Question

The length and breadth of the rectangular field given in a ratio as 11:4. If the cost of fencing is Rs.100 per meter and it cost us Rs.75000 to fence the rectangular field then find the actual length and breadth of the field.

Answer 1

Hint: We are given that, length to breadth ratio is 11:4, so we consider firstly that, common term cancel is x. So we get a length of 11x and breadth as 4x. Then, we know total cost is given as \[\text{Total cost}=\text{cost per meter}\times \text{perimeter}\]. So from here, we find perimeter, then for rectangle, we have perimeter formula as $2\left( L+B \right)$ so we use this to find the value of x, as we have x we will now find 11x to get the length and solve 4x to get the breadth.

Complete step by step answer:
We are given a rectangular field whose length to breadth ratio is given to us as 11.4. We know that whenever we take ratio we cut the greatest common divisor of both the terms. As we also have the length to breadth ratio given to us. So there must be some common term divided by both length and breadth.
Let x be the term got canceled from length and breadth.
Now, as we have \[\text{length}:\text{breath}=\text{11}:\text{4}\] so,
The actual length is 11x while the actual breadth is 4x.
Now, we have that total cost of fencing is 75000.
We know the total cost of fencing is given as the product of cost per minute and length of total fencing.
As we know, total length of fencing is given as perimeter. So,
\[\text{Total cost}=\text{cost per meter}\times \text{perimeter}\]
As total cost is 75000 and cost per meter of fencing is 100. So, we get:
\[\Rightarrow 75000=100\times \text{perimeter}\]
Solving further we get:
\[\Rightarrow \text{Perimeter}=\dfrac{75000}{100}\]
Hence, we get perimeter as 750 meter.
Now, we know for rectangle, perimeter is given as twice as the sum of length and breadth, that means,
\[\text{Perimeter of rectangle}=2\left( L+B \right)\]
As we have length = 11x and breadth = 4x. Also, perimeter is 750. So, we get:
\[750=2\left( 11x+4x \right)\]
Now, simplifying we get:
\[750=2\left( 15x \right)\]
Now solving further we get:
\[750=30x\]
Solving for x, we get:
\[\begin{align}
  & x=\dfrac{750}{30} \\
 & \therefore x=25 \\
\end{align}\]

So we get x as 25.
Means common term canceled from length and breadth is x = 25.
Now, as we have length = 11x. So, by putting x = 25, we get:
\[\text{Length}=11\times 25=275m\]
And as breadth we have 4x. So, by putting x = 25, we get:
\[\text{Breadth}=4\times 25=100m\]
So, we get length of rectangular field as 275 and breadth as 100m.

Note:
 Some mistakes like $2\left( L+B \right)=2L+B$ may happen, whenever we have terms in a bracket that term outside bracket with each term of brackets we have ratio L:B as 11:4 so, it doesn't mean length is 11 and breadth is 4.
Also, when we add two variables, the only the coefficient is added up and not the power that is $\left( 11x+4x \right)\ne 15{{x}^{2}}$ it is simplified as $11x+4x=15x$.