Question

If the height of a triangle is decreased by $40\%$ and its base is increased by $40\%$ what will be the effect on its area?
A) No change
B) $8\%$ decrease
C) $16\%$ decrease
D) $16\%$ increase
E) None of these

Answer 1

Hint: The area of the triangle is given by $A = \dfrac{1}{2} \times b \times h$ where b is the base and h is the height of the triangle get the area of the triangle by reducing the height by $40\%$ and increasing the base by $40\%$ then compare with the known formula to get the final results.
Complete Step by Step Solution:
We know that the area of a triangle is given by $A = \dfrac{1}{2} \times b \times h$ where b is the base and h is the height of the triangle.
Let us assume that the new triangle formed by reducing the height and increasing the base is given by $A' = \dfrac{1}{2} \times b' \times h'$ Where $b'$ is the new base, $h'$ is the new height and $A'$ is the new area.
Now using this let us try to get the height when it will be reduced by $40\%$

\therefore b' = b + 40\% \times b\\\ = b + \dfrac{{40}}{{100}}b\\\ = 1.4b \end{array}$$ Similarly for height we will have $$\begin{array}{l} \therefore h' = h - 40\% \times h\\\ = h - \dfrac{{40}}{{100}}h\\\ = 0.6h \end{array}$$ Now putting all of this in the area formula i.e., $$\begin{array}{l} A' = \dfrac{1}{2} \times b' \times h'\\\ = \dfrac{1}{2} \times \left( {1.4b} \right) \times \left( {0.6h} \right)\\\ = 0.84 \times \dfrac{1}{2} \times b \times h \end{array}$$ Now we know that $$A = \dfrac{1}{2} \times b \times h$$ Putting this we get it as $$A' = 0.84A$$ Clearly $$A > A'$$ Therefore if we subtract A’ from A we will get that how much less it is $$\begin{array}{l} \therefore A - A' = A - 0.84A\\\ = 0.16A \end{array}$$ So if we convert it into percent we will get that $$16\%$$ of A Therefore A’ is reduced by 16% of A Which means option C is correct. It is a 16% decrease. Note: Do note that getting a relation between the old area and the new area is a key step. The ultimate goal in these types of questions must be to establish a relation between the new and the old value.