   SEARCH HOME Math Central Quandaries & Queries  Question from Robert, a student: A rectangle is three times as long as it is wide. If the length is decreased by 2 cm and the width is increased by 3 cm, the new rectangle has an area of 42 cm. What are the dimensions of the original rectangle? I am having trouble setting this problem up correctly. Rob Hi Rob,

Suppose that the rectangle has length L cm and width W cm. Look at the first two sentences in the question.

A rectangle is three times as long as it is wide.

This says that L = 3W

If the length is decreased by 2 cm and the width is increased by 3 cm, the new rectangle has an area of 42 cm.

Decreasing the length by 2 cm makes the new rectangle L - 2 cm long. Increasing its width by 3 cm makes it W + 3 cm wide. This new rectangle has an area of 42 square cm and hence length × width = 42 or

(L - 2)(W + 3) = 42

From the first equation you know that L = 3W and thus you can substitute 3W for L in the second equation and solve for W.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.