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# Metric System

## Understand metric units of length, mass and volume.

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Metric System

Jonathan is visiting his cousin Katie who lives in Australia. He is helping her to make pies to sell at a bake sale at her school. Katie has a 2 kilogram bag of flour. Jonathan reads the recipe for the pie dough and it says they will need 315 grams of flour for each pie. They were hoping to make 5 pies. How can Jonathan figure out if they will have enough flour to do this?

In this concept, you will learn to convert between common metric units of measure.

### Converting Between Common Metric Units of Measure

The metric system is the system of measurement primarily used in science and in countries outside of the United States. The metric system includes units of length (meters), mass (grams), and capacity (liters).

The base unit of length is the meter. The table below shows some of the most common metric units of length and how they are related to the meter.

 \begin{align*}\text{millimeter (mm)} = 0.001 \ \text{meter}\end{align*} \begin{align*}\text{centimeter (cm)} = 0.01 \ \text{meter}\end{align*} \begin{align*}\text{meter (m)} = 1 \ \text{meter}\end{align*} \begin{align*}\text{kilometer (km)} = 1000 \ \text{meters}\end{align*}

Notice that all metric units of length include “meter”. The prefix of each unit of measurement indicates how that unit relates to the meter.

• milli” means one thousandth. There are 1000 millimeters in 1 meter.
• centi” means one hundredth. There are 100 centimeters in 1 meter.
• kilo” means one thousand. There are 1000 meters in 1 kilometer.

The same prefixes are used throughout the metric system. The base unit of mass is the gram. The base unit of capacity is the liter. The tables below show some of the most common metric units of mass and capacity and how they are related to the gram and the liter.

 \begin{align*}\text{milligram (mg)} = 0.001 \ \text{gram}\end{align*} \begin{align*}\text{centigram (cg)} = 0.01 \ \text{gram}\end{align*} \begin{align*}\text{gram (g)} = 1 \ \text{gram}\end{align*} \begin{align*}\text{kilogram (kg)} = 1000 \ \text{grams}\end{align*}
 \begin{align*}\text{milliliter (mL)} = 0.001 \ \text{liter}\end{align*} \begin{align*}\text{centiliter (cL)} = 0.01 \ \text{liter}\end{align*} \begin{align*}\text{liter (L)} = 1 \ \text{liter}\end{align*} \begin{align*}\text{kiloliter (kL)} = 1000 \ \text{liters}\end{align*}

Remembering the metric prefixes can help you to remember how the different units of measurement are related.

Notice that the relationship between the units of measurement are all based on powers of 10. This is because the metric system is based on powers of 10 just like our number system. To move between different units of length, mass, and capacity all you need to do is move the decimal point.

• Any time you are going from a smaller unit of measure to a larger unit of measure you will need to divide or move the decimal point to the left.
• Any time you are going from a larger unit of measure to a smaller unit of measure you will need to multiply or move the decimal point to the right.

Here is an example.

\begin{align*}340 \ \text{centiliters} = \underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{liters}\end{align*}

First, notice that you are converting centiliters to liters. The prefix “centi” means one hundredth. That means 100 centiliters make up 1 liter.

Next, notice you are going from a smaller unit to a larger unit. This means to determine how many liters you have, you will need to divide 340 by 100. This is the same as moving the decimal point 2 spaces to the left.

\begin{align*}\frac{340}{100}=3.4\end{align*}

The answer is \begin{align*}340 \ \text{centiliters} = 3.4 \ \text{liters}\end{align*}.

Here is another example.

\begin{align*}5 \ \text{meters} = \underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{millimeters}\end{align*}

First, notice that you are converting meters to millimeters. The prefix “milli” means one thousandth. That means 1000 millimeters make up 1 meter.

Next, notice you are going from a larger unit to a smaller unit. This means to determine how many millimeters you have, you will need to multiply 5 by 1000. This is the same as moving the decimal point 3 spaces to the right.

\begin{align*}5 \times 1000 = 5000\end{align*}

The answer is \begin{align*}5 \ \text{meters} = 5000 \ \text{millimeters}\end{align*}.

Here is one more example.

\begin{align*}15 \ \text{milligrams} = \underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{centigrams}\end{align*}

First, notice that you are converting milligrams to centigrams. The prefix “milli” means one thousandth which means there are 1000 milligrams in one gram. The prefix “centi” means one hundredth which means there are 100 centigrams in one gram. One way to complete this problem is to first convert milligrams to grams, and then convert grams to centigrams. While doing it this way takes two steps, it allows you to practice using the prefixes.

Convert milligrams to grams. You are going from a smaller unit to a larger unit. To determine how many grams you have, you will need to divide 15 by 1000. This is the same as moving the decimal point 3 spaces to the left.

\begin{align*}\begin{array}{rcl} \frac{15}{1000}&=&0.015 \\ 15 \ \text{milligrams} &=& 0.015 \ \text{grams} \end{array}\end{align*}

Now, convert grams to centigrams. You are going from a larger unit to a smaller unit. To determine how many centigrams you have, you will need to multiply 0.015 by 100. This is the same as moving the decimal point 2 spaces to the right.

\begin{align*}0.015 \times 100=1.5\end{align*}

The answer is \begin{align*}15 \ \text{milligrams} = 1.5 \ \text{centigrams}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Jonathan, who is helping Katie make pies.

They have a 2 kilogram bag of flour. They are hoping to make 5 pies and they need 315 grams of flour for each pie crust. Jonathan wants to see if they will have enough flour to do this.

First, Jonathan should figure out how much flour they need total. They need 315 grams of flour for each pie and they want to make 5 pies. To figure out the total amount of flour they need he should multiply.

\begin{align*}315 \times 5=1575\end{align*}

They need 1575 grams of flour to make 5 pies.

