Approximations and Accuracy

Grade 12 Math - Approximations and Accuracy

Lesson Objectives

Define and identify significant figures in numbers.
Round numbers to a specified number of significant figures.
Calculate percentage error in measurements and approximations.
Apply concepts of accuracy and error in real-life contexts.
Differentiate between absolute error and percentage error.

Lesson Introduction

In daily life and scientific measurements, we often need to round numbers for simplicity and estimate values within a reasonable margin of accuracy. This is where concepts like significant figures and percentage error become essential. For instance, when measuring the length of a table, we might not always get an exact value, and knowing how close our value is to the actual measurement is critical.

Core Lesson Content

1. Significant Figures

Significant figures are the digits in a number that carry meaningful contributions to its accuracy. They include all non-zero digits, any zeros between them, and trailing zeros in a decimal number.

2. Rules for Counting Significant Figures

All non-zero digits are significant.
Zeros between non-zero digits are significant.
Leading zeros are not significant.
Trailing zeros in a decimal number are significant.

3. Percentage Error

Percentage error is used to compare the difference between an approximate (measured or calculated) value and an exact (true) value. It is given by:

$\text{Percentage Error} = \left( \frac{|\text{Measured Value} - \text{Actual Value}|}{\text{Actual Value}} \right) \times 100\%$

Worked Examples

Example 1: Write 0.004509 to 3 significant figures.

\text{Answer: } 0.00451

Example 2: Write 13870 to 2 significant figures.

\text{Answer: } 1.4 \times 10^4

Example 3: Round 9.87654 to 4 significant figures.

\text{Answer: } 9.877

Example 4: Find the percentage error if the measured length is 48 cm and the actual length is 50 cm.

\text{Percentage Error} = \left( \frac{|48 - 50|}{50} \right) \times 100 = 4\%

Example 5: Round 0.009678 to 2 significant figures.

\text{Answer: } 0.0097

Example 6: The actual value is 100 m, but a student measured it as 92 m. Find the percentage error.

\text{Percentage Error} = \left( \frac{8}{100} \right) \times 100 = 8\%

Example 7: Express 57000 in 2 significant figures in standard form.

\text{Answer: } 5.7 \times 10^4

Example 8: A rod is measured as 7.5 cm but its actual length is 7.52 cm. Find the percentage error.

\text{Percentage Error} = \left( \frac{0.02}{7.52} \right) \times 100 \approx 0.27\%

Example 9: Find the number of significant figures in the number 0.006070.

\text{Answer: 4 significant figures}

Example 10: Round 5462 to 3 significant figures.

\text{Answer: } 5460

Exercises

Round 0.00045678 to 2 significant figures. $\text{Answer: } ?$
Find the percentage error if the measured value is 85 and the actual value is 100. $\text{Answer: } ?$
Write 203000 in 3 significant figures using standard form. $\text{Answer: } ?$
How many significant figures are in 0.07080?
[NECO] Round 123.456 to 4 significant figures. (Past Question)
[WAEC] A boy measured the length of a stick to be 120 cm. The actual length was 118 cm. What is the percentage error? (Past Question)
Write 0.0003895 to 3 significant figures.
[WAEC] Find the number of significant figures in the result of the calculation: $2.75 \times 4.0$ . (Past Question)
[JAMB] The actual value of a mass is 50 kg. If measured as 48.5 kg, what is the percentage error? (Past Question)
Round 96.387 to 3 significant figures.

Conclusion/Recap

In this lesson, you learned how to identify and round numbers to a required number of significant figures and calculate percentage error to understand measurement accuracy. These skills are essential in scientific and real-life estimations. In the next lesson, we will focus on Standard Form and Scientific Notation, where you will learn how to express very large or very small numbers efficiently.