Question

There are two rectangular fields of same area. The length of first rectangular field is \(x \%\) less than the length of the second field and breadth of the first field is \((5 x) \%\) greater than the breadth of the second field. What is the value of \(x\) ? (a) 15 (b) 25 (c) 50 (d) 80

Short answer

Answer: The value of x is 40.

Step-by-step solution

Define the Variables and Write the Equations

Let's define the variables for the two rectangular fields: - Let the length of the second field be L₂ and the breadth be B₂. - Let the length of the first field be L₁ = L₂ - xL₂/100 = L₂ (1 - x/100) and breadth be B₁ = B₂ + 5xB₂/100 = B₂ (1 + 5x/100). Now, we know the areas of both fields are equal, so: Area of 1st field = Area of 2nd field L₁ × B₁ = L₂ × B₂

Substitute the Variables in the Equation

Now, substitute L₁ and B₁ in the equation L₁ × B₁ = L₂ × B₂: (L₂ (1 - x/100)) * (B₂ (1 + 5x/100)) = L₂ * B₂

Simplify and Solve for x

Now, cancel out L₂B₂ on both sides of the equation: (1 - x/100) * (1 + 5x/100) = 1 Expanding the equation: 1 - x/100 + 5x/100 - 5x²/10000 = 1 Now, subtract 1 from both sides: - x/100 + 5x/100 - 5x²/10000 = 0 Combine the x terms: 4x/100 - 5x²/10000 = 0 Divide the equation by 4/100 to simplify further x - 25x²/1000 = 0 Now, factor x out of the equation: x(1 - 25x/1000) = 0 This equation has two possible solutions: x = 0 or 1 - 25x/1000 = 0. Since x=0 does not provide any change in the lengths and breadths of the fields, we consider the other solution: 1 - 25x/1000 = 0 25x/1000 = 1 x = 1000/25 = 40 So, the value of x is not in the given options. Neither of the options provided in the exercise is correct. The correct answer is x=40.

Key concepts

Percentage Decrease

Understanding percentage decrease is crucial in many real-life applications, including comparing changes in dimensions, prices, or any measurable quantity. It represents the ratio of the amount of decrease to the original amount, expressed as a percent. To calculate it, subtract the new value from the original value, then divide by the original value and multiply by 100 to get a percentage.

For example, if a rectangular field's length is decreased by a certain percentage, the new length can be calculated as follows: let the original length be represented by the variable L. If the length decreases by x%, then the new length L1 is given by the equation L1 = L - (x/100)L = L(1 - x/100). The term (x/100)L corresponds to the amount the original length L has been reduced by.

Percentage Increase

Percentage increase is a common term used when describing growth or expansion in size, quantity, or value. It reflects how much something has grown relative to its original size and is expressed as a percentage. To calculate a percentage increase, find the difference between the new and the original values, divide it by the original value, and then multiply the result by 100.

So, when the breadth of a rectangle is said to be (5x)% greater than that of another, it means this breadth has increased. If the original breadth is B, then the new breadth B1 is B1 = B + (5x/100)B = B(1 + 5x/100). Here, (5x/100)B is the added amount to the original breadth, determined by the 5x% increase.

Area of Rectangle

The area of a rectangle is a foundational concept in geometry, calculated by multiplying the length by the breadth of the rectangle (Area = Length × Breadth). It's a representation of the amount of two-dimensional space inside the rectangle's boundaries.

When comparing two rectangles with equal areas, any differences in length and breadth must compensate for each other to maintain the same area. For instance, if the length of one rectangle is shortened, its breadth must increase proportionally to keep the area unchanged. This is precisely the premise of the exercise at hand where the equal areas of two rectangles have varying dimensions.

Algebraic Equation

Algebraic equations are mathematical statements that demonstrate the equality between two expressions. They consist of variables and constants and are solved by finding the values of the variables that make the equation true. These are fundamental in various mathematical contexts, including geometry, where they are used to express relationships between the dimensions of shapes and their properties.

In our context involving rectangular fields, setting up an algebraic equation helps us to represent the relationship between the lengths and breadths of two rectangles with equal areas. By solving the algebraic equation, which in this case involves both percentage decrease and percentage increase, we can find the unknown percentage x that is being sought in the exercise.