Limits of accuracy are essential in mathematics, particularly when dealing with rounded data. This concept helps us understand the range within which the true value of a number lies, considering it has been rounded to a specific accuracy. Grasping this concept is crucial for precise calculations and assessments in various fields of study.

Introduction to Limits of Accuracy

When numbers are rounded to a specified accuracy, we lose the exact value of the number. However, by determining the upper and lower bounds of this number, we can find the range within which the true value lies.

Key Concepts

• Rounding: Adjusting numbers to make them simpler, typically to a specified level of accuracy (whole numbers, decimal places, significant figures).
• Upper Bound: Maximum possible value before rounding.
• Lower Bound: Minimum possible value before rounding.

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Calculating Upper and Lower Bounds

The method to calculate bounds depends on how the number was rounded: to the nearest whole number, decimal place, or significant figure.

Whole Numbers

• Upper Bound: Add 0.5.
• Lower Bound: Subtract 0.5.

Decimal Places

• Upper Bound: Add 0.5 of the next smallest decimal place.
• Lower Bound: Subtract 0.5 of the next smallest decimal place.

Significant Figures

• The approach depends on the significant figure's place value and requires a nuanced understanding.

Impact of Rounding on Calculations

Rounding affects calculations, especially in multi-step problems and statistical analysis, potentially leading to significant errors or variations.

Worked Examples

Example 1: Whole Number Rounding

For a number rounded to 24, calculate the upper and lower bounds.

Solution

• Upper Bound: $24 + 0.5 = 24.5$
• Lower Bound: $24 - 0.5 = 23.5$

This indicates the true value is between 23.5 and 24.5.

Example 2: Decimal Place Rounding

For a number rounded to 3.6 (one decimal place), find the bounds.

Solution

• Upper Bound: $3.6 + 0.05 = 3.65$
• Lower Bound: $3.6 - 0.05 = 3.55$

The actual value lies between 3.55 and 3.65.

Example 3: Rounding Impact on Circle Area

Examine the effect of rounding a circle's radius to 5cm on calculating its area.

Radius Information:

• Rounded: 5 cm
• Actual Bounds: 4.5 cm (lower) to 5.5 cm (upper)

Area Calculations:

1. Using Rounded Value:

2. Lower Bound:

3. Upper Bound:

The variance in areas highlights the impact of rounding on accuracy.

Practice Questions

1. Length Rounded to 150 mm (nearest 10 mm):

Question: Find upper and lower bounds.

Solution:

Length Bounds:

• Upper: $150 + 5 = 155 \, \text{mm}$
• Lower: $150 - 5 = 145 \, \text{mm}$

2. Mass Rounded to 2.5 kg (nearest 0.1 kg):

Question: Determine the actual mass range.

Solution:

Mass Bounds:

• Upper: $2.5 + 0.05 = 2.55 \, \text{kg}$
• Lower: $2.5 - 0.05 = 2.45 \, \text{kg}$

3. Effect of Rounding a Square's Side to 7 cm on Area:

Question: Analyze the impact.

Solution:

Square Area Analysis:

• Rounded Side: 7 cm
• Area with Rounded Side: $A = 7^2 = 49 \, \text{cm}^2$
• Bounds: ($6.95 \, \text{cm}$ (lower) and $7.05 \, \text{cm}$ (upper)
• Area Using Bounds:
  • Lower Bound Area: $6.95^2 \approx 48.30 \, \text{cm}^2$
  • Upper Bound Area: $7.05^2 \approx 49.70 \, \text{cm}^2$