# Unit Conversions

#### Common Questions

#### What is a unit of measurement?

Glad you asked! A unit is a defined quantity used to express the amount of something. It tells us what type of measurement we’re dealing with and its relative size.

For example, running $3yards$ is very different from running $3miles$. Both the yard and the mile are units that measure distance; however, a mile is much larger than a yard.

Read below to learn more.

#### Common Questions

Glad you asked! A unit is a defined quantity used to express the amount of something. It tells us what type of measurement we’re dealing with and its relative size. For example, running $3yards$ is very different from running $3miles$. Both the yard and the mile are units that measure distance; however, a mile is much larger than a yard. Read below to learn more. | |

Brace yourself! 😱 Just kidding! 😅 Dimensional analysis is just a sophisticated name for a simple method that uses multiplication to convert between units. Other names for dimensional analysis include: unit analysis, factor-label method, and unit factor method. Read below to learn more. | |

Great question! A conversion factor compares two quantities that are equivalent and can be arranged as a ratio fraction. Since the numerator is equivalent to the denominator, the conversion factor equals 1: $12inches 1foot =12inches12inches =1foot 1foot =1$ It’s important to remember that conversion factors ALWAYS equal $1$. Read below to learn more. | |

For you?! Absolutely! Let’s convert $3yards$ to $inches$ using the following conversion factors: $1yard=3feet$ and $1foot=12inches$ $13 yds =1 yd 3 ft =1 ft 12in =1×1×13×3×12in =108in$ Notice that we only cancel out the units not the numbers they’re associated with! Read below to learn more. | |

Great question! The main difference between the two systems is that the metric system is made up of units based on powers of $10$ (ex: $1m=10_{2}cm=10_{3}mm$). English units don’t have a predictable relationship to each other, so converting units within the English system is a bit more complex. Read below to learn more. |

### What are units of measurement and why do they matter?

Simply put, a unit tells us what type of measurement we’re dealing with and its relative size.If you were to give someone directions to your favorite bodega, you would probably say something like, “It’s $2$ blocks down this way.” You wouldn’t just say, “It’s $2$ this way”.

Leaving off the unit (blocks in this case) makes it unclear how much further this person needs to keep walking. A block doesn’t necessarily have a defined size, but if you’ve lived in a city, you have a general sense of the distance it represents.

However, not everyone has lived in a city, which leaves room for confusion. This is why we have more standardized systems of measurement like the metric and English systems.

### Metric vs. English

The main difference between the two systems is that the metric system is made up of units based on powers of $10$:

English units don’t have a predictable relationship to each other, so converting units within the English system is a bit more complex.

This is why most of the world uses metric units... except for us (🇺🇸) and two other countries (Liberia and Myanmar). Check out this link to learn more about why the US hasn’t adopted the metric system.

Despite their differences we can convert metric and English units using a single method called dimensional analysis 🔥.

Side note: the US customary (or standard) system is the updated version of the English system used in America. The international System of Units (Système international d’unités or SI) is the current version of the metric system.

### How do we convert between units?

We use dimensional analysis to convert between units. It sounds super fancy and complicated, but really it’s just the process of multiplying a measurement by one (or more) conversion factors in order to express it using a different set of units.

A conversion factor is a ratio that compares two quantities that are equivalent. If we look at measuring tape, we can see that $1foot= 12inches$. This is an example of a conversion factor.

I can rewrite the ratio, $12in.1ft. $, by replacing $1ft.$ with $12in.$

$12in.1ft. =12in.12in. =1$Since the numerator is equivalent to the denominator, conversion factors ALWAYS equal $1$. This means we can multiply any measurement by a conversion factor and it won’t change the value of the measurement, it just changes how it looks.

This is similar to how we can express our thoughts and/or feelings using emojis in place of words. If we’re not in the mood to type out a text that reads “but that’s none of my business,” we can simply send “🐸☕” and call it a day.

Both options carry the same meaning; however, the way that meaning is expressed changes as we switch from one system (words) to another (emojis).

