For a rectangle, the perimeter is calculated by using the formula P = 2l + 2b, where l represents the length and b represents the breadth.
In our case, the initial perimeter is given to be 420 cm.
Therefore, 420 = 2l + 2b.
We can simplify this equation to 210 = l + b ( Equation 1).
Now, we are told that the length is increased by 20% and the breadth was reduced by 40% to form a new rectangle that has the same perimeter. This means that the new rectangle has length 1.2l and breadth 0.6b.
The new perimeter, which is still 420 cm, is now calculated by 420 = 2(1.2l) + 2(0.6b).
This simplifies to 420 = 2.4l + 1.2b, and further simplifies to 350 = 2l + b ( Equation 2).
We now have two equations, and we need to solve them simultaneously.
Subtract Equation 2 from Equation 1, we have:
210 - 350 = l + b - 2l - b, which simplifies to
-140 = -l, therefore l = 140cm.
Substitute l = 140 into Equation 1, we have 210 = 140 + b, so b = 70 cm.
So, the initial length and breadth of the rectangle are 140 cm and 70 cm, respectively.
After, increasing the length by 20% and decreasing the breadth by 40%, we get the new length, l_new as 1.2 * 140 = 168 cm, and the new breadth, b_new as 0.6 * 70 = 42 cm.
So, among the options given, option B, that is 168 cm and 42 cm, matches the calculated length and breadth for the new rectangle. Therefore, the correct answer is option B.
Answer: B. 168 cm and 42 cm