Let's assume the original length of the rectangle is "L" units and the original breadth is "B" units.
According to the first condition, if the length is reduced by 5 units and the breadth is increased by 3 units, the new length becomes (L - 5) units and the new breadth becomes (B + 3) units. The area of the new rectangle is reduced by 9 square units, so we can write the equation:
L * B - (L - 5)(B + 3) = 9
Expanding the equation:
LB - (L * B + 3L - 5B - 15) = 9
Simplifying the equation:
LB - LB - 3L + 5B + 15 = 9
2B - 3L = -6 ----(equation 1)
Now let's consider the second condition. If the length is increased by 3 units and the breadth is increased by 2 units, the new length becomes (L + 3) units, and the new breadth becomes (B + 2) units. The area of the new rectangle is increased by 67 square units, so we can write the equation:
(L + 3)(B + 2) - LB = 67
Expanding the equation:
LB + 2L + 3B + 6 - LB = 67
2L + 3B + 6 = 67
2L + 3B = 61 ----(equation 2)
Now we have a system of two equations:
2B - 3L = -6 ----(equation 1)
2L + 3B = 61 ----(equation 2)
We can solve these equations simultaneously to find the values of L and B.
Multiplying equation 1 by 2:
4B - 6L = -12 ----(equation 3)
Adding equation 3 and equation 2:
4B - 6L + 2L + 3B = -12 + 61
7B - 3L = 49
7B = 3L + 49 ----(equation 4)
Now we can substitute equation 4 into equation 1:
2(3L + 49) - 3L = -6
6L + 98 - 3L = -6
3L + 98 = -6
3L = -6 - 98
3L = -104
L = -104 / 3
L ≈ -34.67
Since length cannot be negative, we can conclude that there is no valid solution for this problem. None of the given options (17 and 10, 17 and 19, 17 and 39) is correct.
