Home > C The Unit Factor Method and Conversion Problems Many assignments in this course will require you to perform a mathematical calculation. For ex

**The Unit Factor
Method and Conversion Problems**

Many assignments in this course will
require you to perform a mathematical calculation. For example, ��Dr.
Gomez orders 0.08 grams of medication for a patient. Each pill contains
0.003 grams of the medicine. How many pills will the patient need?��
This problem may seem difficult at first but there is a problem-solving
method called the **unit factor method** that will guide you, step
by step, to the correct answer.

**A) Numbers and units**

Any value or quantity in a problem has two parts: A number and a unit. For example,

0.08 grams

The number The unit

The unit tells you what *type of thing*
you are talking about. The number tells you how much of that thing you
are talking about.

**B) Making
unit conversion factors**

What is a unit conversion factor? A unit conversion factor is anything that relates one quantity to another quantity. For example, the phrase ��60 seconds per minute�� is a unit conversion factor because it relates seconds to minutes. ��Peaches are $1.50 a pound�� is unit conversion factor that relates pounds of peaches to dollars. ��Four legs per dog�� is a unit conversion factor that relates legs to dogs. Unit conversion factors are important because they are what you use to solve medication problems, like the Dr. Gomez medication problem given at the start of this handout.

All unit conversion factors can be written as a fraction, with one quantity on top of the fraction and the other quantity on the bottom of the fraction. For example, the unit conversion factor ��60 seconds per minute�� can be written as:

__60 seconds__

1 minute

Notice that the word ��per�� tells
you what goes above and what goes below the fraction line. Whatever
comes before the word ��per�� always goes above the fraction line
and whatever comes after the word ��per�� always goes below the fraction
line. You can think of the word ��per�� as meaning ��a fraction line��
or ��divided by.�� In fact, the unit conversion factor written above
is read aloud by saying ��60 seconds **per** minute.��

Also notice the number 1 was put before the minute unit. Each unit must always have a number in front of it. If the phrase doesn��t tell you the number for a unit, assume the number is one.

The word ��equals�� works the same way as the word ��per.�� For example, you know the phrase ��12 inches equals one foot.�� From this you can make a conversion factor:

__12 inches__

1 foot

Just like the word ��per,�� the word ��equals�� means you should draw a fraction line. What comes before the word ��equals�� always goes above the fraction line and what comes after the word ��equals�� always goes below the fraction line.

Often the unit conversion factor you
need to solve a problem is not given to you directly in the problem.
So where does it come from? From you! You can make a unit conversion
factor from any phrase you know that contains the word ��per�� or
the word ��equals.�� You can even make unit conversion factors out
of phrases that don��t contain the words ��per�� or ��equals.�� __
Any__ phrase that relates one quantity to another is a unit conversion
factor. Some examples are given below:

Gasoline
costs $1.90 a gallon. __1.90
dollars__

(��1.90 dollars per gallon��) 1 gallon

Nickels
weigh 4.5 grams each. __4.5
grams__

(��4.5 grams per nickel��) 1 nickel

Every
day the cat eats 3 cups of cat food. __3
cups cat food____ __

(��3 cups cat food per day��) 1 day

When you look at a written unit conversion factor, whatever value is on top of the fraction line is called the numerator. Whatever value is below the fraction line is called the denominator. In the previous example, 3 cups cat food is the numerator and 1 day is the denominator.

The last thing you need to know about unit conversion factors is that you are always free to flip them upside down whenever you need to. This is called inverting them.

__12 inches__ can become��
__1 foot__

1 foot 12 inches

So unit conversion factors always come in pairs. Each member of the pair is made by inverting the numerator and denominator of the other.

__Exercise 1:__ For each unit below,
write a pair of unit conversion factors that relate it to some other
unit. The first one is done for you as an example

a)
Days __7 days__ and __1
week__ b) Pennies

1 week 7 days

c) Minutes d) Feet

**C) The unit factor conversion
method of problem solving**

As an example of how to use the unit conversion factor method to solve a problem, we will use the following problem: ��An experiment is done to see the health effect of large amounts of aspirin on rats. Each rat gets 22 grams of aspirin. If 60 rats are used in the experiment, how many grams of aspirin will be used?��

**Step 1: Write the answer unit **
Read the problem and find what units the answer must have. The answer
units are usually the units after the words ��how many�� or ��how
much.�� In our example, the problem asks ��how __many__ grams of
aspirin�� so grams of aspirin are the answer unit. Write ��= ___________
grams aspirin�� on the right side of the paper:

= __________grams aspirin

**Step 2: Find a unit conversion factor
that involves the answer unit **Read the problem and find a sentence
that relates the answer unit to something else. In our example, one
sentence reads ��Each rat gets 22 grams of aspirin.�� This sentence
relates the answer unit (grams of aspirin) to something else (rats).

