Question

if the base of a triangle is decreased by 30% and its height is increased by 25% then the percentage decrease in the area is​

Answer 1

let b and h be the original measures of the base and height respectively.

so, new base

= original base - 30% of original base

$= b - \frac{30}{100} \times b\\ = \frac{100b- 30b}{100} = \frac{70b}{100} \\ = \frac{7b}{10}$

and, new height

= original height + 25% of original height

$= h + \frac{25}{100} h = h + \frac{1}{4} h \\ = \frac{4h + 1h}{4} \\ = \frac{5h}{4}$

now, original area of the triangle

$= \frac{1}{2} \times b \times h$

and, new area of the triangle

$= \frac{1}{2} \times \frac{7}{10} b \times \frac{5}{4} h \\ = \frac{35}{40} \times \frac{1}{2} \times b \times h \\ = \frac{7}{8} \times original \: area \\ \\ hence \: \% \: of \: new \: area \: with \: respect \: \\ to \: original \: area \: \\ = \frac{7}{8} \times 100 = \frac{700}{8} = \frac{175}{2} = 87.5 \\ \\ hence \: \% \: decrease \\ = 100\% \: - \: 87.5\% \\ = 12.5\%$

Hence, percentage decrease in the area of the triangle is 12.5%