In the joined States, both the U.S. Customary measure up system and also the metric system room used, particularly in medical, scientific, and also technical fields. In many other countries, the metric device is the major system the measurement. If you travel to other countries, you will view that road indicators list ranges in kilometers and milk is marketed in liters. People in countless countries use words prefer “kilometer,” “liter,” and also “milligram” to measure up the length, volume, and weight of different objects. These measurement units are part of the metric system.

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Unlike the U.S. Customary device of measurement, the metric system is based on 10s. Because that example, a liter is 10 times larger than a deciliter, and also a centigram is 10 times larger than a milligram. This idea the “10” is not existing in the U.S. Customary system—there space 12 customs in a foot, and also 3 feet in a yard and also 5,280 feet in a mile!

So, what if you have actually to uncover out how plenty of milligrams room in a decigram? Or, what if you want to transform meters to kilometers? Understanding just how the metric system works is a an excellent start.

In this section we will find the an easy units provided in the metric system, and also show how to convert in between them. We will also explore temperature scales. In the unified States, temperatures room usually measured making use of the Fahrenheit scale, if most nations that usage the metric device use the Celsius range to document temperatures. Learning around the various scales—, including just how to convert between them— will aid you figure out what the weather is walking to it is in like, no issue which country you uncover yourself in.


Learning Objectives

In this lesson you will learn exactly how to carry out the following:

Describe the general relationship between the U.S. Customary units and also metric units of length, weight/mass, and volumeDefine the metric prefixes and also use them to perform straightforward conversions among metric unitsSolve application problems using metric unitsState the freezing and also boiling point out of water top top the Celsius and also Fahrenheit temperature scales.Convert native one temperature scale to the other, utilizing conversion formulas

Metric system Basics

What Is Metric?

The metric mechanism uses devices such as meter, liter, and also gram to measure up length, fluid volume, and mass, just as the U.S. Customary device uses feet, quarts, and ounces to measure up these.

In addition to the difference in the simple units, the metric mechanism is based upon 10s, and also different measures for length include kilometer, meter, decimeter, centimeter, and also millimeter. Notification that the word “meter” is part of all of these units.

The metric system also applies the idea that devices within the system get larger or smaller sized by a power of 10. This means that a meter is 100 times bigger than a centimeter, and also a kilogram is 1,000 times heavier 보다 a gram. You will discover this idea a little later. Because that now, an alert how this idea that “getting bigger or smaller by 10” is an extremely different than the relationship between units in the U.S. Customary system, where 3 feet equals 1 yard, and 16 ounces equals 1 pound.

Length, Mass, and Volume

The table listed below shows the an easy units the the metric system. Note that the names of all metric units follow from this three an easy units.

LengthMassVolume
basic units
metergramliter
other units you may see
kilometerkilogramdekaliter
centimetercentigramcentiliter
millimetermilligrammilliliter

In the metric system, the basic unit of size is the meter. A meter is slightly larger than a yardstick, or just over 3 feet.

The simple metric unit of mass is the gram. A regular-sized paperclip has a massive of about 1 gram.

Among scientists, one gram is identified as the massive of water that would certainly fill a 1-centimeter cube. Friend may an alert that the word “mass” is used here instead the “weight.” In the sciences and also technical fields, a difference is made between weight and mass. Weight is a measure up of the traction of heaviness on one object. Because that this reason, an object’s weight would certainly be various if it to be weighed on earth or top top the moon because of the difference in the gravitational forces. However, the object’s mass would remain the same in both places because mass actions the lot of substance in an object. As lengthy as you space planning on just measuring objects on Earth, you deserve to use mass/weight reasonably interchangeably—but the is worth noting that there is a difference!

Finally, the basic metric unit the volume is the liter. A liter is slightly larger than a quart.

The manage of a shovel is about 1 meter.A paperclip weighs around 1 gram.A medium-sized container the milk is about 1 liter.

Though the is rarely crucial to convert in between the customary and metric systems, sometimes it help to have a mental picture of how big or small some units are. The table listed below shows the relationship in between some common units in both systems.

Common measurements in Customary and also Metric Systems
Length1 centimeter is a small less than fifty percent an inch.
1.6 kilometers is about 1 mile.
1 meter is around 3 inches much longer than 1 yard.
Mass1 kilogram is a little more than 2 pounds.
 28 grams is around the very same as 1 ounce.
Volume1 liter is a little much more than 1 quart.
 4 liters is a little much more than 1 gallon.

