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The Metric System

Identify units of measurement in the metric system and their uses.

Learning Objectives

After completing this section, you should be able to:

Identify units of measurement in the metric system and their uses.
Order the six common prefixes of the metric system.
Convert between like unit values.

Units of Measurement in the Metric System

Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system:

The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch.
The metric unit for area is the square meter (m²). The area of a professional soccer field is 7,140 square meters (7,140 m²).
The metric base unit for volume is the cubic meter (m³). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles.
The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g).
The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C.

While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the relationship between smaller or larger units.

Determining the Correct Base Unit

Try it.

Which base unit would be used to express the following?

amount of water in a swimming pool
length of an electrical wire
weight of one serving of peanuts

Solution

Liquid volume is generally expressed in units of liters (L).
Length is measured in units of meters (m).
Weight is commonly expressed in units of grams (g).

While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Metric Prefixes

Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is \({10}^{-2}\), or 100 times smaller than a meter, while the kilometer is 10³, or 1,000 times larger than a meter.

The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The most used prefixes are shown in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died From Drinking Chocolate Milk.

Prefix	kilo–	hecto–	deca–	base unit	deci–	cent–	milli–
Abbreviation	\(\text{k}\)	\(\text{h}\)	\(\text{da}\)		\(\text{d}\)	\(\text{c}\)	\(\text{m}\)
Magnitude	\({10}^{3}\)	\({10}^{2}\)	\({10}^{1}\)	\({10}^{0}\text{or}1\)	\({10}^{-1}\)	\({10}^{-2}\)	\({10}^{-3}\)

Ordering the Magnitude of Units

Try it.

Order the measures from smallest unit to largest unit.

centimeter, millimeter, decimeter

Solution

Looking at the metric prefixes, we can see that the prefix order from smallest unit to largest unit is milli-, centi-, deci-, so the order of the units from smallest to largest is millimeter, centimeter, decimeter.

Determining Reasonable Values for Length

Try it.

What is a reasonable value for the length of a person’s thumb: 5 meters, 5 centimeters, or 5 millimeters?

Solution

Given that a meter is slightly longer than a yard, 5 meters is not a reasonable value for the length of a person’s thumb. Since a millimeter is 10 times smaller than a centimeter, which is approximately \(\frac{1}{2}\) inch, 5 millimeters is not a reasonable estimate for the length of a person’s thumb. The correct answer is 5 centimeters.

Converting Metric Units of Measure

Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is \({10}^{3}\)or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams.

The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the measure is \({10}^{-2}\text{or}\frac{1}{100}\) of a meter. One meter is equivalent to 100 centimeters.

A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number that when used with multiplication or division converts from one metric unit to another, both having the same base unit. In the metric system, these conversion factors are directly related to the powers of 10. The most common used conversion factors are shown in .

Converting Metric Distances Using Multiplication

Try it.

The firehouse is 13.45 km from the library. How many meters is it from the firehouse to the library?

Solution

When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from kilometer to the base unit of meters is 1,000.

\[13.45\times 1,000=13,450\]

So, the firehouse is 13,450 meters away from the firehouse.

Converting Metric Capacity Using Division

Try it.

How many liters is 3,565 milliliters?

Solution

When converting from a smaller unit to a larger unit, use division. The conversion factor from milliliter to the base unit of liters is 1,000.

\[3,565\div 1,000=3.565\]

So, 3,565 milliliters is 3.565 liters.

Converting Metric Units of Mass to Solve Problems

Try it.

Caroline and Aiyana are working on a chemistry experiment together and must perform calculations using measurements taken during the experiment. Due to miscommunication, Caroline took measurements in centigrams and Aiyana used milligrams. Convert Caroline’s measurement of 125 centigrams to milligrams.

Solution

When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from centigrams to the milligrams is 10.

\[125\times 10=1,250\]

So, the 125 centigrams are 1,250 milligrams.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

The metric system is decimal system of weights and measures based on base units of meter, liter, and gram. The system was first proposed in 1670 and has since been adopted as the International System of Units and used in nearly every country in the world.
Each successive unit on the metric scale is 10 times larger than the previous one. To convert between units with the same base unit, you must either multiply or divide by a power of 10.
The most used prefixes are listed in . An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died from Drinking Chocolate Milk.

