Question

If the height of a triangle is decreased by 40% and its base is increased by 40% what will be the percentage change in its area.

Answer 1

Hint: To solve this question we will use the formula of the area of triangle which is given by \[A=\dfrac{1}{2}(b)(h)\], where b is the base of the triangle and h is the height of the triangle.

Complete Step-by-Step solution:
Then try to find the value of the new base and new height and substitute the new values to obtain the new area.

We know that the area of triangle is given as,
\[A=\dfrac{1}{2}(b)(h)\], where b is the base of the triangle and h is the height of the triangle.
Given that the height of a triangle is decreased by 40%, this was the percentage increase in the height.
We will calculate the new height now.
Let the new height be h’ then h’ would be equal to,
\[\begin{align}
  & h'=h-40(h)\\
 & \Rightarrow h'=h\left( 1-\dfrac{40}{100} \right)\\
 & \Rightarrow h'=h(1-0.4)\\
 & \Rightarrow h'=0.6h\\
\end{align}\]
Now given that the base is increased by 40%, this was the percentage increase in the base.
We will calculate the new base now.
Let the new height be b’ then b’ would be equal to,
\[\begin{align}
  & b'=b+(40)b\\
 & \Rightarrow b'=b+0.4b\\
 & \Rightarrow b'=1.4b\\
\end{align}\]
Now we will proceed to calculate the new area of the triangle with the base b’ and the height h’.
Let the new area be A’ then,
\[\begin{align}
  & A'=\dfrac{1}{2}(1.4b)(0.6h) \\
 & \Rightarrow A'=(0.84)\dfrac{1}{2}(b)(h) \\
 & \Rightarrow A'=0.84A \\
\end{align}\]
Therefore, the new area would be 0.84 times the old area A.
Converting the obtained area, A’ into percentage we get,
The new area is 84% of the original area.
Considering the original area to be equal to A as 100% then the percentage decreased area becomes equal to 84%.
Therefore, the percentage change in area of the triangle is equal to 16%.

Note: The possibility of error in this question can be not converting the given percentages of the base and the height and directly putting them in the value of the new area which would be wrong because we always first convert the percentages in number then proceed to solve the question.