In the united States, both the U.S. Customary measure system and the metric system room used, specifically in medical, scientific, and also technical fields. In many other countries, the metric system is the main system the measurement. If you take trip to other countries, girlfriend will check out that road signs list distances in kilometers and milk is offered in liters. Human being in many countries usage words choose kilometer, liter, and milligram to measure the length, volume, and also weight of various objects. These measurement systems are component of the metric system.

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Unlike the U.S. Customary device of measurement, the metric mechanism is based upon 10s. Because that example, a liter is 10 times bigger than a deciliter, and a centigram is 10 times larger than a milligram. This idea of 10 is not existing in the U.S. Customary systemthere room 12 inches in a foot, and also 3 feet in a yard and also 5,280 feet in a mile!

So, what if you have to discover out how countless milligrams are in a decigram? Or, what if you want to convert meters to kilometers? Understanding how the metric system works is a good start.

In this section we will uncover the basic units offered in the metric system, and show how to convert in between them. We will also explore temperature scales. In the united States, temperatures are usually measured utilizing the Fahrenheit scale, if most nations that usage the metric system use the Celsius range to record temperatures. Learning around the different scales, including exactly how to convert between them will assist you number out what the weather is walk to it is in like, no issue which nation you discover yourself in.

### Learning Objectives

In this class you will learn how to perform the following:

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

## Metric device Basics

### What Is Metric?

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

In addition to the difference in the simple units, the metric mechanism is based upon 10s, and different measures for length encompass kilometer, meter, decimeter, centimeter, and millimeter. Notice that the word meter is component of all of these units.

The metric system additionally applies the idea that devices within the system get larger or smaller sized by a power of 10. This method that a meter is 100 times larger than a centimeter, and also a kilogram is 1,000 times heavier than a gram. You will explore this idea a little later. Because that now, notice how this idea that getting larger or smaller by 10 is really different than the relationship in between units in the U.S. Customary system, wherein 3 feet equates to 1 yard, and 16 ounces equates to 1 pound.

### Length, Mass, and Volume

The table listed below shows the basic units that the metric system. Note that the surname of every metric units follow from these three basic units.

 Length Mass Volume basic units meter gram liter other units you might see kilometer kilogram dekaliter centimeter centigram centiliter millimeter milligram milliliter

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

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

Among scientists, one gram is identified as the massive of water that would fill a 1-centimeter cube. Friend may notice that words mass is used below instead the weight. In the sciences and technical fields, a distinction is made between weight and also mass. Weight is a measure of the traction of gravity on an object. For this reason, one objects weight would certainly be various if it to be weighed on earth or ~ above the moon due to the fact that of the difference in the gravitational forces. However, the objects mass would continue to be the exact same in both places since mass actions the quantity of problem in an object. As lengthy as you room planning on just measuring objects top top Earth, you can use mass/weight reasonably interchangeablybut that is precious noting the there is a difference!

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

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

Though the is rarely important to convert between the customary and also metric systems, occasionally it help to have actually a mental picture of how large or small some systems are. The table below shows the relationship in between some usual units in both systems.

 Common measurements in Customary and also Metric Systems Length 1 centimeter is a little less than fifty percent an inch. 1.6 kilometers is around 1 mile. 1 meter is about 3 inches longer than 1 yard. Mass 1 kilogram is a little more than 2 pounds. 28 grams is about the same as 1 ounce. Volume 1 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 device is a base 10 system. This means that each successive unit is 10 times bigger than the previous one.

The surname of metric units are developed by including a prefix come the straightforward unit that measurement. Come tell how big or small a unit is, friend look at the prefix. Come tell whether the unit is measure length, mass, or volume, you look at the base.

 Prefixes in the Metric System kilo- hecto- deka- metergramliter deci- centi- milli- 1,000 times larger than base unit 100 time larger 보다 base unit 10 time larger than base unit base units 10 time smaller than base unit 100 time smaller 보다 base unit 1,000 times smaller 보다 base unit
Using this table as a reference, you have the right to see the following:

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

Here is a comparable table that just shows the metric devices of measurement because that mass, along with their size relative to 1 gram (the base unit). The usual abbreviations because that these metric units have actually been included as well.

