Question

Carry out the following conversions: (a) \(22.6 \mathrm{~m}\) to decimeters, (b) \(25.4 \mathrm{mg}\) to kilograms, (c) \(556 \mathrm{~mL}\) to liters, (d) \(10.6 \mathrm{~kg} / \mathrm{m}^{3}\) to \(\mathrm{g} / \mathrm{cm}^{3}\).

Short answer

a) 226 dm, b) 0.0000254 kg, c) 0.556 L, d) 0.0106 g/cm³.

Step-by-step solution

Converting Meters to Decimeters

To convert from meters to decimeters, we need to know that 1 meter is equal to 10 decimeters. Therefore, we multiply the given value in meters by 10 to get the value in decimeters.Given: \(22.6 \text{ meters}\)Calculation: \[ 22.6 \text{ m} \times 10 = 226 \text{ dm} \]

Converting Milligrams to Kilograms

To convert from milligrams to kilograms, we note that there are 1,000,000 milligrams in a kilogram. Thus, we divide the value in milligrams by 1,000,000.Given: \(25.4 \text{ mg}\)Calculation: \[ \frac{25.4 \text{ mg}}{1,000,000} = 0.0000254 \text{ kg} \]

Converting Milliliters to Liters

To convert from milliliters to liters, we use the fact that 1 liter is 1,000 milliliters. Therefore, we divide the milliliter value by 1,000.Given: \(556 \text{ mL}\)Calculation: \[ \frac{556 \text{ mL}}{1,000} = 0.556 \text{ L} \]

Converting Kilograms per Cubic Meter to Grams per Cubic Centimeter

To convert from kg/m³ to g/cm³, we use the conversion that 1 kg/m³ equals 0.001 g/cm³.Given: \(10.6 \text{ kg/m}^3\)Calculation: \[ 10.6 \text{ kg/m}^3 \times 0.001 = 0.0106 \text{ g/cm}^3 \]

Key concepts

Metric System

The metric system is a universal decimal measurement system used worldwide. It is simple, easy to apply, and based on multiples of ten. The base units include the meter for length, the kilogram for mass, and the liter for volume. These units make calculations straightforward:

• Length is measured in meters, which can be converted into kilometers, centimeters, and millimeters.
• Mass is measured in kilograms, easily convertible into grams, milligrams, or metric tons.
• Volume is generally measured in liters, convertible into milliliters and cubic meters.

By knowing these base units, students can perform unit conversions smoothly in the metric system.

Measurement Conversion

Measurement conversion involves changing one unit to another within the same system. In the original exercise, we see conversions between units like meters to decimeters and milligrams to kilograms. Understanding the relationship is key:

• Meters to Decimeters: Since 1 meter equals 10 decimeters, multiply meters by 10 to get decimeters.
• Milligrams to Kilograms: Recognize that 1 kilogram equals 1,000,000 milligrams. Hence, divide milligrams by 1,000,000.
• Milliliters to Liters: 1 liter equals 1,000 milliliters, so divide milliliters by 1,000.
• Kilograms/Meters³ to Grams/Centimeters³: Use the conversion factor 1 kg/m³ equals 0.001 g/cm³, multiplying the original value appropriately.

These conversion rules are consistent across various measurements, simplifying complex calculations.

Dimensional Analysis

Dimensional analysis is a technique used to ensure that calculations yield the correct units. It involves the systematic conversion between units using conversion factors. In solving measurement conversion problems, dimensional analysis helps confirm the correctness of unit transformations. To employ dimensional analysis, follow these steps:

• Identify the units you have and the units you want to obtain.
• Determine the conversion factor(s) needed to link the two units.
• Apply the conversion factor by multiplying or dividing, according to the relationship.
• Ensure units cancel appropriately, leaving only the desired units.

By consistently applying dimensional analysis, students reduce errors and deepen their understanding of unit conversions.