given that b:a = 3/4 which means 4b = 3a that is 4b-3a = 0 ----(1)
longer side increased by 6, the shorter side decreased by 2, the area of the rectangle increases by 38
The new length = (a+6) and the new width is (b-2)
Therefore the area of the new rectangle
= new lenght X new width
= (a+6)(b-2)
The old area (ab) is increased by 38 to give the new area (a+6)(b-2)
This implies [(a+6)(b-2)-(ab)] = 38
[a(b-2)+6(b-2)-ab] = 38
ab-2a+6b-12-ab = 38
(ab-ab)-2a+6b = 38+12
[by additive commutativity and associativity on the LHS and transferring (-12) from the left to the right(change side, then change sign)]
0 -2a+6b = 50
That is -2a + 6b = 50
That is -a +3b = 25 ----(2)
which implies (3b-25)= a ----(*)
Putting (*) in (1) that is putting a = (3b-25) in 4b-3a = 0 ----(1)
4b-3(3b-25) = 0
4b-9b+75 = 0
-5b + 75 = 0
75 = 5b
b= 75/5 = (5X15)/5 = 15
Putting b = 15 in (*),
we get a = (3b-25) = 3X15 - 25 = 45-25 = 20
Answer: length = 20 units and width =15 units
Verification: width: legth = 15:20 = 3:4 which is correct
longer side increased by 6, the shorter side decreased by 2, the area of the rectangle should increase by 38
The old rectangle has area = 20X15 = 300 sq units.
The new area is (a+6)(b-2) = (20+6)X(15-2) = 26X13 = 338 which is of course 38 more than the old area.
Therefore our values are correct

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