Converting Between Measurement Systems - Examples

Comparing Quantities with Different Units

Sometimes you will be given quantities with different units of measure (e.g. metric units), in different measurement systems (e.g the imperial system of measure, the International System of Units, the metric system, the customary system), and asked to compare them. For these types of problems, use unit rates to follow these two steps:

1. Do a conversion between measurement systems so that all quantities have the same units.
2. Compare!

Here's an example where we convert between miles and kilometers.

Let's do another example!

Jane is $162.56 \text{ cm}$ tall. Marie is $65.75 \text{ inches}$ tall. Who is taller? $(\text{Hint: } 1 \text{ inch} = 2.54 \text{ cm})$

Step One: Convert to the Same Units

Let's convert $65.75 \text{ inches}$ to $\text{centimeters}$. (Note: we could also convert from centimeters to inches if we wanted to.)

$$\begin{align} &65.75 \text{ inches}=65.75 \text{ inches} \times \dfrac{2.54 \text{ cm}}{1 \text{ inch}}\\ &=65.75 \cancel{\text{ inches}} \times \dfrac{2.54 \text{ cm}}{1 \cancel{\text{ inch}}} \\ &=65.75 \times 2.54 \text{ cm}\\ &=167.005\text{ cm} \end{align}$$

Step Two: Compare Values

Marie is $65.75 \text{ inches} = 167.005\text{ cm}$ tall, and Jane is $162.56\text{ cm}$ tall. $$167.005\text{ cm} > 162.56\text{ cm}$$ Thus, Marie is taller! More importantly, Jane is shorter!

Converting Measurements in Word Problems

Most of the time, a measurement problem is going to ask you a more involved question than just a conversion. This means that you might have to do a conversion or two and then think about what the problem is asking you in order to find the correct answer. Here is an example.

Example $1$

You buy $\color{green}{30}\text{ pounds}$ of ground beef. You make $\color{Blue}{20\text{ hamburgers}}$, using $\color{purple}{80\text{ ounces}}$ of the meat. How many hamburgers can you make with the remaining ground beef?

Solution

First, we should convert pounds into ounces. There are $16\text{ ounces}$ in $1\text{ pound}$. That means, we can make the following conversion. $${\color{green}{30 \text{ pounds}}} \times \frac{16 \text{ ounces}}{1 \text{ pound}} = 480 \text{ ounces}$$

Now, we know there are $480\text{ ounces}$ of ground beef in total. You made $\color{Blue}{20\text{ hamburgers}}$ using $\color{purple}{80\text{ ounces}}$ of beef. We want to find how many ounces of beef are used to make one hamburger. We see, $$\frac{\color{purple}{80 \text{ ounces}}}{\color{Blue}{20 \text{ hamburgers}}} = 4 \text{ ounces per hamburger}$$

Now, find how many ounces of beef we have left after those $\color{Blue}{20\text{ hamburgers}}$. We find that by subtracting $\color{purple}{80}$ from $480$. $$480-{\color{purple}{80}}=400$$

Finally, to find how many hamburgers we can make with the remaining meat, we see, $$400 \text{ ounces of meat} \times \frac{1 \text{ hamburger}}{4 \text{ ounces}} = \color{Red}{100 \text{ hamburgers}}$$

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Example $2$

Let's try another similar problem.

Solution

To solve this problem, the first thing we should do is convert $\color{Blue}{5\text{ miles}}$ into inches. $${\color{Blue}{5 \text{ miles}}}\times \frac{5{,}280 \text{ feet}}{1 \text{ mile}} \times \frac{12 \text{ inches}}{1 \text{ foot}}=316{,}800 \text{ inches}$$ The road is $316{,}800$ inches. Now, we divide that number by the length of the car to see how many cars fit on the road. $$316{,}800 \text{ inches}\times \frac{1 \text{ car}}{\color{green}{177.2 \text{ inches}}} = \color{Red}{1{,}787.8 \text{ cars}}$$

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Now, let's do one more kind of problem where we are comparing different measurements.

