## Making Unit Conversions in the Metric System

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In the metric system, units are related by powers of \(10.\) The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is \(1000\) meters; the prefix *kilo-* means thousand. One centimeter is \(\frac{1}{100}\) of a meter, because the prefix *centi-* means one one-hundredth (just like one cent is \(\frac{1}{100}\) of one dollar).

The equivalencies of measurements in the metric system are shown in the table below. The common abbreviations for each measurement are given in parentheses.

Metric Measurements | ||
---|---|---|

Length | Mass | Volume/Capacity |

\(1\) kilometer (km) = \(1000\) m \(1\) hectometer (hm) = \(100\) m \(1\) dekameter (dam) = \(10\) m \(1\) meter (m) = \(1\) m \(1\) decimeter (dm) = \(0.1\) m \(1\) centimeter (cm) = \(0.01\) m \(1\) millimeter (mm) = \(0.001\) m | \(1\) kilogram (kg) = \(1000\) g \(1\) hectogram (hg) = \(100\) g \(1\) dekagram (dag) = \(10\) g \(1\) gram (g) = \(1\) g \(1\) decigram (dg) = \(0.1\) g \(1\) centigram (cg) = \(0.01\) g \(1\) milligram (mg) = \(0.001\) g | \(1\) kiloliter (kL) = \(1000\) L \(1\) hectoliter (hL) = \(100\) L \(1\) dekaliter (daL) = \(10\) L \(1\) liter (L) = \(1\) L \(1\) deciliter (dL) = \(0.1\) L \(1\) centiliter (cL) = \(0.01\) L \(1\) milliliter (mL) = \(0.001\) L |

\(1\) meter = \(100\) centimeters \(1\) meter = \(1000\) millimeters | \(1\) gram = \(100\) centigrams \(1\) gram = \(1000\) milligrams | \(1\) liter = \(100\) centiliters \(1\) liter = \(1000\) milliliters |

To make conversions in the metric system, we will use the same technique we did in the U.S. system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a \(\text{5 k}\) or \(\text{10 k}\) race? The lengths of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

## Example

Nick ran a \(\text{10-kilometer}\) race. How many meters did he run?

### Solution

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in the table below.

10 kilometers | |

Multiply the measurement to be converted by 1. | |

Write 1 as a fraction relating kilometers and meters. | |

Simplify. | |

Multiply. | 10,000 m |

Nick ran 10,000 meters. |

## Example

Eleanor’s newborn baby weighed \(3200\) grams. How many kilograms did the baby weigh?

### Solution

We will convert grams to kilograms.

Multiply the measurement to be converted by 1. | |

Write 1 as a fraction relating kilograms and grams. | |

Simplify. | |

Multiply. | |

Divide. | 3.2 kilograms |

The baby weighed \(3.2\) kilograms. |

Since the **metric system** is based on multiples of ten, conversions involve multiplying by multiples of ten. In Mathematics 105, we learned how to simplify these calculations by just moving the decimal.

To multiply by \(10,100,\text{or}\phantom{\rule{0.2em}{0ex}}1000,\) we move the decimal to the right \(1,2,\text{or}\phantom{\rule{0.2em}{0ex}}3\) places, respectively. To multiply by \(0.1,0.01,\text{or}\phantom{\rule{0.2em}{0ex}}0.001\) we move the decimal to the left \(1,2,\text{or}\phantom{\rule{0.2em}{0ex}}3\) places respectively.

We can apply this pattern when we make measurement conversions in the metric system.

In the example above, we changed \(3200\) grams to kilograms by multiplying by \(\frac{1}{1000}\phantom{\rule{0.2em}{0ex}}\left(\text{or}\phantom{\rule{0.2em}{0ex}}0.001\right).\) This is the same as moving the decimal \(3\) places to the left.

## Example

Convert: \(\phantom{\rule{0.2em}{0ex}}350\) liters to kiloliters \(\phantom{\rule{0.2em}{0ex}}4.1\) liters to milliliters.

### Solution

We will convert liters to kiloliters. In the table below, we see that \(\text{1 kiloliter}=\text{1000 liters}.\)

350 L | |

Multiply by 1, writing 1 as a fraction relating liters to kiloliters. | |

Simplify. | |

Move the decimal 3 units to the left. | |

0.35 kL |

We will convert liters to milliliters. In the table below, we see that \(\text{1 liter}=1000\phantom{\rule{0.2em}{0ex}}\text{milliliters.}\)

4.1 L | |

Multiply by 1, writing 1 as a fraction relating milliliters to liters. | |

Simplify. | |

Move the decimal 3 units to the left. | |

4100 mL |