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# 4.7: Metric Units of Measure

Difficulty Level: At Grade Created by: CK-12

## Introduction

Everest in Metrics

Josh and his sister Karen were working on homework when the topic of Everest and the metric system came up.

“What about meters?” Karen asked. “How many meters high is Mount Everest?”

“Why are you always thinking of things that cause me more work?” Josh asked, but then he smiled at Karen. “It’s alright. I was thinking of that today anyway.”

“How can we figure it out?” Karen asked.

“Well first, we need to know how many feet are in 1 meter. I already looked that up online, and I found out that there are 3.28 feet in 1 meter. Now I know that the height of Mount Everest is 29,035 feet high, so we can work from there,” Josh explained.

“Yeah, but how?”

“Well, we can use proportions.”

Let’s stop right there. With everything that you have learned in this chapter, do you know how to use a proportion to figure out this metric conversion? Well, pay attention to this lesson and if you aren’t sure how to do it now, you will be by the end of the lesson.

What You Will Learn

In this lesson you will learn how to do the following skills.

• Convert among metric units of measure.
• Revise or interpret scale drawings, maps, recipes and models involving conversion of metric units of measure.
• Solve real-world problems involving scaling of metric units of measure.
• Translate between metric and customary units of measure in scale drawings, maps, recipes and models using common benchmarks.

Teaching Time

I. Convert among Metric Units of Measure

The metric system of measurement is the primary measurement system in many countries; it contains units such as meters, kilometers and liters. You can remember the conversions by learning the prefixes: Milli-means thousandth, centi-means hundredth, and kilo-means thousand. So a millimeter is one-thousandth of a meter, and a kilometer is one thousand meters. You should know these common units of measurement.

Write these units of measurement down in your notebooks.

Now that you have reviewed these units of measurement, we can look at converting among the different units of measurement. Just like we used proportions when we converted among customary units of measurement, we can use proportions and ratios here too.

How do we use proportions to convert among metric units of measure?

First, set up the proportion in the same way you used to find actual measurements from scale drawings. Use the conversion factor as the first ratio, and the known and unknown units in the second ratio.

Let’s look at an example.

Example

How many centimeters are in 5 meters?

First, set up a proportion. The conversion factor is the number of centimeters in 1 meter. We can look at the chart above and see that there are 100 centimeters in 1 meters. That is our first ratio: \begin{align*}\frac{100 \ centimeters}{1 \ meter}\end{align*}.

Now write the second ratio. The known unit is 5 meters. The unknown unit is \begin{align*}x\end{align*} centimeters. Make sure that the second ratio follows the form of the first ratio: centimeters over meters.

\begin{align*}\frac{100 \ centimeters}{1 \ meters} = \frac{x \ centimeters}{5 \ meters}\end{align*}

Now cross-multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}(1)x &= 100(5)\\ x &= 500\end{align*}

There are 500 centimeters in 5 meters.

Example

Henry is making a recipe for lemonade that uses 2 liters of water. If he makes 3 batches of the recipe, how many milliliters of water will she need?

First find the total number of liters she needs. If there are 2 liters in one batch, and she is making 3 batches, then she will need \begin{align*}2 \times 3 = 6 \ liters\end{align*}.

Next, set up a proportion. The conversion factor is the number of milliliters in a liter.

\begin{align*}\frac{1000 \ milliliters}{1 \ liter}\end{align*}

Now write the second ratio, making sure it follows the form of the first ratio.

\begin{align*}\frac{1000 \ milliliters}{1 \ liter} = \frac{x \ milliliters}{6 \ liters}\end{align*}

Cross-multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}(1)x &= 1000(6)\\ x &= 6000\end{align*}

She will need 6000 milliliters of water.

Once you know what you are comparing, you can easily set up a proportion and solve for the missing measurement. Now let’s look at how we can apply conversions to scale.

II. Revise or Interpret Scale Drawings, Maps, Recipes and Models Involving Conversion of Metric Units of Measure

In the last lesson, we used customary units of measurement when working with scale drawings, maps, recipes and models. Sometimes, especially when not in the United States, these same things will have metric units of measurement. We can use our conversions to work with the metric units of measure and figure out the missing measurements.

Let’s look at an example.

Example

A scale model of a building has a height of 1.5 meters. The scale of the model is 1 cm = 0.5 m. What is the actual height of the building?