Now, Jonathan should convert 1575 grams to kilograms. The prefix “kilo” means one thousand. That means 1000 grams make up 1 kilogram.

Next, Jonathan should notice he is going from a smaller unit to a larger unit. This means to determine how many kilograms he has, he will need to divide 1575 by 1000. This is the same as moving the decimal point 3 spaces to the left.

\begin{align*}\frac{1575}{1000}=1.575\end{align*}

They need 1.575 kilograms of flour to make 5 pies.

The answer is because they need 1.575 kilograms of flour and they have 2 kilograms of flour, they have enough flour to make their pies.

#### Example 2

Convert 25,000 centimeters to kilometers.

First, notice that you are converting centimeters to kilometers. The prefix “centi” means one hundredth which means there are 100 centimeters in one meter. The prefix “kilo” means one thousand which means there are 1000 meters in one kilometer.

To complete this problem, you will first convert centimeters to meters, and then convert meters to kilometers.

Convert centimeters to meters. You are going from a smaller unit to a larger unit. To determine how many meters you have, you will need to divide 25,000 by 100. This is the same as moving the decimal point 2 spaces to the left.

\begin{align*}\begin{array}{rcl} \frac{25000}{100} &=& 250 \\ 25,000 \ \text{centimeters} &=& 250 \ \text{meters} \end{array}\end{align*}

Now, convert meters to kilometers. You are going from a smaller unit to a larger unit. To determine how many kilometers you have, you will need to divide 250 by 1000. This is the same as moving the decimal point 3 spaces to the left.

\begin{align*}\begin{array}{rcl} \frac{250}{1000} &=& 0.25 \\ 250 \ \text{meters} &=& 0.25 \ \text{kilometers} \end{array}\end{align*}

The answer is \begin{align*}25,000 \ \text{centimeters} = 0.25 \ \text{kilometers}\end{align*}.

#### Example 3

\begin{align*}87 \ \text{centigrams} = \underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{grams}\end{align*}

First, notice that you are converting centigrams to grams. The prefix “centi” means one hundredth. That means 100 centigrams make up 1 gram.

Next, notice you are going from a smaller unit to a larger unit. This means to determine how many grams you have, you will need to divide 87 by 100. This is the same as moving the decimal point 2 spaces to the left.

\begin{align*}\frac{87}{100}=0.87\end{align*}

The answer is \begin{align*}87 \ \text{centigrams} = 0.87 \ \text{grams}\end{align*}.

#### Example 4

\begin{align*}2.4 \ \text{meters} = \underline{\;\;\;\;\;\;\;\;\;\;} \ \text{millimeters}\end{align*}

First, notice that you are converting meters to millimeters. The prefix “milli” means one thousandth. That means 1000 millimeters make up 1 meter.

Next, notice you are going from a larger unit to a smaller unit. This means to determine how many millimeters you have, you will need to multiply 2.4 by 1000. This is the same as moving the decimal point 3 spaces to the right.

\begin{align*}2.4 \times 1000=2400\end{align*}

The answer is \begin{align*}2.4 \ \text{meters} = 2400 \ \text{millimeters}\end{align*}.

#### Example 5

\begin{align*}15 \ \text{kilometers} = \underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{meters}\end{align*}

First, notice that you are converting kilometers to meters. The prefix “kilo” means one thousand. That means 1000 meters make up 1 kilometer.

Next, notice you are going from a larger unit to a smaller unit. This means to determine how many meters you have, you will need to multiply 15 by 1000. This is the same as moving the decimal point 3 spaces to the right.

\begin{align*}15 \times 1000=15,000\end{align*}

The answer is \begin{align*}15 \ \text{kilometers} = 15,000 \ \text{meters}\end{align*}.

### Review

Fill in the blanks with the equivalent measurement.

1. \begin{align*}1,000 \ \text{centimeters} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{meters}\end{align*}
2. \begin{align*}10 \ \text{kiloliters} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{centiliters}\end{align*}
3. \begin{align*}1,000 \ \text{milligrams} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{centigrams}\end{align*}
4. \begin{align*}100 \ \text{milliliters} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{centiliters}\end{align*}
5. \begin{align*}200 \ \text{milligrams} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{kilograms}\end{align*}
6. \begin{align*}20 \ \text{centimeters} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{meters}\end{align*}
7. \begin{align*}2 \ \text{liters} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{kiloliters}\end{align*}
8.  \begin{align*}2,000 \ \text{centigrams} = \underline{\;\;\;\;\;\;\;\;\;\;}\text{kilograms}\end{align*}

Fill in the blanks with the equivalent measurements for 180.76 centimeters.

1. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{meters}\end{align*}
2. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{millimeters}\end{align*}
3. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{kilometers}\end{align*}

Fill in the blanks with the equivalent measurements for 0.4909 kiloliters.

1. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{liters}\end{align*}
2. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{centiliters}\end{align*}
3. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;} \ \text{milliliters}\end{align*}
4. How many liters in one kiloliter?

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Color Highlighted Text Notes

### Vocabulary Language: English

Centi

Centi is a prefix used in the metric system meaning 100.

Equivalence

Equivalence is the condition of being equal in value or meaning.

Kilo

Kilo is a prefix meaning 1,000. It is used in the metric system.

Measurement

A measurement is the weight, height, length or size of something.

Metric System

The metric system is a system of measurement commonly used outside of the United States. It contains units such as meters, liters, and grams, all in multiples of ten.

Milli

Milli is a prefix used in the metric system meaning $\frac{1}{1000}$ or 0.001.