If the area of a square with the side a is equal to the area of a triangle with base a, then the altitude of the triangle is.

Answer: Option D

Explanation:

As far as questions of Area or Volume and Surface area are concerned, it is all about formulas and very little logic. So its a sincere advice to get all formulas remembered before solving these questions.

Lets solve this,

$$\begin{aligned} \text{Area of rectangle =}l*b\\ \text{Area of triangle =}\frac{1}{2}l*b\\ \text{Ratio =}l*b:\frac{1}{2}l*b \\ = 1:\frac{1}{2} \\ = 2:1 \end{aligned}$$

One little thing which should be taken care in this type of question is, be sure you are calculating ration in the given order of the question.
If it is ratio of triangle and rectangle then we have to write triangle formula first. cheers :)

Answer: Option B

Explanation:

Let breadth =x metres.
Then length =(115x/100)metres.
$$\begin{aligned} =x*\frac{115x}{100}= 460\\ x^2=(460 x 100/115) \\ x^2=400 \\ x= 20 \\ \end{aligned}$$

Answer: Option D

Explanation:

Let original length = x
and original width = y
Decrease in area will be
$$\begin{aligned} = xy-\left( \frac{80x}{100}\times\frac{90y}{100}\right) \\ = \left(xy- \frac{18}{25}xy\right) \\ = \frac{7}{25}xy \\ \text{Decrease = }\left(\frac{7xy}{25xy} \times100\right) \% \\ = 28\% \end{aligned}$$

Answer: Option A

Explanation:

Let the triangle and parallelogram have common base b,
let the Altitude of triangle is h1 and of parallelogram is h2(which is equal to 100 m), then
$$\begin{aligned} \text{Area of triangle =}\frac{1}{2}*b*h1\\ \text{Area of rectangle =}b*h2\\ \text{As per question }\\ \frac{1}{2}*b*h1 = b*h2 \\ \frac{1}{2}*b*h1 = b*100 \\ h1 = 100*2 = 200 m \\ \end{aligned}$$

Answer: Option A

Explanation:

$$\begin{aligned} \text{Ratio of sides =}\frac{1}{2}:\frac{1}{3}:\frac{1}{4} \\ =6:4:3\\ Perimeter = 52 cm \\ \text{So sides are =} \\ \left( 52*\frac{6}{13}\right)cm,\left( 52*\frac{4}{13}\right)cm, \left( 52*\frac{3}{13}\right)cm \end{aligned}$$
a = 24 cm, b = 16 cm and c = 12 cm
Length of the smallest side = 12 cm