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Study Guides > Mathematics for the Liberal Arts Corequisite

Applications of Metric Conversions

Learning Outcomes

  • Solve application problems using metric units
  • Convert between the U.S. customary units and metric units of length, weight/mass, and volume
Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.
TIP: To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.

Understanding Context and Performing Conversions

The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.

Example

Marcus bought at 2 meter board, and cut off a piece 1 meter and 35 cm long. How much board is left?

Answer: To answer this question, we will need to subtract. First convert all measurements to one unit. Here we will convert to centimeters. [latex-display]2\text{ meters}-1\text{ meter and }35\text{ cm}[/latex-display] Use the factor label method and unit fractions to convert from meters to centimeters. [latex-display] \displaystyle \frac{2\text{ m}}{1}\cdot \frac{100\text{ cm}}{1\text{ m}}=\text{ cm}[/latex-display] Cancel, multiply, and solve. Convert the 1 meter to centimeters, and combine with the additional 35 centimeters. Subtract the cut length from the original board length. [latex-display] \displaystyle \frac{2\ \cancel{\text{m}}}{1}\cdot \frac{100\text{ cm}}{1\ \cancel{\text{ m}}}=\text{ cm}[/latex-display] [latex-display] \displaystyle \frac{200\text{ cm}}{1}=200\text{ cm}[/latex-display] [latex-display]1\text{ meter}+35\text{ cm}[/latex-display] [latex-display]100\text{ cm}+35\text{ cm}[/latex-display] [latex-display]135\text{ cm}[/latex-display] [latex-display]200\text{ cm}-135\text{ cm}[/latex-display] [latex-display]65\text{ cm}[/latex-display] There is 65 cm of board left.

An example with a different context, but still requiring conversions, is shown below.

Example

A faucet drips 10 ml every minute. How much water will be wasted in a week?

Answer: Start by calculating how much water will be used in a week using the factor label method to convert the time units. [latex-display] \displaystyle \frac{10\ ml}{1\text{ minute}}\cdot \frac{60\text{ minute}}{1\text{ hour}}\cdot \frac{24\text{ hours}}{1\text{ day}}\cdot \frac{7\text{ days}}{1\text{ week}}[/latex-display] Cancel, multiply and solve. [latex-display] \displaystyle \frac{10\ ml}{1\text{ }\cancel{\text{minute}}}\cdot \frac{60\text{ }\cancel{\text{minute}}}{1\text{ }\cancel{\text{hour}}}\cdot \frac{24\text{ }\cancel{\text{hours}}}{1\text{ }\cancel{\text{day}}}\cdot \frac{7\text{ }\cancel{\text{days}}}{1\text{ week}}[/latex-display] [latex-display] \displaystyle \frac{10\centerdot 60\centerdot 24\centerdot 7\ ml}{1\centerdot 1\centerdot 1\centerdot 1\text{ week}}[/latex-display] To give a more useable answer, convert this into liters. [latex-display] \displaystyle \frac{100800\ ml}{1\text{ week}}[/latex-display] Cancel, multiply and solve. [latex-display] \displaystyle \frac{100800\text{ ml}}{1\text{ week}}\centerdot \frac{1\text{ L}}{1000\text{ ml}}[/latex-display] [latex-display] \displaystyle \frac{100800\text{ }\cancel{\text{ml}}}{1\text{ week}}\centerdot \frac{1\text{ L}}{1000\text{ }\cancel{\text{ml}}}[/latex-display] [latex-display] \displaystyle \frac{100800\text{ L}}{1000\text{ week}}[/latex]=[latex] \displaystyle 100.8\frac{\text{L}}{\text{week}}[/latex-display] The faucet wastes about 100.8 liters each week.

This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.

Try It

A bread recipe calls for 600 g of flour. How many kilograms of flour would you need to make 5 loaves?

Answer: Multiplying 600 g per loaf by the 5 loaves, [latex-display]600\text{g}\cdot5=3000\text{g}[/latex-display] Using factor labels or the move the decimal method, convert this to 3 kilograms. You will need 3 kg of flour to make 5 loaves.

[ohm_question]1002-117516[/ohm_question]

Conversions between U.S. and Metric Measurement Systems

Sometimes we are given a measurement in metric units but needs to know that value in U.S. customary units or vice versa. Maybe you purchased a fancy French cookbook that asks you to measure out flour in grams, but you only have a bag labeled with the weight in pounds. Or maybe you're having a custom sari sewn for you in India for an upcoming wedding but you only know your height in feet and inches and the seamstress would like it in centimeters. Below are some common conversion factors between U.S. and metric units. Notice that each value notes "approximately" because the decimal value has been rounded to make the math accurate enough but not tedious and unreasonable.
Feet (ft) Meter (m) There are approximately 3.28 feet in a meter.
Inch (in) Centimeter (cm) There are approximately 2.54 centimeters in an inch.
Pound (lb) Kilogram (kg) There are approximately 2.2 pounds in a kilogram.
Gallon (gal) Liter (L) There are approximately 3.785 liters in a gallon.

Try It

[ohm_question]1009[/ohm_question]

Checking your Conversions

Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.

Example

A bottle contains 1.5 liters of a beverage. How many 250 mL servings can be made from that bottle?

Answer: To answer the question, you will need to divide 1.5 liters by 250 milliliters. To do this, convert both to the same unit. You could convert either measurement. [latex-display]1.5\text{ L}\div250\text{ mL}[/latex-display] Convert 250 mL to liters [latex-display]250\text{ mL}=\text{ ___ L}[/latex-display] [latex-display] \displaystyle \frac{250\text{ mL}}{1}\cdot \frac{1\text{ L}}{1000\text{ mL}}=\text{___ L}[/latex-display] [latex-display] \displaystyle \frac{250\text{ L}}{1000=0.25\text{ L}}[/latex-display] Now we can divide using the converted measurement [latex-display]1.5\text{ L}\div250\text{ mL}=\frac{1.5\text{ L}}{250\text{ mL }}=\frac{1.5\text{ L}}{0.25\text{ L}}[/latex-display] [latex-display] \displaystyle \frac{1.5\text{ L}}{0.25\text{ L}}=6[/latex-display] The bottle holds 6 servings.

Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the “move the decimal” method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.

Summary

The metric system is an alternative system of measurement used in most countries, as well as in the United States. The metric system is based on joining one of a series of prefixes, including kilo-, hecto-, deka-, deci-, centi-, and milli-, with a base unit of measurement, such as meter, liter, or gram. Units in the metric system are all related by a power of 10, which means that each successive unit is 10 times larger than the previous one. This makes converting one metric measurement to another a straightforward process, and is often as simple as moving a decimal point. It is always important, though, to consider the direction of the conversion. If you are converting a smaller unit to a larger unit, then the decimal point has to move to the left (making your number smaller); if you are converting a larger unit to a smaller unit, then the decimal point has to move to the right (making your number larger). The factor label method can also be applied to conversions within the metric system. To use the factor label method, you multiply the original measurement by unit fractions; this allows you to represent the original measurement in a different measurement unit.

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