 Measuring mass in the Metric System kilogram(kg) hectogram(hg) dekagram(dag) gram(g) decigram(dg) centigram(cg) milligram(mg) 1,000 grams 100 grams 10 grams gram 0.1 gram 0.01 gram 0.001 gram
Since the prefixes remain consistent through the metric system, you might create comparable charts because that length and volume. The prefixes have actually the same interpretations whether they space attached come the units of length (meter), massive (gram), or volume (liter).

### Try that Now

Which the the following sets of 3 units space all metric dimensions of length?

A) inch, foot, yard

B) kilometer, centimeter, millimeter

C) kilogram, gram, centigram

D) kilometer, foot, decimeter

B) kilometer, centimeter, millimeter

All that these dimensions are indigenous the metric system. You deserve to tell they are dimensions of length due to the fact that they every contain the word meter.

### Converting units Up and Down the Metric Scale

Converting in between metric units of measure requires expertise of the metric prefixes and also an knowledge of the decimal systemthats about it.

For instance, you can figure out how many centigrams are in one dekagram by making use of the table above. One dekagram is larger than one centigram, so you intend that one dekagram will equal countless centigrams.

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

Identify locations of milligrams and decigrams.

 kg hg dag g dg cg mg ^ ^

Decigrams (dg) are bigger than milligrams (mg), so you suppose there to be plenty of mg in one dg.

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

 imes10 imes10 kg hg dag g dg cg mg downarrow uparrow ightarrow

Since you are going native a larger unit come a smaller sized unit, multiply.

Multiply: 1 · 10 · 10, to uncover the variety of milligrams in one decigram.

1 ext dgcdot10cdot10=100 ext mg

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

Identify places of kilometers and centimeters.

 km hm dam m dm cm mm ^ ^

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

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

Since you are going indigenous a smaller unit to a bigger unit, divide.

 div10 div10 div10 div10 div10 km hm dam m dm cm mm ^ leftarrow leftarrow leftarrow leftarrow ^

Divide: 1div10div10div10div10div10, to discover the number of kilometers in one centimeter.

1 ext cmdiv10div10div10div10div10=0.00001 ext km

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

Once you start to recognize the metric system, you have the right to use a faster way to convert amongst different metric units. The size of metric units rises tenfold as you go up the metric scale. The decimal system works the very same way: a tenth is 10 times larger than a hundredth; a hundredth is 10 times larger than a thousandth, etc. By applying what girlfriend know around decimals to the metric system, converting among units is as straightforward as moving decimal points.

Here is the first problem native above: How numerous milligrams room in one decigram? You have the right to recreate the stimulate of the metric devices as displayed below:

This question asks you to begin with 1 decigram and also convert that to milligrams. As shown above, milligrams is two places to the right of decigrams. You deserve to just move the decimal suggest two places to the best to convert decigrams come milligrams: displaystyle 1 dg=1underbrace0_1underbrace0_2. mg.

The same method works once you are converting indigenous a smaller to a larger unit, together in the problem: transform 1 centimeter come kilometers.

Note that instead of relocating to the right, you space now relocating to the leftso the decimal allude must carry out the same:

displaystyle 1 cm=0.underbrace0_5underbrace0_4underbrace0_3underbrace0_2underbrace1_1 km.

There room 10 milliliters in a centiliter, 10 centiliters in a deciliter, and also 10 deciliters in a liter. Multiply: 10cdot10cdot10, to uncover the variety of milliliters in a liter, 1,000.

### Factor brand Method

There is however another an approach that you deserve to use to convert metric measurementsthe element label method. You provided this technique when you were converting measurement systems within the U.S. Customary system.

The element label technique works the very same in the metric system; it depends on the use of unit fractions and also the cancelling of intermediary units. The table below shows few of the unit equivalents and unit fractions for length in the metric system. (You should notice that all of the unit fountain contain a element of 10. Remember the the metric device is based upon the notion that each unit is 10 times larger than the one that came prior to it.)

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

 Unit Equivalents Conversion 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 variable label technique in the metric system, be sure to inspect that you space not skip over any kind of intermediate units of measurement!

Meters is bigger than centimeters, therefore you suppose your answer come be less than 7,225.

