Unit Conversions (Part 1):

Unit Conversions with multiple conversion factors

Sometimes multiple conversions are needed before we end up with the units we want. For example, How many minutes are there in 2 days? The process is the same, we just repeat the steps for every unit conversion we make.

Steps for Unit Conversions

Look at the units you have.
Figure out the units you want.
Find the conversion factors that will help you step by step get to the units you want.
Arrange conversion factors so that unwanted units cancel out.

The following video will show an example of using two conversion factors.

Video Source (04:34 mins) | Transcript

Time is one of the most commonly used conversions. Depending on what you are converting between, it is also a good example of sometimes needing more than one conversion factor.

Example:

How many minutes are there in two days?

Unless you know the conversion for minutes to days, this takes two conversion factors.

We can use the following equivalence statements to make our conversion factors.

1 day = 24 hours

1 hour = 60 minutes

First, we start with what we have which is two days.

Next, we use one of our equivalence statements to make the conversion factor that will allow us to cancel out “days.”

$2\text{ days} \times \frac{24\text{ hours}}{1\text{day}}$

The conversion factor $\frac{24\:\text{hours}}{1\:\text{day}}$ allows us to cancel “days” in the numerator with “days” in the denominator.

$2\cancel{{\color{Red} \text{ days}}} \times \frac{24\text{ hours}}{1\cancel{{\color{Red} \text{day}}}}$

This leaves us with “hours” in the numerator. We still need another conversion factor to cancel out the “hours.” The conversion factor $\frac{60\text{ minutes}}{1\text{ hour}}$ allows us to cancel out “hours” in the numerator with “hours” in the denominator.

$2\cancel{{\color{Red} \text{ days}}} \times \frac{24\cancel{{\color{Blue} \text{ hours}}}}{1\cancel{{\color{Red} \text{day}}}} \times \frac{60\text{ minutes}}{1\cancel{{\color{Blue} \text{ hour}}}}$

This now leaves us with just “minutes” as our only units left.

Now that we have arrived at the units we want, we do the calculation.

$\frac{2\times24\times60 \text{ minutes}}{1}= 2880 \text{ minutes}$

Thus, two days = 2880 minutes.

Even though it took two conversion factors, we were still able to get to the answer.

The following video shows another example of multiple unit conversions, focusing on following the 4 step process.

Video Source (06:39 mins) | Transcript

Steps for Unit Conversions

Look at the units you have

Figure out the units you want

Find the conversion factors that will help you step by step get to the units you want

Arrange conversion factors so unwanted units cancel out

These are the general steps for doing unit conversions. Let’s use them in an example to demonstrate how it works.

Example: How many days is 5000 minutes?

Step 1: Look at the units you have.

We have 5000 minutes; the units we currently have are minutes.

Step 2: Figure out the units you want.

Our question wants us to find out how many days this is equivalent to. Days are the units we want to get to.

Step 3: Find the conversion factors that will help you step by step get to the units you want.

60 minutes = 1 hour

24 hours = 1 day

We use these equivalence statements to create our conversion factors to help us cancel out the unwanted units.

Step 4: Arrange the conversion factors so unwanted units cancel out.

$\left ( 5000{\cancel{\color{Magenta} \text{ minutes}}} \right ) \left ( \frac{1\cancel{{\color{DarkOrange} \text{ hour}}}}{60\cancel{{\color{Magenta} \text{ minutes}}}} \right ) \frac{1\text{ day}}{24\cancel{{\color{DarkOrange} \text{ hours}}}}$

The next key to getting the correct answer is to do the calculations correctly. This is a place where students sometimes make mistakes.

There are two ways to go about doing this calculation.

Multiply Numerators and Denominators and then divide.

The first method is to multiply everything in the numerator together, and multiply everything in the denominator together. Then divide.

(This is demonstrated in the video: Unit Conversions for Time (4:34 mins; Transcript))

$\frac{\text{Everything in numerator}}{\text{Everything in denominator}}=\frac{5000\times1\times1\:\text{day}}{60\times24}=\frac{5000}{1440}= 3.47222\text{...days}$

So 5000 minutes = 3.5 days (rounded to nearest tenth.)

Zig-Zag Method

The other method to properly calculate several fractions being multiplied together is to use the zig-zag method. The zig-zag method says to calculate the numbers going in a zig-zag pattern starting with the first numerator.