The scale is in centimeters, but the scale model height is given in meters. First convert the scale height to centimeters. Then find the height of the building.

\begin{align*}\frac{1 \ meter}{100 \ centimeters} = \frac{1.5 \ meters}{x \ centimeters}\end{align*}

Now cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}(1)x &= 100(1.5)\\ x &= 150\end{align*}

The height of the scale model is 150 centimeters. Now find the height of the actual building.

\begin{align*}\frac{1 \ centimeter}{0.5 \ meter} = \frac{150 \ centimeters}{x \ meters}\end{align*}

Next cross multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}(1)x &= 150(0.5)\\ x &= 75\end{align*}

The actual building is 75 meters tall.

Example

Jessica works in a science lab. She needs to convert the liquid measure that she is working with from liters to milliliters. She has been given 3.5 liters to convert. If each container that Jessica has holds 100 milliliters, how many containers will she need?

First, notice that there are two parts to this problem. First, let’s figure out how many milliliters are equal to 3.5 liters.

There are 1000 milliliters in one liter.

\begin{align*}\frac{1000 \ mL}{1 \ L}= \frac{x}{3.5 \ Liters}\end{align*}

Next, we cross multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}3500 \ mL = x\end{align*}

Now we can work on figuring out the number of containers Jessica will need. Each container holds 100 mL. We can divide 3500 mL by 100 mL.

\begin{align*}3500 \div 100 = 35 \ containers\end{align*}

Jessica will need 35 containers to hold all of the liquid.

Once you know how to complete the conversions using proportions and ratios, you can apply this skill to any item with a scale. Notice that even when went from units of measurement to units of capacity that the method of solving the problem did not change. Just be sure that you apply the correct units of measure to each problem you are solving.

III. Solve Real – World Problems Involving Scaling of Metric Units of Measure

When we work with real – world problems, we sometimes need to create a scale to measure. This happens whether we use customary units of measurement or metric units of measurement. To create a scale, we first need to determine the correct unit of measurement that we need to use.

Look at this example.

Example

Kyle is going to be traveling with his family over the winter holidays. He wants to figure out how many kilometers it is from his home in Cincinatti to his grandparents home in Chicago. Which unit of measurement should Kyle use?

First, let’s think about the correct unit of measurement for Kyle to use. If Kyle is measuring a far distance, he needs a measure of length. We know that the metric units for measuring length are millimeters, centimeters, meters and kilometers. Kyle is measuring the distance between two cities. It makes the most sense for him to use the largest unit for measuring length, and that is kilometers.

Kyle would use kilometers to measure the distance.

Example

Marcy is making beef stew. Her recipe calls for 900 grams of beef. She looks in the refrigerator and sees that she has 1.5 kilograms of beef wrapped in a package. Marcy isn’t sure how much of the beef she should use. Figure out how much of the beef Marcy needs for her recipe.

First, let’s think about the difference between grams and kilograms. We can call this scaling because we are comparing one measurement to another.

There are 1000 grams in 1 kilogram. We can write that as our first ratio.

\begin{align*}\frac{1000 \ grams}{1 \ kilogram}\end{align*}

Now we know that Marcy has 1.5 kilograms of beef and she needs 900 grams. Next, we need to convert the kilograms that Marcy has to grams so we can figure out how much of the whole she will need.

We write a proportion.

\begin{align*}\frac{1000 \ g}{1 \ kg} = \frac{x}{1.5}\end{align*}

Next, cross multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}1500 \ g = x\end{align*}

Now let’s think about Marcy. She has 1500 grams of meat, but only needs 900 grams. She will have 600 grams of meat left over.

IV. Translate between Metric and Customary Units of Measure in Scale Drawings, Maps, Recipes and Models using Common Benchmarks

You should also know how to convert between the metric and the customary systems of measurement. Any of these conversions will be estimates, because you cannot make an exact measurement when converting between systems of measurement.

There are some common benchmarks you can use.

Length

An inch is about 2.5 centimeters.

A meter is slightly longer than a yard.

A kilometer is about 0.6 of a mile.

Capacity

A liter is about the same as a quart.

Mass

A kilogram is a little more than 2 pounds.

Write these benchmarks down in your notebook.

Now let’s look at how we can apply these benchmarks to some real – life examples.

Example

Omar has a dog that weighs 30 pounds. About how many kilograms does Omar’s dog weigh?

Use an estimated conversion factor to write a ratio. We will compare kilograms to pounds. One kilogram is about 2 pounds, or \begin{align*}\frac{1 \ kilogram}{2 \ pounds}\end{align*}.

