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Chapter 5: Problem 11

The actual area of a rectangle is \(60 \mathrm{~cm}^{2}\), but while measuring its length a student decreases it by \(20 \%\) and the breadth increases by \(25 \%\). The percentage error in area, calculated by the student is : (a) \(5 \%\) (b) \(25 \%\) (c) can't be determined (d) none of these

Short Answer

Expert verified
Answer: (d) none of these

Step by step solution

01

Find the actual length and breadth

Let the actual length and breadth be \(l\) and \(b\), respectively. We are given that the actual area is \(60 \mathrm{~cm}^{2}\): \(A = lb = 60 \mathrm{~cm}^{2}\) We don't have sufficient information to find the exact values of \(l\) and \(b\). However, since we only need the percentage error in the calculated area, we don't need the exact values.
02

Apply the percentage errors in length and breadth

The student decreases the length by \(20\%\): \(l' = 80\% \cdot l = 0.8l\) And increases the breadth by \(25\%\): \(b' = 125\% \cdot b = 1.25b\)
03

Calculate the area with the corrected length and breadth

The student calculates the area using the corrected length and breadth: \(A' = l' \cdot b' = 0.8l \cdot 1.25b = (0.8)(1.25)lb = 1.0 \cdot lb\)
04

Calculate the percentage error in area

We can see that the calculated area is equal to the actual area (\(A' = A\)), meaning that the student's percentage error in area calculations is \(0\%\). However, this option is not available in the given choices, so the answer is: (d) none of these

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area Calculation
Calculating the area of a rectangle is a straightforward process. The area is the space contained within the rectangle's boundaries. It is calculated as the product of its length and breadth (width). The formula is:
  • Area = Length 脳 Breadth
For our rectangle, it's given that this area is exactly 60 cm虏. This means the product of its length (let's call it 'l') and its breadth ('b') equals 60. Knowing this helps us understand any changes or errors in measuring the rectangle.
Rectangle Dimensions
The dimensions of a rectangle are its length and breadth. These measurements define its shape and area.
When changes occur in these dimensions, it affects the area proportionally. In our exercise, the length is decreased by 20%, and the breadth is increased by 25%. To understand the changes, we apply the specific percentage adjustments:
  • New length = 80% of the original length = 0.8l
  • New breadth = 125% of the original breadth = 1.25b
Even though the dimensions changed, the new area calculation shows that the effects balanced out, maintaining the original area.
Percentage Change
A percentage change in measurement refers to how much a quantity has increased or decreased in percentage terms. Here, the length decreased by 20%, meaning it became 80% of what it was before. In mathematical terms:
  • Decreased Value = Original Value 脳 (1 - 0.20)
Similarly, the breadth increase of 25% means it grew to 125% of its original size:
  • Increased Value = Original Value 脳 (1 + 0.25)
These changes usually alter the total area, but in this unique case, they counterbalance each other.
Mathematical Problem Solving
Solving mathematical problems often involves understanding concepts and applying them logically. With the given problem, we first set up what we know: the original area, the changes in dimensions, and what those mean. By substituting changes into our formula for area calculation, we discover that the new area is exactly the original area:
  • Calculated Area = New Length 脳 New Breadth = 0.8l 脳 1.25b = 1.0 脳 lb = 60 cm虏
This shows that the percentage error is indeed 0%, which isn't an option in the provided choices. Careful problem-solving unveils how changes can offset each other perfectly, a valuable lesson in analytical thinking.

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