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Unit Conversions with Multiple Conversion Factors
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Introduction


In this lesson, you will learn how to perform a unit conversion when it requires more than one unit conversion.

Steps for Unit Conversions

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

These videos illustrate the lesson material below. Watching the videos is optional.


Unit Conversions for Time

Time is one of the most commonly used conversions. Sometimes, it can also be a good example of needing more than one conversion factor.

Example 1
How many minutes are in two days?

Unless you know the conversion for minutes to days, this takes two conversion factors. You can use the following equivalence statements to make the conversion factors.

  • \(1\; day = 24 \;hours\)
  • \(1\; hour = 60\; minutes\)

First, start with known information, which is 2 days. Next, use one of the equivalence statements to make the conversion factor that will allow you to cancel out “days.”
\begin{align*} 2\text{ days} \times \frac{24\text{ hours}}{1\text{ day}} \end{align*}

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

\begin{align*} 2\cancel{{\color{Red} \text{ days}}} \times \frac{24\text{ hours}}{1\cancel{{\color{Red} \text{day}}}} \end{align*}

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

\begin{align*} 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}}}} \end{align*}

This leaves you with just “minutes” as the only units left. Now that you have arrived at the units you want, do the calculation.

\begin{align*} \frac{2\times24\times60 \text{ minutes}}{1}= 2880 \text{ minutes} \end{align*}

Thus, \(2\; days = 2880 \; minutes\).

Example 2
How many days is 5000 minutes?

Here are the equivalence statements that will help you get to the units you want.
\begin{align*} 60\; minutes = 1 \;hour\\\\ 24\; hours = 1\; day \end{align*}

Use these equivalence statements to create conversion factors in a way that unwanted units will cancel out. Then, add the known information and multiply it by the conversion factors:
\begin{align*} \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}}}} \end{align*}

Be sure to do the calculations correctly. This is a place where students sometimes make mistakes. There are two methods to calculate:

  • Method 1: Multiply Numerators and Denominators and then Divide
  • Method 2: Zig-Zag Method

The first method is to multiply everything in the numerator together, and multiply everything in the denominator together, and then divide.
\begin{align*} \frac{\text{Everything in numerator}}{\text{Everything in denominator}}=\frac{5000\times1\times1\:\text{day}}{60\times24}=\frac{5000}{1440}= 3.47222\text{...days} \end{align*}

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

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.

Figure 1

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.

Figure 2

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.

Figure 3

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

\begin{align*} 5000\div 1 \times 1\div 60\times 1 \div 24 = 3.47222\text{...} \end{align*}

If you round to the nearest tenth, this means \(5000\; min = 3.5\; days\).


Things to Remember


  • There must be a unit in the denominator and the same unit in the numerator of the next fraction for the units to cancel out.
  • When calculating, use one of the following methods:
    • Multiply the numerator. Multiply the denominator. Divide.
    • Use the zig-zag method: Divide numerator and denominator. Multiply when going up from the denominator to the numerator of the next fraction.

Practice Problems

Video Solution for Problems 1 - 4
| Transcript
1. 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
(
Solution
x
Solution: \(\displaystyle\frac{4 \text{ days}}{1} \times \frac{24\text{ hours}}{1 \text{ day}} = 96\text{ hours}\)
)
2. How many seconds are there in 2.5 hours? Use the following information to convert this time to seconds:
1 hour = 60 minutes
1 minute = 60 seconds
(
Solution
x
Solution: \(\displaystyle\frac{2.5 \:\text{hours}}{1} \times \frac{60\:\text{minutes}}{1\:\text{hour}} \times \frac{60\:\text{seconds}}{1\:\text{minute}} = 9000\:\text{seconds}\)
)
3. 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
(
Solution
x
Solution: \(\displaystyle\frac{18\:\text{weeks}}{1} \times\frac{7\:\text{days}}{1\:\text{week}} \times \frac{1\:\text{hr}}{1\:\text{day}} \times \frac{60\:\text{min}}{1\:\text{hr}} = 7560\:\text{min}\)
Details:
Step 1: Find the units you have. In the problem you are told that Sara is running 18 weeks for one hour per day, so the units you have are weeks.

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

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

1 week running = 7 days running

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

1 hour running = 60 minutes running

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

\(\dfrac{18\:\text{weeks}}{1}\times\dfrac{7\:\text{days}}{1\:\text{week}}\)

Now cancel out the weeks:

\(\dfrac{18\:\cancel{{\color{Red} \text{weeks}}}}{1} \cdot\dfrac{7\:\text{days}}{1\:\cancel{{\color{Red} \text{week}}}}\)

Since Sara is training for 1 hour per day, you can convert days to hours by multiplying by \(\dfrac{1\:\text{hour of running}}{1\:\text{day}}\), so now you have the following:

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

Now cancel out the days:

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

The last conversion factor you need to include is multiplying by \(\dfrac{60\:\text{min}}{1\:{\color{Blue} \text{hr}}}\) to change hours to minutes. Now you have the following:

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

Now cancel out the hours:

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

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

\(\dfrac{18\: \cdot \: 7\: \cdot \:1 \:\cdot\: 60\:\text{min}}{1\: \cdot\: 1 \:\cdot \:1 \:\cdot\: 1}\)

This simplifies to the following:

\(\dfrac{7560\:\text{min}}{1}\)

Sara has trained \(7,560\) minutes over the past 18 weeks.
)
4. 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
(
Solution
x
Solution: \(\displaystyle\frac{6600\:\text{sec}}{1} \times \frac{1\:\text{min}}{60\:\text{sec}} \times \frac{1\:\text{hr}}{60\:\text{min}} = 1.83\:\text{hrs}\)
)

Video Solution for Problems 5 - 7
| Transcript
5. 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
(
Solution
x
Solution:
First, convert minutes into seconds.
\(\displaystyle\frac{1\:\text{min}}{1}\times\frac{60\:\text{sec}}{1\:\text{min}}=60\:\text{sec}\)

Next, add the seconds together.
\(60 {\text{sec}} +54 {\text{sec}} =114 {\text{sec}}\)
)
6. 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
(
Solution
x
Solution: \(\displaystyle\frac{8\:\text{weeks}}{1} \times \frac{7\:\text{days}}{1\:\text{week}} \times \frac{24\:\text{hours}}{1\:\text{day}} = 1344\:\text{hours}\)
)
7. 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
(
Solution
x
Solution: \(\displaystyle\frac{15\:\text{lb}}{1} \times \frac{12\:\text{min}}{1\:\text{lb}} \times \frac{1\:\text{hour}}{60\:\text{min}} = 3\:\text{hours}\)
Details:
Step 1: Find the units you have. In this problem, you have a 15-pound turkey, so the units you have are pounds.

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

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

Step 4: Arrange conversion factors so unwanted units cancel out. You know that the turkey is 15 lbs, which needs to be converted to minutes, then to hours. Set it up like the following:

\(\displaystyle\frac{15\:\text{lbs}}{1}\times\frac{12\:\text{min}}{1\:\text{lb}}\times\frac{1\:\text{hour}}{60\:\text{min}}\)

Next, cancel out the pounds and the minutes:

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

Then multiply straight across, which gives you the following:

\(\dfrac{15\times12\times1\:\text{hours}}{1\times1\times60}\)

Then simplify the top and bottom of the fraction:

\(\dfrac{180\:\text{hours}}{60}\)

Then divide 180 by 60 to get 3 hours.

\(\dfrac{180\:\text{hours}}{60}=3\:\text{hours}\)

It takes 3 hours to cook a 15 lb turkey.
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