7,225 ext cm= ext___ m

Using the element label method, create 7,225 cm as a fraction and usage unit fountain to transform it to m.

displaystyle frac7,225 cm1cdot frac1 m100 cm=\_\_\_ m

Cancel similar units, multiply, and simplify.

displaystyle frac7,225 cancelcm1cdot frac1 ext m100 cancel extcm=\_\_\_m

displaystyle frac7,2251cdot frac1 ext m100=frac7,225100 extm

displaystyle frac7,225 ext m100=72.25 ext m

7,225 ext centimeters=72.25 ext meters

Convert mg come g.

5,000 ext mg= ext___ g

displaystyle frac5,000 ext mg1cdot frac1 ext g1,000 ext mg= ext g

displaystyle frac5,000cancel extmg1cdot frac1 ext g1,000 cancel extmg= ext g

displaystyle frac5,000cdot 1 ext g1cdot 1,000=frac5,000 ext g1,000

displaystyle frac5,000 ext g1,000=5 ext g

5 ext g=5,000 ext mg, for this reason the pharmacist did no make a mistake.

To answer this question, we will have to subtract.

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

2 ext meters-1 ext meter and also 35 ext cm

Use the aspect label method and unit fountain to convert from meters to centimeters.

displaystyle frac2 ext m1cdot frac100 ext cm1 ext m= ext cm

Cancel, multiply, and solve.

Convert the 1 meter to centimeters, and also combine with the added 35 centimeters.

Subtract the reduced length indigenous the original board length.

displaystyle frac2 cancel extm1cdot frac100 ext cm1 cancel ext m= ext cm

displaystyle frac200 ext cm1=200 ext cm

1 ext meter+35 ext cm

100 ext cm+35 ext cm

135 ext cm

200 ext cm-135 ext cm

65 ext cm

There is 65 centimeter of board left.

Start through calculating just how much water will be supplied in a week utilizing the variable label an approach to convert the time units.

displaystyle frac10 ml1 ext minutecdot frac60 ext minute1 ext hourcdot frac24 ext hours1 ext daycdot frac7 ext days1 ext week

Cancel, multiply and also solve.

displaystyle frac10 ml1 ext cancel extminutecdot frac60 ext cancel extminute1 ext cancel exthourcdot frac24 ext cancel exthours1 ext cancel extdaycdot frac7 ext cancel extdays1 ext week

displaystyle frac10centerdot 60centerdot 24centerdot 7 ml1centerdot 1centerdot 1centerdot 1 ext week

To give a more useable answer, convert this right into liters.

displaystyle frac100800 ml1 ext week

Cancel, multiply and solve.

displaystyle frac100800 ext ml1 ext weekcenterdot frac1 ext L1000 ext ml

displaystyle frac100800 ext cancel extml1 ext weekcenterdot frac1 ext L1000 ext cancel extml

displaystyle frac100800 ext L1000 ext week= displaystyle 100.8frac extL extweek

The faucet wastes about 100.8 liters each week.

To answer the question, you will must divide 1.5 liters by 250 milliliters. To carry out this, convert both to the very same unit. You could transform either measurement.

1.5 ext Ldiv250 ext mL

Convert 250 mL come liters

250 ext mL= ext ___ L

displaystyle frac250 ext mL1cdot frac1 ext L1000 ext mL= ext___ L

displaystyle frac250 ext L1000=0.25 ext L

Now we have the right to divide making use of the converted measurement

1.5 ext Ldiv250 ext mL=frac1.5 ext L250 ext mL =frac1.5 ext L0.25 ext L

displaystyle frac1.5 ext L0.25 ext L=6

The bottle holds 6 servings.

Understanding the paper definition of real-life application problems is important. Look because that words within the problem that aid you recognize what operations are needed, and then use the correct unit conversions. Checking your last answer by using an additional conversion an approach (such as the move the decimal method, if you have used the factor label technique to resolve the problem) can reduced down ~ above errors in your calculations.

### Summary

The metric system is an alternative system the measurement provided in many 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 also milli-, v a basic unit the measurement, such as meter, liter, or gram. Devices in the metric system are all associated by a power of 10, which way that each successive unit is 10 times bigger than the ahead one.

This provides converting one metric measure to an additional a simple process, and also is frequently as simple as moving a decimal point. The is always important, though, to think about the direction that the conversion. If you space converting a smaller sized unit come a bigger unit, climate the decimal point has to relocate to the left (making your number smaller); if you are converting a bigger unit to a smaller unit, then the decimal point has to move to the appropriate (making her number larger).