Prefixes in the Metric System

The metric system is a base 10 system. This way that each successive unit is 10 times bigger than the ahead one.

The names of metric units are created by including a prefix to the basic unit the measurement. To tell how huge or little a unit is, you look in ~ the prefix. Come tell whether the unit is measure length, mass, or volume, girlfriend look in ~ the base.

Prefixes in the Metric System
kilo-hecto-deka-meter

gram

liter

deci-centi-milli-
1,000 times larger 보다 base unit100 times larger than base unit10 time larger than base unitbase units10 time smaller 보다 base unit100 time smaller 보다 base unit1,000 time smaller 보다 base unit
Using this table as a reference, you have the right to see the following:

A kilogram is 1,000 times larger than one gram (so 1 kilogram = 1,000 grams).A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters).A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).

Here is a comparable table that just shows the metric systems of measurement because that mass, in addition to their size relative to 1 gram (the base unit). The typical abbreviations for these metric units have been consisted of as well.

Measuring mass in the Metric System
kilogram

(kg)

hectogram

(hg)

dekagram

(dag)

gram

(g)

decigram

(dg)

centigram

(cg)

milligram

(mg)

1,000 grams100 grams10 gramsgram0.1 gram0.01 gram0.001 gram
Since the prefixes remain constant through the metric system, you can create similar charts for length and volume. The prefixes have the same definitions whether they are attached to the devices of size (meter), massive (gram), or volume (liter).


Try that Now

Which that the following sets of three units are all metric measurements of length?

A) inch, foot, yard

B) kilometer, centimeter, millimeter

C) kilogram, gram, centigram

D) kilometer, foot, decimeter


B) kilometer, centimeter, millimeter

All the these measurements are from the metric system. You have the right to tell castle are measurements of length because they every contain words “meter.”


Converting devices Up and Down the Metric Scale

Converting between metric systems of measure requires expertise of the metric prefixes and an knowledge of the decimal system—that’s about it.

For instance, you can number out how many centigrams room in one dekagram by making use of the table above. One dekagram is larger than one centigram, so you expect that one dekagram will equal plenty of centigrams.

In the table, each unit is 10 times larger than the one to its immediate right. This way that 1 dekagram = 10 grams; 10 grams = 100 decigrams; and 100 decigrams = 1,000 centigrams. So, 1 dekagram = 1,000 centigrams.


Identify areas of milligrams and also decigrams.

kghgdaggdgcgmg
^^

Decigrams (dg) are larger than milligrams (mg), therefore you expect there come be many mg in one dg.

Dg is 10 times bigger than a cg, and a cg is 10 times larger than a mg.

\times10\times10
kghgdaggdgcgmg
\downarrow\uparrow
\rightarrow

Since you space going native a bigger unit come a smaller sized unit, multiply.

Multiply: 1 · 10 · 10, to discover the number of milligrams in one decigram. 

1\text dg\cdot10\cdot10=100\text mg

There space 100 milligrams (mg) in 1 decigram (dg).


Identify places of kilometers and centimeters.

kmhmdammdmcmmm
^^

Kilometers (km) are larger than centimeters (cm), for this reason you suppose there to be less than one kilometres in a cm.

Cm is 10 times smaller than a dm; a dm is 10 times smaller than a m, etc.

Since you space going from a smaller unit to a larger unit, divide.

\div10\div10\div10\div10\div10
kmhmdammdmcmmm
^\leftarrow\leftarrow\leftarrow\leftarrow^

Divide: 1\div10\div10\div10\div10\div10, to uncover the variety of kilometers in one centimeter. 

1\text cm\div10\div10\div10\div10\div10=0.00001\text km

1 centimeter (cm) = 0.00001 kilometers (km).


Once you start to recognize the metric system, you deserve to use a faster way to convert amongst different metric units. The dimension of metric units rises tenfold together you go up the metric scale. The decimal device works the exact same way: a tenth is 10 times larger than a hundredth; a hundredth is 10 times bigger than a thousandth, etc. By applying what girlfriend know about decimals come the metric system, converting among units is as basic as moving decimal points.