$This image shows 5000 min (1 hour over 60 min)(1 day over 24 hours) equal. The parentheses next to each other indicate multiplication between the fractions. The units of min of the first number and the units of min on the bottom of the second fraction are both crossed out. The units of hour on the top of the second fraction and the units of hours on the bottom of the third fraction are both crossed out. There is an hour pointing down from 500 to 1 on the first fraction. There is an hour pointing from 1 in the first fraction to the 1 in the second fraction. A third arrow points from the 1 in the second fraction to the 60 on the bottom of the second fraction. There is a fourth arrow pointing from the 60 in the second fraction to the 1 on the top of third fraction. A fifth arrow points from the 1 on the top of the third fraction to the 24 on the bottom of the third fraction. The last arrow points from the 24 to the equal sign.$

Any time you go down to the denominator you divide.

$This image shows the first part of the equation with the 5000 over 1. An arrow is pointing from the 5000 to the 1 on the bottom of the fraction. A box next to the fraction reads divide.$

Any time you go up to the next numerator you multiply.

$This image shows the next arrow pointing from the 1 in the first equation to the 1 in the second equation. A box next to the fractions reads multiply.$

This method makes putting the numbers into your calculator very quick. In this case, we enter the following into our calculator going from left to right:

$5000\div 1 \times 1\div 60\times 1 \div 24 = 3.47222\text{...}$

If we round to the nearest tenth, this means 5000 min = 3.5 days.

Additional Resources

Khan Academy: Converting Units: Minutes to Hours (4:53 mins | Transcript)
Khan Academy: Multi-Step Unit Conversion Examples (5:37 mins | Transcript)

Practice Problems

A trip from Los Angeles to New York by car is expected to take about four days of driving time (non-stop). How many hours will a person drive if they make this trip? Use the following information to convert this trip to hours:
1 day = 24 hours

How many seconds are there in 2.5 hours? Use the following information to convert this time to seconds:
1 hour = 60 minutes
1 minutes = 60 seconds

Sara trained for a 10-kilometer race for 18 weeks by running one hour every day. How many minutes did she run altogether during her training? Use the following information to convert her running time to minutes:
1 week = 7 days
1 hour = 60 minutes

The running time for a new children’s movie is 6600 seconds. What is the running time for the movie in hours? Use the following information to convert the running time to hours. Round to the nearest hundredth.
1 minute = 60 seconds
1 hour = 60 minutes

Michael Phelps swam the 200-meter individual medley in 1 minute and 54 seconds. How long did it take him to swim this race using seconds only? Use the following information to convert his time to seconds:
1 minutes = 60 seconds

How many hours are there in eight weeks? Use the following information to convert this time to hours:
1 week = 7 days
1 day = 24 hours

I have a 15 pound turkey. The instructions say to cook it for 12 minutes per pound. The timer uses hours. How many hours should I set the timer for? Use the following information to find how many hours to set the timer for:
60 minutes = 1 hour

Solutions

$\frac{4 \text{ days}}{1} \times \frac{24\text{ hours}}{1 \text{ day}} = 96\text{ hours}$
$\frac{2.5 \:\text{hours}}{1} \times \frac{60\:\text{minutes}}{1\:\text{hour}} \times \frac{60\:\text{seconds}}{1\:\text{minute}} = 9000\:\text{seconds}$
$\frac{18\:\text{weeks}}{1} \times\frac{7\:\text{days}}{1\:\text{week}} \times \frac{1\:\text{hr}}{1\:\text{day}} \times \frac{60\:\text{min}}{1\:\text{hr}} = 7560\:\text{min}$ (Written Solution)

Step 1: Find the units we have. In the problem we are told that Sara is running 18 weeks for one hour per day, so the units we have are weeks.

Step 2: Figure out what units we want. The problem asks us to convert weeks to minutes, so the units we want are minutes.

Step 3: Find conversion factors that will help get the units we want. We need to convert weeks to days to hours to minutes so we will need the following conversions:

1 week running = 7 days running

1 day running = 1 hour running (If we were solving for the total number of hours in 18 weeks then we would use the conversion 1 day = 24 hours. Since the runner only runs for 1 hour per day, that is the unit conversion we use instead.)

1 hour running = 60 minutes running

Step 4: Arrange conversion factors so unwanted units cancel out. We know that Sara has been training for 18 weeks, so we need to change weeks to days. To do that we multiply $\frac{18\:\text{weeks}}{1}$ by $\frac{7\:\text{days}}{1\:\text{week}}$:

$\frac{18\:\text{weeks}}{1}\times\frac{7\:\text{days}}{1\:\text{week}}$

We can now cancel out the weeks:

$\frac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\frac{7\:\text{days}}{1\:\cancel{{\color{Red} \text{week}}}}$

Since Sara is training for 1 hour per day, we can convert days to hours by multiplying by $\frac{1\:\text{hour of running}}{1\:\text{day}}$, so now we have:

$\frac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\frac{7\:\text{days}}{1\:\cancel{{\color{Red} \text{week}}}} \cdot \frac{1\:\text{hr}}{1\:\text{day}} $