Example $3$

Alexis can run $1\text{ mile}$ in $15\text{ minutes}$. Ryan drives $70\text{ yards}$ in $60\text{ seconds}$. Derek rides his scooter $3.41\text{ miles}$ in $1\text{ hour}$. They are trying to get to school on time.

Solution

Alexis arrives to school first. To solve this problem, convert each distance per time into the same units, so they are easier to compare.

First, we will calculate Alexis's distance per time in miles per hour. $$\frac{1 \text{ mile}}{15 \cancel{\color{Red}{\text{ minute}}}}\times \frac{60 \cancel{\color{Red}{\text{ minutes}}}}{1 \text{ hour}} = \frac{4\text{ mile}}{\text{ hour}}$$

Now, let's calculate Ryan's distance per time in miles per hour. $$\frac{70 \cancel{\color{green}{\text{ yards}}}}{60 \cancel{\color{Red}{\text{ seconds}}}}\times \frac{3{,}600 \cancel{\color{Red}{\text{ seconds}}}}{1 \color{Blue}{\text{ hour}}} \times \frac{3 \cancel{\color{purple}{\text{ feet}}}}{1 \cancel{\color{green}{\text{ yard}}}}\times \frac{1 \text{ mile}}{5{,}280 \cancel{\color{purple}{\text{ feet}}}}= \frac{2.39\text{ miles}}{\text{hour}}$$

Derek's distance per time is given as $3.41\text{ miles per hour}$. The person going the fastest is Alexis, so Alexis arrives at school first.

Converting Units

There are many unit conversions and rates that we know and use every day, probably without thinking about them.

If someone asked you to wait $60\text{ seconds}$, you'd know they were asking you to wait $1\text{ minute}$. This is because you know that there are $60$ seconds per minute.

Also, if someone said that their baby was exactly $1\text{ year}$ old, you'd know that the child has been alive for $365\text{ days}$. This is because you know that there are $365\text{ days per year}$.

To solve a problem like this, it's important to write down all of the conversions that will be useful:

Notice how we can write these rates as fractions. Think of the line separating the numerator and denominator as meaning "per."

Next, what we need to do is line up the value in the original question ($4.5\text{ hours}$) with each of our conversions. We'll do this in a way that allows us to cancel out units.

Take a look:

If you look below, you'll see how we can cancel out some of the units. Notice how the only units that remain after all of the canceling (in red and in yellow) are seconds—the units we need in our answer.

Now, we can clean up and re-write what we've got, so we can finish up:

This means there are $16{,}200\text{ seconds}$ in $4.5\text{ hours}$.

Let's try another example.

Now, let's gather up and write down any conversions that will be helpful.

Time to line up the value in our question ($1.2\text{ feet}$) with our conversions.

Like before, we'll cancel out as many units as we can. That will leave us with:

This means that there are $36.6\text{ cm}$ in $1.2\text{ ft}$.

Example: Two Methods for Conversion

Say you need to convert between units in two measurement systems. You can do this conversion by using ratios!

Let's walk through an example of converting between kilometers and miles. We need to first figure out how many kilometers are in a mile (or vice versa), then set up a strategy to convert between the two units. $$1.609\text{ kilometers is }1\text{ mile}$$ There are different approaches to converting between the two. Figure out which one you like best and use it! I will present the ratio + cross product method.

Ratio + Cross Product

This method includes expressing the relationship in a ratio and setting up a cross product.

First, express the relationship with a ratio: $$\frac{1\text{ mile}}{1.609\text{ km}}$$ Set up the cross product. All you have to do is put in actual numbers or variables in front of $\text{miles}$ and $\text{km}$ in your second ratio! If you want to know how many miles is $5\text{ km}$, you write:

$$\frac{1\text{ mile}}{1.609\text{ km}} = \frac{x\text{ miles}}{5\text{ km}}$$

Then solve the cross product! You can dismiss the words to make the ratios simpler, if you want. $$\begin{align} \frac{1}{1.609} &= \frac{x}{5}\\ \\ x\cdot 1.609&= 5\cdot 1\\ \\ 1.609x &= 5\\ \\ x&=\dfrac{5}{1.609}\\ \\ x&\approx 3.11 \end{align}$$

So, $5\text{ km}\approx 3.11\text{ miles}.$

Canceling Units

Another method I use is "canceling units." Think about what unit you need and what unit you have, and try canceling units out (with the right ratio, of course). I use this for more complicated unit conversion problems as well.