### How do we use dimensional analysis?

We can summarize dimensional analysis into five steps:

Step 1: Identify the given units: (1) original unit (2) desired unit

Step 2: Identify the fewest number of conversion factor(s) that will help you get from your original unit to your desired unit

Step 3: Set up your equation so that your undesired units cancel out to give you your desired unit

Step 4: Multiply across the top and the bottom to get values for the numerator and denominator

Step 5: Divide the numerator by the denominator to get your final answer

Pause ✋🏾✋🏽✋🏻 I feel like an example would be super clutch in helping us better understand this.

### Dimensional Analysis Example Problem

Absolutely! Let’s walk you through one example first, and then we’ll do a second one together!

How many minutes are in $1year$?

Step 1: Identify the given units: (1) original unit (2) desired unit

Original: year (yrs)

Desired: minutes (min)

Step 2: Identify the fewest number of conversion factor(s) from the options below that will help you get from your original unit to your desired unit

In order to go from years to minutes using the fewest number of conversion factors from below, we need to take the following path:

years → days → hours → minutes → seconds

This requires **4** conversion factors to go from years to days, days to hours, hours to minutes, and minutes to seconds. The green factors below are the factors we need to do this. See how each one matches with one of our steps in the path?

It’s important to note that this isn’t the only path that can take you from years to minutes. In general, you want to take the shortest path using what you know, which can look differently from person to person.

Step 3: Set up your equation so that your undesired units cancel out to give you your desired unit

Remember, in order for a unit to cancel out, it needs to show up on the top and bottom an equal number of times.

Step 4: Multiply across the top and the bottom to get values for the numerator and denominator

Notice that we were able to leave off some of the units because they canceled out in the previous step.

Step 5: Divide the numerator by the denominator to get your final answer

In this case, the denominator was equal to one but that’s not always going to be the case. It always depends on how we arrange the conversion factors in order to cancel out undesired units.

Huh, turns out there really are $525,600minutes$ in a year 🎵

Pause ✋🏾✋🏽✋🏻 If dimensional analysis doesn’t change the value of a measurement, how did we go from $1$ to $525,600$???

Sharp observation! Let’s take a step back and consider something we’re all familiar with: language. America is known as a melting pot because it consists of people who come from all over the world. This means that you’re likely to run into someone who speaks a different language than you do.

Luckily we have translating resources like Google Translate that help us convert words, as simple as “hello,” into a language we can understand. Though the words change from language to language, they still carry the same meaning or in math terms, value.

English | Hello |

Mohawk | Sago |

Spanish | Hola |

French | Bonjour |

Chinese | Nǐ hǎo 你好 |

Korean | Annyeong 안녕 |

Emoji | 👋🏻 |

Similarly, when we convert measurements from one unit to another, we’re not changing the meaning or value. We’re still dealing with the same amount of time, length, mass, temperature, etc., we’re just expressing it in another way. In the case of the previous example, the timespan of $1year$ can also be expressed as $525,600minutes$.

Let’s tackle some more conversion problems using our interactive unit conversion calculator! Learning is doing 💡

### Unit Conversion Calculator

What type of measurement do you want to convert?

Convert to

Step 1: Identify the two givens:

Original: ()

Desired: ()

Based on what you entered, we need to convert to . If everything looks correct, let’s move on to step 2. Otherwise, go back up and adjust your input so that the givens correctly display what we need to convert.

Step 2: Identify the fewest number of conversion factor(s) that will help you get from your original unit to your desired unit:

In order to convert ** ** to , using the fewest number of conversion factors from below, we need to take the following path:

Select the conversion factors that match this path. Hint: the first factor should include the first two units in our path, and we need **0** total factors.

Step 3: Set up your equation so that your undesired units cancel out to give you your desired unit:

Remember, in order for a unit to cancel out, it needs to show up on the **top and bottom** an equal number of times. Start by choosing the conversion factor that lets you cancel out .

Below you can see how the undesired units cancel out:

You’ve just completed the hardest step!

There’s only two steps left, but since you’ve taken care of the setup, we’ll take care of the multiplication and division in steps 4 and 5.

Step 4: Multiply across the top and the bottom to get values for the numerator and denominator:

Notice that we were able to leave off some of the units because they canceled out in the previous step.

Step 5: Divide the numerator by the denominator to get your final answer:

We'll take care of the division for you, but if you’re interested in reviewing how you can use long division to solve this, check out our long division calculator ▶️.

Congratulations! You just successfully converted to ! When you take it step by step, dimensional analysis is not as bad as it seems 😉