Question

If the base of a rectangle is increased by 10% and the area is unchanged then the altitude is decreased by
(a) 9%
(b) 10%
(c) 11%
(d) \[11\dfrac{1}{9}\]%
(e) \[9\dfrac{1}{11}\]%

Answer 1

Hint:At first consider base as l and altitude as b. Then find the original area of the rectangle. As the new base of the rectangle is given so by the fact that areas remain unchanged, find the value of new altitude in terms of decrease using the formula, $\dfrac{\text{decrease}}{\text{original value}}\times100\%$.

Complete step-by-step answer:
In the question we are told that the base of a rectangle is increased by 10 % but the area of the rectangle is unchanged or remains constant then we have to find the percentage by which altitude decreases.
First let’s assume the base of the rectangle is l and the altitude of the rectangle is b. So the area of the rectangle will be l \[\times \] b or lb.
Now in the question it is further said that base is increased by 10%. So, let the new base of the rectangle be l. Hence \[l'=l+\dfrac{10}{100}l=l+\dfrac{l}{10}\] or \[\dfrac{11}{10}l\].
As the altitude is also changed so let’s new altitude be b or k be where k be the rational number.
So, the area of new rectangle will be \[l'\times b'\] or \[\dfrac{11l}{10}\times kb\] or \[\dfrac{11k}{10}lb\].
As the area said in the question remains unchanged so we can say that, \[\dfrac{11klb}{10}=lb\].
Or, \[\dfrac{11k}{10}=1\]
So, the value of k is \[\dfrac{10}{11}\].
Hence the new altitude b of the rectangle will be \[b'=\dfrac{10}{11}b\]. The original measure of altitude was b. So, the decrease in altitude is \[\left( b-\dfrac{10}{11}b \right)\] or \[\dfrac{b}{11}\].
The percentage in decrease is to be found out using formula,
$\dfrac{\text{decrease}}{\text{original value}}\times100\%$.
So, here decrease is \[\dfrac{b}{11}\] and original value is b so the percentage in decreases is \[\dfrac{\dfrac{b}{11}}{b}\times 100\] % or \[\dfrac{1}{11}\times 100\] which is \[\dfrac{100}{11}\]% or \[9\dfrac{1}{11}\] %.
So the altitude is decreased by \[9\dfrac{1}{11}\] %.
Hence the correct option is (e).

Note: We can also find the percentage in decrease by unitary method by considering b as 100 %. Hence the \[\dfrac{b}{11}\] quantity will be of \[\dfrac{b}{11}\times \dfrac{100}{b}\] % or \[\dfrac{100}{11}\] %.