Question

The length and breadth of a rectangle are directly proportional. If length increases from \[6\] cm to\[21\] cm and breadth is now \[14\] cm, then what was the breadth before any change in the length occurred?

Answer 1

Hint:Here, we are given that the length and breadth of a rectangle are directly proportional. This means that\[L \propto B\], where L is the length of the rectangle and B is the breadth of the rectangle. Also, we need to find the breadth of the rectangle before any change in length occurs, which means we need to find the breadth of the rectangle when the length of the rectangle is \[6\]cm given.

Complete step by step answer:
Given that, the length and breadth of a rectangle are directly proportional.
Also, given that,
Length of the rectangle be $L_1$ = \[6\] cm
Length of the rectangle be $L_2$ = \[21\] cm
And, the breadth of the rectangle be $B_2$ = \[14\] cm
Let the original breadth of the rectangle be $B_1$ = x cm.
As we are given that, the length and breadth are directly proportional, means
\[L_1 \propto B_1\] and \[L_2 \propto B_2\]
\[ \Rightarrow L_1 = B_1\] and \[L_2 = B_2\]

Thus,
\[\dfrac{{L1}}{{B1}} = \dfrac{{L2}}{{B2}}\]
Substituting the values, we get,
Taking all the values one side and x on the other side, we get,
\[ \Rightarrow \dfrac{{6 \times 14}}{{21}} = x\]
Rearranging this equation, we get,
\[ \Rightarrow x = \dfrac{{6 \times 14}}{{21}}\]
Simplify the above equation, we get,
\[x = \dfrac{{6 \times 2}}{3} \\
\Rightarrow x = 2 \times 2 \\
\therefore x = 4\,cm \\ \]
Thus, the original breadth of the rectangle was \[4\,cm\].

Note: One should know the meaning of directly proportional and inversely proportional. Directly proportional means\[L \propto B\], where length is directly proportional to breadth. And, inverse or indirect proportional means\[L \propto \dfrac{1}{B}\], where length is inverse proportional to breadth.
Here, we suppose given than length is inverse proportional to breadth, then we have,
\[L \propto \dfrac{1}{B}\]
\[ \Rightarrow LB \propto 1\]
So, now according to the data given in the question, we get,
\[L_1 \times B_1 = L_2 \times B_2\]
\[ \Rightarrow \dfrac{{L_1}}{{L_2}} = \dfrac{{B_2}}{{B_1}}\]
Substitute the values and solve accordingly.