We can now cancel out the days:

$\frac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\frac{7\:\cancel{{\color{Orchid} \text{days}}}}{1\:\cancel{{\color{Red} \text{week}}}} \cdot \frac{1\:\text{hr}}{1\:\cancel{{\color{Orchid} \text{day}}}} $

The last conversion factor we need to include is multiplying by $\frac{60\:\text{min}}{1\:{\color{Blue} \text{hr}}}$ to change hours to minutes. Now we have:

$\frac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\frac{7\:\cancel{{\color{Orchid} \text{days}}}}{1\:\cancel{{\color{Red} \text{week}}}} \cdot \frac{1\:{\color{Blue} \text{hr}}}{1\:\cancel{{\color{Orchid} \text{day}}}} \cdot \frac{60\:\text{min}}{1\:{\color{Blue} \text{hr}}}$

Now we can cancel out the hours:

$\frac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\frac{7\:\cancel{{\color{Orchid} \text{days}}}}{1\:\cancel{{\color{Red} \text{week}}}} \cdot \frac{1\:\cancel{{\color{Blue} \text{hr}}}}{1\:\cancel{{\color{Orchid} \text{day}}}} \cdot \frac{60\:\text{min}}{1\:\cancel{{\color{Blue} \text{hr}}}}$

At this point we can multiply the fractions together, straight across:

$\frac{18\: \cdot \: 7\: \cdot \:1 \:\cdot\: 60\:\text{min}}{1\: \cdot\: 1 \:\cdot \:1 \:\cdot\: 1}$

This simplifies to:

$\frac{7560\:\text{min}}{1}$

So Sara has trained 7,560 minutes over the past 18 weeks.
$\frac{6600\:\text{sec}}{1} \times \frac{1\:\text{min}}{60\:\text{sec}} \times \frac{1\:\text{hr}}{60\:\text{min}} = 1.83\:\text{hrs}$

Solution Video #1 - #4 | Transcript

First, convert minutes into seconds.$\frac{1\:\text{min}}{1}\times\frac{60\:\text{sec}}{1\:\text{min}}=60\:\text{sec}$
- Next, add the seconds together. 60 sec+54 sec=114 sec
$\frac{8\:\text{weeks}}{1} \times \frac{7\:\text{days}}{1\:\text{week}} \times \frac{24\:\text{hours}}{1\:\text{day}} = 1344\:\text{hours}$
$\frac{15\:\text{lb}}{1} \times \frac{12\:\text{min}}{1\:\text{lb}} \times \frac{1\:\text{hour}}{60\:\text{min}} = 3\:\text{hours}$ (Written Solution)

Step 1: Find the units we have. In this problem, we have a 15-pound turkey, so the units we have are pounds.

Step 2: Figure out what units we want. The problem asks us to find out how many hours we need to cook the turkey for, so the unit we want is hours.

Step 3: Find conversion factors that will help get the units we want. We are given the conversion factor 12 minutes = 1 pound, and we will also need 60 minutes=1 hour.

Step 4: Arrange conversion factors so unwanted units cancel out. We know that our turkey is 15 lbs, which need to be converted to minutes, then to hours. We can set it up like this:

$\frac{15\:\text{lbs}}{1}\times\frac{12\:\text{min}}{1\:\text{lb}}\times\frac{1\:\text{hour}}{60\:\text{min}}$

Next, we need to cancel out the pounds and the minutes:

$\frac{15\:\cancel{{\color{Red} \text{lbs}}}}{1}\times\frac{12\:\cancel{{\color{Blue} \text{min}}}}{1\:\cancel{{\color{Red} \text{lb}}}}\times\frac{1\:\text{hour}}{60\:\cancel{{\color{Blue} \text{min}}}}$

Then we can multiply straight across, which gives us:

$\frac{15\times12\times1\:\text{hours}}{1\times1\times60}$

Then we simplify the top and bottom of the fraction:

$\frac{180\:\text{hours}}{60}$

Then we divide 180 by 60 to get 3 hours.

$\frac{180\:\text{hours}}{60}=3\:\text{hours}$

So it takes 3 hours to cook a 15 lb turkey.

Solution Video #5 - #7 | Transcript

Unit Conversions (Part 1):

Unit Conversions with multiple conversion factors

Steps for Unit Conversions

Look at the units you have Figure out the units you want Find the conversion factors that will help you step by step get to the units you want Arrange conversion factors so unwanted units cancel out

Multiply Numerators and Denominators and then divide.

Zig-Zag Method

Additional Resources

Solutions

Look at the units you have

Figure out the units you want

Find the conversion factors that will help you step by step get to the units you want

Arrange conversion factors so unwanted units cancel out