Here is the first problem indigenous above: How countless milligrams space in one decigram? You deserve to recreate the stimulate of the metric systems as presented below:

\displaystyle kg\quad hg\quad dag\quad g\quad d\underbraceg\quad c_1\underbraceg\quad m_2g

This concern asks girlfriend to begin with 1 decigram and also convert that to milligrams. As shown above, milligrams is two locations to the best of decigrams. You can just move the decimal point two areas to the best to convert decigrams to milligrams: \displaystyle 1\ dg=1\underbrace0_1\underbrace0_2.\ mg.

The same method works when you space converting from a smaller to a larger unit, together in the problem: transform 1 centimeter come kilometers.

\displaystyle k\underbracem\quad h_5\underbracem\quad d_4\underbraceam\quad _3\underbracem\quad d_2\underbracem\quad c_1m\quad mm

Note that rather of moving to the right, you are now relocating to the left—so the decimal point must execute the same:

\displaystyle 1\ cm=0.\underbrace0_5\underbrace0_4\underbrace0_3\underbrace0_2\underbrace1_1\ km.


There room 10 milliliters in a centiliter, 10 centiliters in a deciliter, and 10 deciliters in a liter. Multiply: 10\cdot10\cdot10, to discover the number of milliliters in a liter, 1,000.


Factor label Method

There is however another technique that you can use to convert metric measurements—the variable label method. You supplied this an approach when you to be converting measurement units within the U.S. Customary system.

The factor label an approach works the exact same in the metric system; it depends on the usage of unit fractions and the cancelling of intermediate units. The table below shows several of the unit equivalents and unit fractions for length in the metric system. (You should an alert that all of the unit fountain contain a factor of 10. Remember that the metric mechanism is based upon the concept that every unit is 10 times bigger than the one that came prior to it.)

Also, an alert that two brand-new prefixes have been added here: mega- (which is really big) and micro- (which is an extremely small).

Unit EquivalentsConversion Factors
1 meter = 1,000,000 micrometers \displaystyle \frac1\ m1,000,000\ \mu m \displaystyle \frac1,000,000\ \mu m1\ m
1 meter = 1,000 millimeters \displaystyle \frac1\ m1,000\ mm \displaystyle \frac1,000\ mm1\ m
1 meter = 100 centimeters \displaystyle \frac1\ m100\ cm \displaystyle \frac100\ cm1\ m
1 meter = 10 decimeters \displaystyle \frac1\ m10\ dm \displaystyle \frac10\ dm1\ m
1 dekameter = 10 meters \displaystyle \frac1\ dam10\ m \displaystyle \frac10\ m1\ dam
1 hectometer = 100 meters \displaystyle \frac1\ hm100\ m \displaystyle \frac100\ m1\ hm
1 kilometer = 1,000 meters \displaystyle \frac1\ km1,000\ m \displaystyle \frac1,000\ m1\ km
1 megameter = 1,000,000 meters \displaystyle \frac1\ Mm1,000,000\ m \displaystyle \frac1,000,000\ m1\ Mm

When using the element label method in the metric system, be certain to examine that you are not skip over any kind of intermediate systems of measurement!


Meters is larger than centimeters, therefore you suppose your answer to be less than 7,225.

7,225\text cm=\text___ m

Using the variable label method, write 7,225 cm together a portion and usage unit fractions to transform it come m.

\displaystyle \frac7,225\ cm1\cdot \frac1\ m100\ cm=\_\_\_ m

Cancel similar units, multiply, and also simplify.

\displaystyle \frac7,225\ \cancelcm1\cdot \frac1\text m100\ \cancel\textcm=\_\_\_m

\displaystyle \frac7,2251\cdot \frac1\text m100=\frac7,225100\textm

\displaystyle \frac7,225\text m100=72.25\text m

7,225\text centimeters=72.25\text meters


Convert mg come g.

5,000\text mg=\text___ g

\displaystyle \frac5,000\text mg1\cdot \frac1\text g1,000\text mg=\text g

\displaystyle \frac5,000\cancel\textmg1\cdot \frac1\text g1,000\ \cancel\textmg=\text g

\displaystyle \frac5,000\cdot 1\text g1\cdot 1,000=\frac5,000\text g1,000

\displaystyle \frac5,000\text g1,000=5\text g

5\text g=5,000\text mg, so the pharmacist did no make a mistake.


To price this question, us will should subtract.

First transform all dimensions to one unit. Right here we will transform to centimeters.

2\text meters-1\text meter and 35\text cm

Use the element label technique and unit fountain to convert from meters to centimeters.

