Translating Problems into Equations (Level
1) In this video we are going to learn how to
translate simple word problems into equations. Word problems are perhaps one of the most
feared and hated problems encountered in a typical mathematics course, this is due to
a variety of reasons, many students don’t bother reading word problems and if they do
happen to read them they have a hard time understand them. The best strategy to tackle
on word problems head on is by having a plan and sticking to it.
In general, word problems describe a situation in which certain numbers are related to each
other. For the most part some of these numbers are given in the problem and are considered
to be known numbers or quantities. Then you have other numbers for which you have no idea
what their values are these numbers are referred to as your unknowns. You must determine their
value by using the facts of the problem. In this video we will practice translating
simple word problems that contain two facts involving two unknowns. The following steps
will serve as a blue print to get you started in learning how to translate word problems
into equations. In a much later video we will learn how to find the solution of these word
problems by actually solving the equation. For now we first need to learn how to translate
the word problem into an equation. The following three steps will serve as a
guideline for you to practice translating problems into equations.
Step 1: Read the problem carefully. (Whenever you see a word problem just start reading
it so you can get an idea of what the problem is about.
Decide what the unknowns are. (Try to figure out what quantities you need to solve for
or try to figure out for which quantities you have little to no information on. This
is usually a tell tell sign that the quantity is an unknown).
Decide what the facts are. (These facts are key in determining how you will relate the
quantities with each other). Step 2: Choose a variable to represent the
unknowns. Choose a variable for one unknown. (Remember
that you can choose any letter of the alphabet to represent this unknown. Choose letters
that are relevant to the problem like the first letter of a person’s name or object
for example you can use the variable N to represent the name Newton or the variable
L to represent the length of a rectangle, or you can just use the letter x to represent
your unknown. Write an expression for the other unknown
using the variable and one of the facts. (In these problems you will have two unknowns
as a result you need to figure out how the two variables are related to each other and
denote them mathematically by using mathematical operations such as addition, subtraction,
multiplication, and division). Step 3: Reread the problem and write an equation.
(I cannot over emphasize the importance of reading the problem over and over again to
make sure you understand all the facts and how they are related to each other). Once
you understand the facts go ahead and use the facts from the problem to write an
equation. Alright, let’s go over some examples and
illustrate how to use these 3 steps. Translate the problem into an equation.
Maria has twice as much money as Helena. Together they have $36.
Alright, step 1 is to read the problem carefully and determine the unknowns and the facts.
The facts of this problem are represented by the numbered sentences. Notice that in
this problem the unknowns are the amount of money that Maria and Helena each have, since
we are not explicitly told how much money they have in their possession. Rather we are
given a relationship of this amount between the two persons. This relationship will be
used in step 2. In step two we go ahead and assign variables to our unknowns. Many students
have a hard time with this step, so let’s focus on this step, we already know that our
unknowns represent the amount of money that Maria and Helena have. The question we should
ask our self’s is how do we algebraically denote them with a variable. From the first
sentence we know that Maria has twice as much money as Helena. We are actually given some
kind of relation between Maria’s amount and Helena’s amount, this is not the case
for the amount of money that Helena has. As a rule of thumb you should always assign the
first variable to the quantity for which you have the least amount of information. In this
case we really don’t know much about Helena’s amount so let’s go ahead and assign this
quantity with the letter h, now if we use this variable along with the first sentence
we can represent Maria’s amount as 2h for twice as much. All we need to do now
is to write an equation relating these two expressions. Notice that in sentence 2 we
are told that together Maria and Helena have $36. This means that if we were to add the
amount of money that Maria and Helena have they would add up to $36. So our equation
becomes h + 2h=36 and this is our final answer. We have successfully translated the
word problem into an equation. In a later video we will learn how to solve this equation,
for now we want to practice using these 3 steps which provide the framework needed so
you can tackle these and future word problems. Alright, let’s try the next example. A piece of wood 50 inches long is sawed into
two pieces. One piece is 5 inches longer than the other. If a word problem involves lengths or distances,
a sketch can help you visualize and analyze the problem. So feel free to use your artistic
skills when solving word problems. Alright let’s start with step 1: we first
need to identify our unknowns, we need to figure out for which quantities we have limited
or no information. In this case we really don’t know how much each of the individual
pieces measure so our unknowns are the lengths of these two pieces of wood.
Step 2: Next we need to assign variables to our unknowns, before we do this we need to
first assign a variable to the unknown that contains the least amount of information.
From sentence 2 we are told that one piece is 5 inches longer than the other piece. This
means that we have one piece that is larger than the other piece. We have a small
and a large piece. Looking at the facts from both sentences we have no information or relation
about the smaller piece so we will assign the variable x to the shorter length. Having
assigned a variable to the shortest length we can use sentence 2 to denote the largest
length as x + 5. Now that we have algebraic expressions for both unknown quantities we
are ready to translate this word problem into an equation, from the first sentence we know
that the original length of this piece of wood was 50 inches this means that if we were
to add the smaller and larger piece it should equal 50 inches. So we go ahead and add the shortest
piece x with the longest piece x + 5 and set it equal to 50 as follows, so our final answer is x + (x+5)=50 and this is our equation. Alright let’s try the final example. Brenda drove three times as far as John.
Brenda drove 24 miles more than John. Once again feel free to make a drawing to
help you visualize and analyze the problem. The first step is to identify our unknowns.
After reading both sentences we see that we don’t know how many miles Brenda or John
drove so these quantities are going to be our unknowns. Next we need to assign variables
to these unknowns, once again let’s determine which of the two unknowns we have little to
no information on. Notice that we have sentences that relate Brenda’s distance to John’s
distance, but not the other way around. We have no idea what john’s distance is or
how it is related to Brenda’s distance so we will assign this quantity as our variable
and we will use the letter J. Now an expression for Brenda’s distance is going to depend
on the sentence that we use. The first sentence says that Brenda’s distance is three times
as far as John this translates to 3J or 3 times J. On the other hand if we use the second sentence we
have that Brenda’s distance is 24 more than John’s distance this translates to 24 + J.
Notice that in this example we have two variable expressions that represent Brenda’s distance.
Now the last step is to translate these expressions into an equation. Because both sentences
relate Brenda’s distance with that of John’s, it only makes sense that these two expressions
are equivalent to each other so we set both expressions equal to each other as follows.
3J=24 + J and this is our final equation. In this example we had two sentences that
related a single unknown quantity with the other unknown quantity in two distinct ways
since both sentences relate the same unknowns it is natural that both expressions are equivalent
to each other so you go ahead and set them equal to one another. Alright in our next
video we will continue going over more challenging word problems.