Question

Which of the following are exact equivalent relationships? (a) \(1 \mathrm{~m}=10 \mathrm{dm}\) (b) \(1 \mathrm{lb}=454 \mathrm{~g}\) (c) \(1 \mathrm{~L}=1000 \mathrm{~mL}\) (d) \(1 \mathrm{qt}=946 \mathrm{~mL}\)

Short answer

(a) and (c) are exact equivalent relationships.

Step-by-step solution

Understanding Exact Equivalence

Exact equivalence means two values are precisely equal with no rounding or approximation involved. This typically occurs when the given units belong to the same system of measurement.

Convert and Compare Each Relationship

Let's analyze each given relationship to determine if they are exact: (a) 1 meter is exactly 10 decimeters because they belong to the metric system, where 1 meter is defined as 10 decimeters. (b) 1 pound is an approximation to 454 grams because the exact value is defined as 453.59237 grams. (c) 1 liter is exactly 1000 milliliters by definition in the metric system. (d) 1 quart approximates 946 milliliters because it actually equals 946.352946 milliliters.

Determine Exact Equivalences

Based on the definitions and analysis: - (a) 1 m = 10 dm is exact. - (b) 1 lb = 454 g is not exact. - (c) 1 L = 1000 mL is exact. - (d) 1 qt = 946 mL is not exact.

Key concepts

Measurement Systems

Measurement systems are frameworks used to quantify physical quantities such as length, mass, and volume. They help us understand and communicate the size, weight, or capacity of objects in a standardized way. There are two main measurement systems in use today:

• The Metric System: Predominantly used worldwide, it is based on powers of ten, which makes it extremely convenient for conversions.
• The Imperial System: Mostly used in the United States, it includes units like pounds, inches, and gallon, which might not always be intuitive for conversions.

Each measurement system has its own set of units and defined relationships between these units. Understanding which system you're dealing with is crucial, as it affects how you perform calculations and conversions.

Metric System

The metric system is a decimal-based system of measurement. It was developed in France during the late 18th century and has since spread as the preferred system of measurement globally. Here are some fundamental units within the metric system:

• Length: Meter (m), with subunits such as centimeters (cm) and millimeters (mm).
• Mass: Kilogram (kg), with subunits like grams (g).
• Volume: Liter (L), with smaller units like milliliters (mL).

Characteristics of the metric system include uniformity and scalability since each unit is a power of ten. This feature makes conversions straightforward, ensuring exact equivalents within the system, such as 1 meter = 10 decimeters or 1 liter = 1000 milliliters. As a globally recognized measurement framework, it simplifies scientific calculations and international collaboration.

Unit Conversion

Unit conversion helps in expressing a physical quantity in different units, often across different systems of measurement. This is essential when comparing measurements from different systems or when a specific unit is preferred for a detailed analysis. Unit conversion typically involves:

• Identifying the current unit and the desired unit.
• Using conversion factors, which are ratios that express the relationship between two units.
• Adjusting the value based on these factors to arrive at an equivalent measurement.

For example, to convert 1 liter to milliliters, use the conversion factor that 1 liter = 1000 milliliters. Unit conversion can occasionally introduce approximations, especially when converting between systems, like pounds to grams. This is why not all conversions are exact, emphasizing the importance of understanding and carefully applying conversion factors.

Approximation in Measurements

Approximation in measurements occurs when a measurement is rounded to a more convenient number or when an exact conversion isn't possible. Approximation is often necessary when translating between different measurement systems, as these systems may define units differently. Approximations are common when dealing with conversions like:

• Pounds to grams: Although 1 lb is approximately 454 grams, the exact conversion gives you 453.59237 grams.
• Quarts to milliliters: Instead of 946 mL exactly, 1 quart actually equals approximately 946.352946 mL.

These approximations are particularly important in scenarios requiring precision, such as scientific experiments or engineering calculations, where the tiny differences could have significant effects. Understanding when a relationship involves approximation helps in choosing the right level of precision for your needs and ensuring results are reliable and accurate.