Say we want to find out how many miles in $5\text{ km}$.

We need to find out what the conversion ratio (or rate) is. Do we multiply $5$ by $1.609$ or do we divide by $1.609$? Looking at the equation, we want to cancel out km, and end up with mile. Therefore:

Putting this in the equation:

Now all you have to do, is calculate! the answer is approximately $3.11\text{ miles}.$

(Video) Conversion Between Feet and Meters

by Timothy Palladino

Here's a good video by Timothy Palladino showing you how to convert between feet and meters.

$\large{\text{Summary}}$

For example he shows how to convert $5\text{ feet}$ to $\text{meters}.$ (Feet are from the imperial system, and meters are from the metric system.) To set up conversion tables, you first list the initial known measurement. $$\begin{array}{l}5\text{ ft}\\ \hline \text{ }\end{array}$$ Then, add a conversion between feet and a different unit. We know $1\text{ foot}$ equals $12\text{ inches}$. Since feet were previously in the numerator, the next conversion has to have feet in the denominator so the units will cancel out. So we use $\frac{12\text{ inches}}{1\text{ foot}}$. $$\begin{array}{l|l}5\text{ ft}&12\text{ in}\\ \hline \text{ }&1\text{ ft}\end{array}$$ Next, add the conversion between inches and centimeters. This is known to be $1\text{ inch}$ per $2.54\text{ centimeters}$. As we did before, since inches were previously in the numerator, the new conversion needs to have inches in the denominator. So $\frac{2.54\text{ centimeters}}{1\text{ inch}}$. $$\begin{array}{l|l|l}5\text{ ft}&12\text{ in}&2.54\text{ cm}\\ \hline \text{ }&1\text{ ft}&1\text{ in}\end{array}$$ Lastly, we need to convert a centimeter to a meter. There are $100\text{ centimeters}$ per $1\text{ meter}$. So $\frac{1\text{ meter}}{100\text{ centimeters}}$. (Notice how $\text{cm}$ are in the denominator since they were previously in the numerator.) $$\begin{array}{l|l|l|l}5\text{ ft}&12\text{ in}&2.54\text{ cm}&1\text{ m}\\ \hline \text{ }&1\text{ ft}&1\text{ in}&100\text{ cm}\end{array}$$

Now that the table is set up, you can begin by canceling out units. Once you do that, simply multiply and divide the numbers. Keep the unit that is not canceled out — that is the unit you are trying to find! $$\begin{align} &\begin{array}{l|l|l|l} 5\cancel{\text{ ft}}&12\cancel{\text{ in}}&2.54\cancel{\text{ cm}}&1\text{ m}\\ \hline \text{ }&1\cancel{\text{ ft}}&1\cancel{\text{ in}}&100\cancel{\text{ cm}}\end{array}\\ &\begin{array}{c|c|c|c}5&12&2.54&1\text{ m}\\ \hline \text{ }&1&1&100\end{array}\\ &=\frac{5\cdot12\cdot2.54\cdot1}{1\cdot1\cdot100}\text{ meters}\\ &=\frac{152.4}{100}\text {meters}\\ &=1.524\text{ meters} \end{align}$$

Let's set up our conversion.

$$\begin{array}{l|l|l}122 \cancel{\text{mi}}&1.609\cancel{\text{ km}}&1000\cancel{\text{ m}}\\ \hline \text{ }&1\cancel{\text{ mile}}&1\cancel{\text{ km}}\end{array}$$

The last step is to multiply across the top $$122 \cdot 1.609 \cdot 1000 = 196298 \text{ m}$$