\displaystyle \frac2\text m1\cdot \frac100\text cm1\text m=\text cm

Cancel, multiply, and also solve.

Convert the 1 meter to centimeters, and combine v the additional 35 centimeters.

Subtract the cut length indigenous the original board length.

\displaystyle \frac2\ \cancel\textm1\cdot \frac100\text cm1\ \cancel\text m=\text cm

\displaystyle \frac200\text cm1=200\text cm

1\text meter+35\text cm

100\text cm+35\text cm

135\text cm

200\text cm-135\text cm

65\text cm

There is 65 cm of board left.


Start by calculating exactly how much water will be supplied in a week utilizing the element label technique to transform the time units.

\displaystyle \frac10\ ml1\text minute\cdot \frac60\text minute1\text hour\cdot \frac24\text hours1\text day\cdot \frac7\text days1\text week

Cancel, multiply and also solve.

\displaystyle \frac10\ ml1\text \cancel\textminute\cdot \frac60\text \cancel\textminute1\text \cancel\texthour\cdot \frac24\text \cancel\texthours1\text \cancel\textday\cdot \frac7\text \cancel\textdays1\text week

\displaystyle \frac10\centerdot 60\centerdot 24\centerdot 7\ ml1\centerdot 1\centerdot 1\centerdot 1\text week

To offer a more useable answer, convert this into liters.

\displaystyle \frac100800\ ml1\text week

Cancel, multiply and also solve.

\displaystyle \frac100800\text ml1\text week\centerdot \frac1\text L1000\text ml

\displaystyle \frac100800\text \cancel\textml1\text week\centerdot \frac1\text L1000\text \cancel\textml

\displaystyle \frac100800\text L1000\text week= \displaystyle 100.8\frac\textL\textweek

The faucet wastes around 100.8 liters every week.


To answer the question, friend will have to divide 1.5 liters by 250 milliliters. To perform this, convert both to the very same unit. You could transform either measurement.

1.5\text L\div250\text mL

Convert 250 mL come liters

250\text mL=\text ___ L

\displaystyle \frac250\text mL1\cdot \frac1\text L1000\text mL=\text___ L

\displaystyle \frac250\text L1000=0.25\text L

Now we deserve to divide utilizing the converted measurement

1.5\text L\div250\text mL=\frac1.5\text L250\text mL =\frac1.5\text L0.25\text L

\displaystyle \frac1.5\text L0.25\text L=6

The bottle holds 6 servings.


Understanding the paper definition of real-life application difficulties is important. Look because that words in ~ the difficulty that assist you determine what operations room needed, and also then apply the exactly unit conversions. Checking your last answer through using another conversion technique (such together the “move the decimal” method, if you have actually used the aspect label an approach to solve the problem) can cut down on errors in her calculations.

Summary

The metric device is an alternate system the measurement provided in most countries, as well as in the unified States. The metric device is based upon joining among a series of prefixes, including kilo-, hecto-, deka-, deci-, centi-, and milli-, through a base unit the measurement, such as meter, liter, or gram. Systems in the metric mechanism are all associated by a power of 10, which way that each succeeding unit is 10 times bigger than the previous one.

This renders converting one metric measurement to an additional a simple process, and is regularly as an easy as moving a decimal point. The is always important, though, to take into consideration the direction that the conversion. If you space converting a smaller sized unit come a bigger unit, then the decimal allude has to move to the left (making her number smaller); if you room converting a bigger unit to a smaller sized unit, climate the decimal suggest has to relocate to the ideal (making your number larger).

The factor label method can likewise be applied to conversions within the metric system. To usage the variable label method, you main point the initial measurement by unit fractions; this permits you to represent the initial measurement in a different measurement unit.

Temperature Scales

Turn ~ above the television any kind of morning and also you will see meteorologists talking about the day’s weather forecast. In addition to informing you what the weather problems will be favor (sunny, cloudy, rainy, muggy), they additionally tell you the day’s forecast for high and low temperatures. A warm summer day might reach 100° in Philadelphia, while a cool spring day may have actually a short of 40° in Seattle.

If you have been to other countries, though, friend may notice that meteorologists measure up heat and cold in different ways outside the the joined States. Because that example, a TV weatherman in mountain Diego may forecast a high of 89°, however a similar forecaster in Tijuana, Mexico—, which is just 20 mile south—, might look in ~ the exact same weather pattern and also say that the day’’s high temperature is walking to be 32°. What’s walk on here? The distinction is the the two countries use various temperature scales.