Practice (9)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Which base unit would be used to express the following?
1. amount of water in a swimming pool
2. length of an electrical wire
3. weight of one serving of peanuts
Kuratidza mhinduro
1. Liquid volume is generally expressed in units of liters (L).
2. Length is measured in units of meters (m).
3. Weight is commonly expressed in units of grams (g).
Order the measures from smallest unit to largest unit.
centimeter, millimeter, decimeter

Kuratidza mhinduro

Looking at the metric prefixes, we can see that the prefix order from smallest unit to largest unit is milli-, centi-, deci-, so the order of the units from smallest to largest is millimeter, centimeter, decimeter.
What is a reasonable value for the length of a person’s thumb: 5 meters, 5 centimeters, or 5 millimeters?

Kuratidza mhinduro

Given that a meter is slightly longer than a yard, 5 meters is not a reasonable value for the length of a person’s thumb. Since a millimeter is 10 times smaller than a centimeter, which is approximately \(\frac{1}{2}\) inch, 5 millimeters is not a reasonable estimate for the length of a person’s thumb. The correct answer is 5 centimeters.
The firehouse is 13.45 km from the library. How many meters is it from the firehouse to the library?

Kuratidza mhinduro

When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from kilometer to the base unit of meters is 1,000.
\[13.45\times 1,000=13,450\]
So, the firehouse is 13,450 meters away from the firehouse.
How many liters is 3,565 milliliters?

Kuratidza mhinduro

When converting from a smaller unit to a larger unit, use division. The conversion factor from milliliter to the base unit of liters is 1,000.
\[3,565\div 1,000=3.565\]
So, 3,565 milliliters is 3.565 liters.
Caroline and Aiyana are working on a chemistry experiment together and must perform calculations using measurements taken during the experiment. Due to miscommunication, Caroline took measurements in centigrams and Aiyana used milligrams. Convert Caroline’s measurement of 125 centigrams to milligrams.

Kuratidza mhinduro

When converting from a larger unit to a smaller unit, use multiplication. The conversion factor from centigrams to the milligrams is 10.
\[125\times 10=1,250\]
So, the 125 centigrams are 1,250 milligrams.
A bottle contains 500 mL of juice. If the juice is packaged in 24-bottle cases, how many liters of juice does the case contain?

Kuratidza mhinduro

Step 1: Multiply the amount of juice in each bottle by the number of bottles.
\[500\text{mL}\times 24=12,000\text{mL}\]
Step 2: Divide by 1,000 to convert from milliliters to liters.
\[\frac{12,000\text{mL}}{1,000}=12L\]
So, there are 12 liters of juice in each case.
The nutrition label on a jar of spaghetti sauce indicates that one serving contains 410 mg of sodium. You have poured two servings over your favorite pasta before recalling your doctor’s advice about keeping your sodium consumption below 1 g per meal. Have you followed your doctor’s recommendation?

Kuratidza mhinduro

Step 1: Multiply the number of servings by the amount of sodium in each serving.
\[410\text{mg}\times 2=820\text{mg}\]
Step 2: Divide by 1,000 to convert from milligrams to grams.
\[\frac{820\text{mg}}{1,000}=0.82g\]
You have followed doctor’s recommendation because 0.82 g is less than 1 gram.
A student carefully measured 0.52 cg of copper for a science experiment, but their lab partner said they need 6 mg of copper total. How many more centigrams of copper does the student need to add?

Kuratidza mhinduro

Step 1: Convert these two measurements into a common unit. Since the question asks for the number of centigrams, convert 6 mg to centigrams, which is 0.6 cg.
Step 2: Find the difference by subtracting \(0.6-0.52\) which is 0.08 cg. This means the student must add another 0.08 cg of copper.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: The Metric System

Identify units of measurement in the metric system and their uses.
Order the six common prefixes of the metric system.
Convert between like unit values.
The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch.
The metric unit for area is the
The metric base unit for volume is the
The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g).
The metric unit for temperature is

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Tarisa yako

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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