Question

Why can two conversion factors be written for the equality \(1 \mathrm{~m}=100 \mathrm{~cm} ?\)

Short answer

Two conversion factors exist to convert between the units—one for meters to centimeters and one for centimeters to meters.

Step-by-step solution

Understanding the Equality

First, recognize that the equality given is \(1 \mathrm{~m} = 100 \mathrm{~cm}\). This indicates that 1 meter is equivalent to 100 centimeters.

Defining Conversion Factors

A conversion factor is a ratio or fraction that represents the relationship between two different units. It is used to convert a quantity from one unit to another without changing its value.

Writing the First Conversion Factor

From the given equality, the first conversion factor can be written as \(\frac{1 \text{ meter}}{100 \text{ centimeters}} \). This means 1 meter per 100 centimeters.

Writing the Second Conversion Factor

The second conversion factor can be written as \(\frac{100 \text{ centimeters}}{1 \text{ meter}} \). This means 100 centimeters per 1 meter.

Understanding Why Two Factors Exist

Both fractions correctly represent the relationship between meters and centimeters. One is used to convert meters to centimeters, and the other to convert centimeters to meters. Hence, two conversion factors can be written for this equality.

Key concepts

Unit Conversion

Unit conversion is essential when dealing with measurements. It helps translate one type of measurement unit into another without changing the quantity's value. For example, converting meters to centimeters involves using a specific relationship between the units. To understand unit conversions, you should know the conversion factor, a fractional ratio that expresses how units relate. If you have 1 meter, and you need to know how many centimeters that is, you use the conversion factor \( 1 \text{ meter} = 100 \text{ centimeters} \). Hence, the conversion factor can be written as \( \frac{1 \text{ meter}}{100 \text{ centimeters}} \) or \( \frac{100 \text{ centimeters}}{1 \text{ meter}} \). This makes it easier to go from one unit system to another, and you can use either factor depending on your need to go from meters to centimeters or vice versa.

Metric System

The metric system is a decimal-based system of measurement. It is used globally due to its simplicity and ease of conversion. The basic units in the metric system include meters for length, grams for weight, and liters for volume. These units can be scaled by powers of ten.
For instance, when converting between metric units for length:

• 1 meter (m) = 100 centimeters (cm)
• 1 meter = 1,000 millimeters (mm)
• 1 kilometer (km) = 1,000 meters

These relationships make conversions straightforward. Using conversion factors like \( \frac{1 \text{ meter}}{100 \text{ centimeters}} \) and vice versa ensures accurate exchanges between the units.

Dimensional Analysis

Dimensional analysis is a powerful mathematical technique used to convert units from one system to another. It involves the use of conversion factors to keep track of units during calculations. This method ensures accuracy and consistency in measurement conversions.
When you have an equality such as \( 1 \text{ meter} = 100 \text{ centimeters} \), it means that both sides of the equation represent the same quantity but in different units. Dimensional analysis allows you to multiply or divide by conversion factors to cross-check the units. For example, converting 2 meters to centimeters involves multiplying by the conversion factor: \[ 2 \text{ meters} \times \frac{100 \text{ centimeters}}{1 \text{ meter}} = 200 \text{ centimeters} \]
Such step-by-step conversions help in maintaining unit consistency and ensuring the accuracy of measurements.