Measuring Temperature on 2 Scales

Fahrenheit and Celsius room two various scales because that measuring temperature.

A thermometer measure up a temperature the 22° Celsius is displayed here.

On the Celsius scale, water freezes in ~ 0° and also boils at 100°.

If the United says were to adopt the Celsius scale, forecast temperatures would seldom go below -30° or above 45°. (A temperature of

-18° may be projection for a cold winter work in Michigan, while a temperature of 43° might be predicted for a warm summer job in Arizona.)

Most office buildings maintain an indoor temperature between 18°C and also 24°C to store employees comfortable.

A thermometer measure a temperature of 72° Fahrenheit is presented here.

On the Fahrenheit scale, water freezes in ~ 32° and boils in ~ 212°.

In the unified States, projection temperatures measure up in Fahrenheit rarely go listed below -20° or above 120°. (A temperature the 0° might be forecast for a cold winter day in Michigan, if a temperature of 110° might be predicted for a warm summer work in Arizona.)

Most office buildings maintain an indoor temperature between 65°F and also 75°F to store employees comfortable.

CelsiusFahrenheit


Try that Now

A cook puts a thermometer into a pot that water to view how hot it is. The thermometer reads 132°, yet the water is no boiling yet. I beg your pardon temperature range is the thermometer measuring?


Converting in between the Scales

By looking in ~ the two thermometers shown, you have the right to make some basic comparisons between the scales. Because that example, many human being tend to be comfortable in outdoor temperatures between 50°F and 80°F (or in between 10°C and also 25°C). If a meteorologist predicts an mean temperature that 0°C (or 32°F), then it is a for sure bet the you will require a winter jacket.

Sometimes, that is important to convert a Celsius measurement come its specific Fahrenheit measure up or angry versa. For example, what if you desire to understand the temperature the your kid in Fahrenheit, and also the only thermometer you have actually measures temperature in Celsius measurement? convert temperature between the equipment is a straightforward process as long as you usage the formulas provided below.


Temperature switch Formulas

To transform a Fahrenheit measurement come a Celsius measurement, use this formula.

C=\frac59(F-32)

To transform a Celsius measurement to a Fahrenheit measurement, use this formula.

F=\frac95C+32


How were these formulas developed? They come from compare the two scales. Due to the fact that the freezing point is 0° in the Celsius scale and 32° ~ above the Fahrenheit scale, we subtract 32 once converting indigenous Fahrenheit come Celsius, and add 32 when converting from Celsius come Fahrenheit.

There is a factor for the fractions \frac59 and \frac95, also. There room 100 degrees in between the freeze (0°) and also boiling points (100°) of water ~ above the Celsius scale and also 180 degrees in between the similar points (32° and 212°) top top the Fahrenheit scale. Creating these 2 scales together a ratio, \fracF^\circ C^\circ , provides \frac180^\circ 100^\circ =\frac180^\circ \div 20100^\circ \div 20=\frac95. If you flip the ratio to it is in \frac\textC^\circ \textF^\circ , you gain \frac100^\circ 180^\circ =\frac100^\circ \div 20180^\circ \div 20=\frac59. Notification how these fractions are provided in the counter formulas.

The example below illustrates the counter of Celsius temperature come Fahrenheit temperature, utilizing the boiling suggest of water, which is 100° C.


Example

The boiling allude of water is 100°C. What temperature walk water boil at in the Fahrenheit scale?

A Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula in ~ the left.

F=\frac95C+32

Substitute 100 because that C and also multiply.

F=\frac95(100)+32

F=\frac9005+32

Simplify \frac9005 by dividing numerator and denominator by 5.

F=\frac900\div 55\div 5+32

F=\frac1801+32

Add 180+32.

F=212

The boiling point of water is 212°F.


A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.

C=\frac59(F-32)

Substitute 32 for F and subtract.

C=\frac59(32-32)

Any number multiplied by 0 is 0.

C=\frac59(0)

C=0

The freezing suggest of water is 0^\circ\textC.


One temperature is provided in °C, and the other is given in °F. To uncover the difference in between them, we have to measure castle on the same scale.

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What is the difference in between 120°C and also 250°F?

Use the conversion formula to transform 120°C to °F.

(You could transform 250°F come °C instead; this is defined in the text after this example.)