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Chapter 2: Problem 49

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

Short Answer

Expert verified
Two conversion factors can be written because they show reciprocal relationships: meters per centimeter and centimeters per meter.

Step by step solution

01

Understand the Equality

The equality given is: 1 meter (m) = 100 centimeters (cm). This means that 1 meter is exactly equal to 100 centimeters.
02

Definition of Conversion Factor

A conversion factor is a ratio or fraction that represents the relationship between two different units. It equals 1 because it represents the same quantity in different units.
03

Write the First Conversion Factor

Using the equality, the first conversion factor can be written as \ \( \frac{1 \text{ meter}}{100 \text{ centimeters}} \). This shows the number of meters per centimeter.
04

Write the Second Conversion Factor

The second conversion factor can be written as \ \( \frac{100 \text{ centimeters}}{1 \text{ meter}} \). This shows the number of centimeters per meter.
05

Reason for Two Possible Conversion Factors

Both conversions represent the same relationship (equality) but in reciprocal forms. One can convert from meters to centimeters, and the other can convert from centimeters to meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Unit conversion involves changing the measurement of a quantity from one unit to another. It is an essential skill in science and everyday life. When we convert units, we use conversion factors. Conversion factors are ratios that express how many of one unit are equal to another unit.

For example, in the equality given in the exercise, we know that 1 meter (m) is exactly equal to 100 centimeters (cm). This allows us to write two conversion factors:

- \( \frac{1 \text{ meter}}{100 \text{ centimeters}} \)
- \( \frac{100 \text{ centimeters}}{1 \text{ meter}} \)

These factors are used to convert between meters and centimeters. By multiplying any given quantity by the appropriate conversion factor, you ensure the units change correctly while maintaining the value of the quantity.
Metric System
The metric system is an international system of measurement used in most countries around the world. It is based on powers of ten, making it easy to convert between units.

The basic units include meters for length, grams for mass, and liters for volume. Prefixes like kilo-, centi-, and milli- help define the size of the unit.

In the exercise, we use meters and centimeters, where:

- 1 meter (m) = 100 centimeters (cm)

Knowing metric relationships makes unit conversion straightforward and eliminates confusion, as the factors of ten are consistent across different quantities.
Ratios and Proportions
Ratios and proportions play a crucial role in unit conversion. A ratio is a comparison between two numbers, and a proportion is an equation that states two ratios are equal.

In the provided solution, both conversion factors \( \frac{1 \text{ meter}}{100 \text{ centimeters}} \) and \( \frac{100 \text{ centimeters}}{1 \text{ meter}} \) are examples of ratios.

These ratios help us set up proportions to convert between units. For example, if you want to convert 3 meters to centimeters, you would set up the proportion:

\[ \frac{1 \text{ meter}}{100 \text{ centimeters}} = \frac{3 \text{ meters}}{X \text{ centimeters}} \]
By solving for X, you find the value in centimeters. Practicing ratios and proportions will make unit conversion more intuitive and effortless.

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Most popular questions from this chapter

Give the abbreviation for each of the following: a. gram b. liter c. degree Celsius d. pound e. second

Solve each of the following problems: a. A urine sample has a density of \(1.030 \mathrm{~g} / \mathrm{mL}\). What is the specific gravity of the sample? b. A \(20.0-\mathrm{mL}\) sample of a glucose IV solution that has a mass of \(20.6 \mathrm{~g}\). What is the density of the glucose solution? c. The specific gravity of a vegetable oil is \(0.92\). What is the mass, in grams, of \(750 \mathrm{~mL}\) of vegetable oil? d. A bottle containing \(325 \mathrm{~g}\) of cleaning solution is used to clean hospital equipment. If the cleaning solution has a specific gravity of \(0.850\), what volume, in milliliters, of solution was used?

On Greg's last visit to his doctor, he complained of feeling tired. His doctor orders a blood test for iron. Sandra, the registered nurse, does a venipuncture and withdraws \(8.0 \mathrm{~mL}\) of blood. About \(70 \%\) of the iron in the body is used to form hemoglobin, which is a protein in the red blood cells that carries oxygen to the cells of the body. About \(30 \%\) is stored in ferritin, bone marrow, and the liver. When the iron level is low, a person may have fatigue and decreased immunity. The normal range for serum iron in men is 80 to \(160 \mathrm{mcg} / \mathrm{dL}\). Greg's iron test showed a blood serum iron level of \(42 \mathrm{mcg} / \mathrm{dL}\), which indicates that Greg has iron deficiency anemia. His doctor orders an iron supplement to be taken twice daily. One tablet of the iron supplement contains \(65 \mathrm{mg}\) of iron. a. Write an equality and two conversion factors for Greg's serum iron level. b. How many micrograms of iron were in the \(8.0-\mathrm{mL}\) sample of Greg's blood?

Write the complete name for each of the following units: a. cL b. \(\mathrm{kg}\) c. ms d. Gm

Solve each of the following problems: a. A glucose solution has a density of \(1.02 \mathrm{~g} / \mathrm{mL}\). What is its specific gravity? b. A \(0.200-\mathrm{mL}\) sample of high-density lipoprotein (HDL) has a mass of \(0.230 \mathrm{~g}\). What is the density of the HDL? c. Butter has a specific gravity of \(0.86 .\) What is the mass, in grams, of \(2.15 \mathrm{~L}\) of butter? d. A \(5.000-\mathrm{mL}\) urine sample has a mass of \(5.025 \mathrm{~g}\). If the normal range for the specific gravity of urine is \(1.003\) to \(1.030\), would the specific gravity of this urine sample indicate that the patient could have type 2 diabetes?
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