Question

Ravi made an error of \(5 \%\) in excess while measuring the length of rectangle and an error of \(8 \%\) deficit was made while measuring the breadth. What is the percentage error in the area? (a) \(-3 \%\) (b) \(-40 \%\) (c) \(-3.4 \%\) (d) can't be determined

Short answer

a) 4% b) -4% c) -3.4% d) 3.4% Answer: (c) -3.4%

Step-by-step solution

Calculate errors in length and breadth

Let the true length be \(L\) and true breadth be \(B\). Then Ravi measures the length as \(L + 5\%L = 1.05L\) and breadth as \(B - 8\%B = 0.92B\).

Calculate the measured area

Now, we can calculate the area of the rectangle as measured by Ravi. The measured area would be equal to the product of the measured length and breadth: \((1.05L)(0.92B) = 0.966LB\).

Calculate the true area

The true area of the rectangle would be the product of the true length and breadth, which is equal to \(LB\).

Calculate the percentage error in the area

To find the percentage error in the area, we can compare the measured area with the true area. The percentage error would be: $$\frac{\text{Measured Area - True Area}}{\text{True Area}} \times 100 = \frac{(0.966LB - LB)}{LB} \times 100$$ This simplifies to:$$\frac{-0.034LB}{LB} \times 100 = -3.4 \%$$

Compare the calculated error with the given options

We calculated the percentage error in the area to be \(-3.4\%\). Comparing this with the given options, we see that it matches option (c). So, the correct answer is (c) \(-3.4 \%\).

Key concepts

Error Calculation

Error calculation is crucial when dealing with measurements, whether in a lab or solving math problems. In percentage error calculation, we determine how far our measured value is from the true value, expressed as a percentage. For example, if you estimate a measurement higher or lower, the percentage error tells you by what percentage you're off.

To calculate error, you subtract the true value from the estimated or measured value, then divide the result by the true value. Finally, multiply by 100 to convert to a percentage. This formula is: \\[\text{Percentage Error} = \frac{(\text{Measured Value} - \text{True Value})}{\text{True Value}} \times 100\%\]

Understanding errors helps make informed decisions about the reliability of your measurements or calculations. Whether you're slightly off or way off, knowing your percentage error is essential for assessing the accuracy of your work.

Measurement Error

Measurement error occurs when there is a discrepancy between a true or actual measurement and the measurement obtained using a particular method or tool. In the problem with Ravi, there are two kinds of errors: an excess in measuring length and a deficit in measuring breadth.

• An **excess error** implies that the measured value is greater than the actual value, in this case, the length is measured as 105% of the actual length.
• A **deficit error** means the measured value is less than the actual value, where the breadth is only 92% of the actual breadth.

Measurement errors can be systematic, meaning they consistently occur in one direction, or random, appearing inconsistently. Recognizing the kind of error is crucial to improving measurement techniques for future accuracy. Understanding and correcting measurement errors help ensure that mathematical and scientific analyses based on these measurements are reliable.

Mathematics Problem Solving

Mathematics problem-solving involves a methodical approach to find solutions, commonly using logical steps and reasoning. In Ravi's scenario, there is a practical application in handling measurement errors.

Steps in solving the problem include:

• Identifying the problem: Recognize the types of errors in the measurements. Here, a 5% excess and an 8% deficit were noted.
• Applying mathematical operations: Use multiplication to find the area with erroneous measurements and with true measurements.
• Using formulas: Apply the percentage error formula to find the overall discrepancy in the area measurement.
• Interpreting results: Understand how the combined impact of length and breadth errors affects the area calculation.

By following these steps, problems become clearer, and solutions are easier to achieve. Practice in solving such problems enhances mathematical dexterity, allowing you to tackle more complex situations in the future.