   SEARCH HOME Math Central Quandaries & Queries  Question from shwetha, a student: The length of a rectangle is greater than the breadth by 18 cm. If both length and breadth are increased by 6 cm then area increases by 168 cm square. Find the length and breadth of the rectangle. Hi there.

Start by assigning variables to what you want to know:

> Find the length and breadth of the rectangle.
Let L = the length of the rectangle.
Let B = the breadth of the rectangle.

Since the question deals with area, let's also define a variable for the area and state the equation relating these variables:

Let A = the area of the rectangle.

Area of a rectangle is the breadth times the length:

A = BL

Now translate the other two sentences into equations:

> The length of a rectangle is greater than the breadth by 18 cm.
L = B + 18
> If both length and breadth are increased by 6 cm then area increases by 168 cm square.
(L + 6)(B + 6) = A + 168

The general principle when working with systems of linear equations is that you need at least as many equations as variables in order to solve the system. Here, we have three variables: L, B, and A. And we have three equations:

i. A = BL
ii. L = B + 18
iii. (L + 6)(B + 6) = A + 168

Now use the substitution method (look that up in our Quick Search if you don't know how to do it) to substitute for A by blending equation i and iii. Then blend the result with ii by substituting for L. This will give you a single-variable equation that you can solve for B. Using that value for B in equation ii, you can solve for L.

Hope this helps,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.