Translating Words Into Symbols (Level 1) To solve problems using algebra it is important
that you become comfortable translating mathematical phrases written in words and rewriting them
using the language of algebra. In other words, we are taking words that have a specific mathematical
meaning and translating it by using mathematical symbols such as numbers and variables.
Alright we are going to systematically go through each mathematical operator and learn
how to translate specific phrases into mathematical symbols let’s start with addition.
The common phrases that correspond to addition are the words: “sum” as in “the sum
of 5 and x”, increased as in “a number increased by 5”, and more as in “5 more
than a number”. Notice that the variable is sometimes explicitly stated like in the
first phrase where we need to use the variable x. Notice that some phrases don’t explicitly
tell you what the variable should be assigned to. In this case you have complete freedom
to choose your favorite letter of the alphabet typically we use the letter n which represents
the word “number”. Alright, let’s move along with the opposite
of addition, subtraction. The common phrases that are associated with subtraction are the
words: difference as in “the difference between a number and 5”, decreased as in
“a number decreased by 5”, less as in “5 less than a number”, and minus as in
“5 minus a number”. When translating phrases that involve subtraction many students have
a hard time translating phrases with the word less, many students translate 5 less than
a number as 5 – x which is completely wrong. The correct way is x – 5 because its 5 less
than a number so you take the number and subtract 5.
Ok, let’s move along to multiplication. The common phrases that are related to multiplication
are the words: product as in “the product of 5 and a number”, times as in “5 times
a number” and words that explicitly state that we are taking a fractional part of a
number as in “one Fifth of a number”. Notice that this last expression can be simplified
as n over 5 but for the most part write it as a product then simplify this is going to be
important when we work on slightly more complicated phrases.
Alright let's transition to our final operator, division the common phrases that are linked
to division are the words: quotient as in “the quotient of a number and 5, and the
word divided as in “a number divided by 5” notice that division can be written using
a fraction bar or with the traditional division symbol for the most part this classical division
symbol will be used less and less in your studies of mathematics nevertheless it will
pop up every now and then. At times a phrase can have more than one mathematical
operation such as the phrase “5 less than half of x” here we have a difference because
of the key word “less” and we have a multiplication because we are taking a fractional part of
a number. This expression can also be simplified as follows. In the same manner, the phrase
“half the difference between x and 5 contains a product because the phrase is denoting that
we are taking a fractional part of a number in this case the number is the difference
between x and 5 or x-5, notice that we used grouping symbols specifically parenthesis
to group the “number” that is composed by the difference between x and 5.
In addition to translating phrases into mathematical expressions you also need to be able to effectively
work with formulas. Formulas are equations that state rules about relationships, a typical
formula that you should be familiar with is the formula to find the area of a rectangle
A = LW where A is the area, l is the length of the rectangle and W is the width of the
rectangle. In addition, the formula to find the perimeter of a rectangle is expressed
as follows: 2 times the length plus 2 times the width, in other words, you
add the lengths of all the sides of the rectangle. Another typical equation that is used in algebra
is the distance formula the distance traveled by an object going at a constant rate for
a specific time interval is given by the formula D = rt where D is the distance, r is the
rate of the object and t is the time interval the object has been traveling. Finally, to
find the monetary cost of a certain number of items we use the formula C = np where n
represents the number of items and p represents the price per item.
When using formulas you need to be comfortable with the two types of answers that formulas
tend to generate, one of them are numerical expressions and the other are variable expressions.
For example say that we would like to find the area of a rectangle whose length is 10
feet and width is 5 feet then we use the formula to find the area of a rectangle and
substitute the value for the length and width using grouping symbols as follows. Then we
evaluate the numerical expression for a final answer of 50 square feet. The other types
of answers that can be obtained from formulas are variable expressions. For example say
we take the same problem the width is still 5 ft but now the length is equal to x. In
this case we once again substitute each variable in the formula with the corresponding value
that we are given in this case the width is equal to 5 and the length is equal to x.
So our final answer is 5x this variable expression represents the area of the rectangle with
the given measurements. Notice that this expression is written in terms of the variable x. At
first, writing the area this way seems a bit confusing and weird but all is says is that
the area of this rectangle is 5 times x. In the same manner the perimeter of the rectangle
with the same measurements is equal to 2 times 5 + 2x, simplifying the variable expression
gives us a final answer equal to 10 + 2x again notice that the answer is left in terms of
the variable x. This expression represents the perimeter of a rectangle with width of
5 and length equal to x. The reason why we want to leave our answer in terms of the variable
is because by doing this we can generate all the areas and perimeters of a rectangle with
width 5 and any length that we would like to have. So if I wanted to know the area of
a rectangle with width 5 and length of 6 I just take the variable expression 5x and substitute
x for 6 and it turns out that it is equal to 30 square feet. In the same manner if I
wanted to know the perimeter I take the expression 10 + 2x and substitute x for 6 and I get 22
feet. In algebra we sometimes don’t want to know the exact value like 30 square feet
or 22 feet we just want to know in general what the perimeter and area would equal to
given a particular width and any length imaginable so we use a variable to represent the varying
nature of the length. Now let’s say that you are riding a bike
and you traveled h + c hours at a constant rate of 20 mph how far did you travel? So since
we are riding at a constant speed we use the distance traveled formula which is equal to
the rate times the time and substitute each expression accordingly. Doing that we have
that the distance traveled is equal to 20 times the quantity h+c and this is our final
answer remember all we want is a general expression for the distance traveled in terms of the
given variables. We don’t care about specific numbers just the general algebraic expression
for the distance. Ok, let’s do one more example using
the cost formula say that you decided to buy 3 hamburgers and each one cost P dollars what
is the total cost of your purchase? So to find the cost we take the number of items
in this case 3 and multiply it by the price per item which is equal to P so the final
answer is 3p. Notice that we don’t actually have a monetary value we have a generalized
variable expression but if we knew what p was then we can find out what the total cost
would be all we need to do is take the price and multiply it by 3.
It is important that you start developing this skill to think algebraically and generalize
phrases in terms of variables even if you don’t know what the particular variable
is equal to we really don’t care, unless the problem is asking us to evaluate the variable
with a particular value in mind. Ok in our next video we will go over even
more examples and practice translating words into symbols.