The factor label an approach can additionally be applied to conversions within the metric system. To usage the factor label method, you multiply the initial measurement by unit fractions; this enables you to stand for the original measurement in a different measurement unit.

## Temperature Scales

Turn top top the television any morning and also you will see meteorologists talking around the days weather forecast. In enhancement to telling you what the weather conditions will be like (sunny, cloudy, rainy, muggy), they likewise tell friend the days estimate for high and low temperatures. A warm summer day might reach 100° in Philadelphia, when a cool feather day may have actually a low of 40° in Seattle.

If you have been to various other countries, though, you may notice that meteorologists measure up heat and cold in different way outside of the joined States. Because that example, a TV weatherman in san Diego may forecast a high that 89°, but a similar forecaster in Tijuana, Mexico, i beg your pardon is only 20 mile south, may look in ~ the exact same weather pattern and say that the day’s high temperature is walking to be 32°. Whats going on here? The difference is the the two countries use various temperature scales.

### Measuring Temperature on 2 Scales

Fahrenheit and also Celsius room two various scales for measuring temperature.

 A thermometer measure up a temperature the 22° Celsius is displayed here.On the Celsius scale, water freezes at 0° and also boils at 100°.If the United claims were to embrace the Celsius scale, estimate temperatures would hardly ever go below -30° or above 45°. (A temperature of-18° may be projection for a cold winter job in Michigan, while a temperature the 43° may be predicted for a hot summer job in Arizona.)Most office structures maintain an indoor temperature in between 18°C and also 24°C to save employees comfortable. A thermometer measure up a temperature that 72° Fahrenheit is presented here.On the Fahrenheit scale, water freezes in ~ 32° and boils in ~ 212°.In the joined States, forecast temperatures measure in Fahrenheit seldom go below -20° or over 120°. (A temperature the 0° may be forecast for a cold winter job in Michigan, if a temperature the 110° might be predicted for a hot summer work in Arizona.)Most office structures maintain an indoor temperature between 65°F and also 75°F to save employees comfortable. Celsius Fahrenheit

### Try it Now

A cook puts a thermometer right into a pot that water to see how warm it is. The thermometer reads 132°, however the water is not boiling yet. I m sorry temperature range is the thermometer measuring?

### Converting between the Scales

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

Sometimes, the is crucial to convert a Celsius measurement come its specific Fahrenheit measure or vice versa. For example, what if you want to know the temperature of your child in Fahrenheit, and also the only thermometer you have actually measures temperature in Celsius measurement? convert temperature in between the equipment is a straightforward process as lengthy as you use the formulas listed below.

### Temperature counter Formulas

To convert a Fahrenheit measurement to a Celsius measurement, usage this formula.

C=frac59(F-32)

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

F=frac95C+32

How were these recipe developed? They come from compare the 2 scales. Since the freezing allude is 0° in the Celsius scale and 32° on the Fahrenheit scale, we subtract 32 as soon as converting indigenous Fahrenheit to Celsius, and include 32 once converting indigenous Celsius come Fahrenheit.

There is a reason for the fractions frac59 and frac95, also. There room 100 degrees in between the freezing (0°) and boiling clues (100°) the water on the Celsius scale and 180 degrees in between the similar points (32° and 212°) on the Fahrenheit scale. Creating these two scales together a ratio, fracF^circ C^circ , offers frac180^circ 100^circ =frac180^circ div 20100^circ div 20=frac95. If you flip the ratio to be frac extC^circ extF^circ , you get frac100^circ 180^circ =frac100^circ div 20180^circ div 20=frac59. Notice how these fractions are offered in the switch formulas.

The example listed below illustrates the counter of Celsius temperature come Fahrenheit temperature, utilizing the boiling allude of water, i beg your pardon is 100° C.

### Example

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

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

F=frac95C+32

Substitute 100 for C and also multiply.

F=frac95(100)+32

F=frac9005+32

Simplify frac9005 by splitting numerator and denominator through 5.

F=frac900div 55div 5+32

F=frac1801+32

F=212

The boiling point of water is 212°F.

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

C=frac59(F-32)

Substitute 32 because that F and also subtract.

C=frac59(32-32)

Any number multiply by 0 is 0.

C=frac59(0)

C=0

The freezing allude of water is 0^circ extC.

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

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

Use the counter formula to convert 120°C to °F.

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