**Step 3: Write a unit conversion factor
based on the sentence.** The unit conversion factor must have the *
answer units on the top*. Remember that sometimes the problem won��t
state the unit conversion factor if it is something that most people
already know (such as ��60 minutes to an hour��).

__22 grams aspirin__ = __________grams
aspirin 1
rat

**Step 4: Find a value in the problem
that has the same units that appear on the
bottom of the unit conversion factor.** In our problem, the
units that appear on the bottom of the unit conversion factor are ��rat,��
so look for a value in the problem that relates to rats. ��If 60 rats
are used in the experiment���� is the value statement relating to
rats.

60
rats x __22 grams aspirin__ = __________grams aspirin
1 rat

**Step 5: Check to see that the units
cancel.** This means that for each unit on the bottom of a fraction,
you must find the same unit at the top of a fraction. Units that are
not shown as fractions (like ��60
rats�� in this problem) are considered
to be at the top of a fraction. You can check that the units cancel
by writing a slash through each unit that appears at the bottom and
at the top of a fraction. In our example, the rat units cancel.

60
rats x __22 grams aspirin__ = __________grams aspirin
1 rat

If you have done this step correctly, the only unit that does not cancel is the answer unit (which is grams aspirin in this problem).

**Step 6:** Carry out the multiplication.
This is the step where you use your calculator. You should multiply
together all the numbers on top of the fractions and divide that answer
by all the numbers on the bottom of the fractions. The buttons you should
press on your calculator for this problem are:

6 0 X 2 2 \ 1 =

The correct answer should be 1320. You may have noticed that the last buttons you pushed (dividing by 1) did not change your answer. Dividing by 1 never changes a number. Nevertheless, it��s good to get in the habit of dividing by all denominator numbers because in other problems it may not be 1 that you are dividing by.

__Exercise 2:__ Use the unit factor
method to solve the following problems. Set up each problem **in the
exact same way (using all six steps) as the example problem on the previous
pages.** Show all your work. If you get stuck, ask your instructor
for help.

a) Apples sell for 43 cents a pound. How much do 3.2 pounds of apples cost?

b) Gold costs $500 per ounce. A solid gold nugget is 23.5 ounces. How much is it

worth?

c) How many feet are there in 58 yards?

d) The new hybrid cars get 65 miles per gallon. How many miles can you drive on 5

gallons of gas?

e) How many nickels are in 126 dollars?

f) How many days are there in two and a half years?

g) Dr. Gomez orders 0.08 grams of medication for a patient. Each pill contains 0.003 grams of the medicine. How many pills will the patient need?��

**D) Problems that require more
than one unit conversion factor**

The problems you have worked so far can be solved using one single unit conversion factor. Some problems require two unit conversion factors to solve. Here is an example of one of these problems:

How many seconds are there in 5 hours?

There is no common phrase that relates seconds per hour, so you can��t solve the problem with a single unit conversion factor. It will require two unit conversion factors. Here is how the problem is set up:

5 hours x __60 minutes__
x __60 seconds__ = ____________ seconds

1 hour 1 minute

First of all, notice that the problem is set up in the exact same way as a single unit conversion factor problem: Starting units (hours) on the left, answer units (seconds) on the right, and conversion factors between them.

And as before, all the units (except the answer units ��seconds��) must cancel out.

How do you pick the right two unit conversion
factors? For the first one, pick a unit conversion factor that will
cancel out the starting units. In other words, your first unit conversion
factor must have the starting units under the fraction line. The second
unit conversion factor must have the answer units on top. On the bottom
it must have the top units *from the first conversion factor.*

To put it simply, each unit conversion factor must cancel out the top units to its left.

__Exercise 3:__ Use the unit factor
method to solve the following problems. Show all your work, including
the unit factors and the cancelled units

a) How many inches are there in 110 yards?

b) How many hours are there in 5 and a half weeks?

c) Sarah can study 20 pages per night. Each page has three homework problems. If

she studies for 3 nights how many homework problems will she do?

d) On the planet Shnoidia, everyone earns 80 Quatloos a day. One Zerumba (a

favorite food item) costs 7 Quatloos. How many Zerumba��s can be purchased if

someone works for 3 days?

e) Dr. Michael prescribes 300 mg a day of medicine to a patent. Each pill has 15 mg

of medicine. How many pills will the patient need for 7 days?

f) A solution contains 5 grams of glucose per 100 milliliters. Each mole of glucose

weighs 180 grams. How many moles are there in 200 milliliters of the glucose

solution?

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