Next, write a proportion and solve.

\begin{align*}\frac{1 \ kilogram}{2 \ pounds} &= \frac{x \ kilograms}{30 \ pounds}\\ x(2) &= 1(30)\\ 2x &= 30\\ x &= 15\end{align*}

Omar’s dog weighs about 15 kilograms.

Example

Randy ran a 20 kilometer race. How many miles did Randy run?

To figure this out, we have to use a comparison between kilometers and miles. A kilometer is about .6 of a mile. That is our first ratio.

\begin{align*}\frac{1 \ km}{.6 \ mile}\end{align*}

Next, we have to look at how many kilometers, Randy actually ran. He ran 20 kilometers. We are looking for his miles. That forms our second ratio, and we can now write a proportion.

\begin{align*}\frac{1 \ km}{.6 \ mile} = \frac{20 \ km}{x \ miles}\end{align*}

Lastly, we cross multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}20 \times .6 = x \ miles\end{align*}

Randy ran 12 miles in his race.

Once again notice that we solved these problems in the same way by using ratios and proportions. As long as you keep in mind what you are comparing, you can solve any measurement conversion problem in this way! Now let’s go and solve the problem from the introduction.

## Real-Life Example Completed

Everest in Metrics

Here is the original problem once again. First reread it. Then write a proportion that you could use to solve the problem. Finally, solve it for the number of meters.

Josh and his sister Karen were working on homework when the topic of Everest and the metric system came up.

“What about meters?” Karen asked. “How many meters high is Mount Everest?”

“Why are you always thinking of things that cause me more work?” Josh asked, but then he smiled at Karen. “It’s alright. I was thinking of that today anyway.”

“How can we figure it out?” Karen asked.

“Well first, we need to know how many feet are in 1 meter. I already looked that up online, and I found out that there are 3.28 feet in 1 meter. Now I know that the height of Mount Everest is 29,035 feet high, so we can work from there,” Josh explained.

“Yeah, but how?”

“Well, we can use proportions.”

Now write a proportion and solve it. Remember, there are two parts to your answer.

Solution to Real – Life Problem

First, let’s write a proportion. Josh told us that there are 3.28 feet in 1 meter. That is our first ratio in the proportion.

\begin{align*}\frac{3.28}{1}\end{align*}

Next, we write the second ratio. That compares the unknown number of meters, our variable with the current height of Everest in feet.

\begin{align*}\frac{29,035}{x}\end{align*}

Our proportion is:

\begin{align*}\frac{3.28}{1}= \frac{29035}{x}\end{align*}

Next, we cross multiply and solve.

The answer is that Mount Everest is about 8852 meters high. We did need to round the answer, so that is why we used the word “about” in our answer.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Metric System
a system of measurement commonly used outside of the United States. It contains units such as meters, milliliters and grams.

## Time to Practice

Directions: Solve each problem.

1. 3 km = _____ m
2. 2000 m = _____ km
3. 5.5 km = _____ m
4. 2500 m = _____ km
5. 12000 m = _____ km
6. 500 cm = _____ m
7. 6000 cm = _____ m
8. 4 m = _____ cm
9. 11 m = _____ cm
10. 50 mm = _____ cm
11. 3 cm = _____ mm
12. 15 cm = _____ mm
13. 2000 g = _____ kg
14. 35000 g = _____ kg
15. 7 kg = _____ g
16. 56 meters = _____ centimeters
17. 41 milliliters = _____ liters
18. 857 grams = _____ kilograms
19. 4.7 kilometers = _____ millimeters

Directions: Solve each problem.

1. A recipe calls for 400 grams of flour. If Leena makes one quarter of the recipe, how many kilograms of flour will she need?
2. Two buildings are 9 centimeters apart on a map. The scale of the map is 0.5 centimeter = 2 kilometers. What is the actual distance between the two buildings in meters?
3. A scale model of a tower is 1.25 meters tall. The scale of the model is 0.5 cm = 5 meters. What is the actual height of the tower in meters?
4. A scale drawing of a conference center includes a meeting room that measures 1.5 centimeters by 2.5 centimeters. If the scale of the drawing is 1 centimeter = 2 meters, what is the area of the meeting room in square centimeters?
5. A shipping company uses boxes that are \begin{align*}2 \frac{1}{2} \ feet\end{align*} long. About how many centimeters long are the boxes?
6. Samir ran a race that was 10 kilometers long. About how many feet did Samir run?

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