Translating Words Into Symbols (level 2) Alright we are going to work on various examples.
It is very important that you understand how to translate phrases into variable expressions;
this is the first step in solving more complicated word problems that you will encounter in your
studies of algebra and other related science classes such as physics and chemistry. Alright
let’s try the following examples: Translate each phrase into a variable expression
our first phrase is: “10 more than a number”
This phrase is describing an addition between two numbers because of the key word, more
so this phrase translates to 10 + n or n + 10 either form represents the same translation
of the phrase. Let’s try this one:
“Five times a number” This phrase is describing a product because
of the key word times; this phrase translates to five times n or 5n.
Let’s try the next one: “A number decreased by 11”
This phrase represents a subtraction between 2 numbers because of the key word decreased,
so this phrase translates to n - 11. Many students have a hard time with phrases that involve
subtraction because they are not sure which numbers are being subtracted from each other.
For example the variable expression 11 - n is read “11 decreased by a number” These
two expressions are completely different because for the most part they yield different results
when you substitute the variable n with a random number. So be extra careful when translating
phrases involving subtraction make sure you are subtracting the numbers in the correct
order. Alright let’s try the next one:
“The difference between a number and 2” Once again we have a subtraction because of
the key word difference, so this phrase translates to n - 2.
Alright let's go to the next phrase:
“The quotient of 15 and d” Here we have a division because of the key
word quotient, this quotient involves the number 15 and the variable d so the phrase
translates to 15 divided by d or 15 over d both answers are acceptable but once again
we usually want the answer with the fraction bar.
Let’s try a slightly more complicated one: “6 more than twice y”
This phrase has two key words, more which means addition and twice which means multiplication
so its 6 more than 2y or 2y + 6. Notice that when we are dealing with addition the order
in which the numbers are added is not important a word of caution this is not the case for
subtraction so be careful with phrases involving the subtraction operator the order
in which the numbers are being subtracted matters.
Let's try the next example: “10 times the sum of a number and 8”
Here we have a multiplication because of the word times and we have an addition because
of the word sum. Also, notice that the product involves the number 10 and the sum of a number
and 8 we are going to need grouping symbols for this one, so the phrase translates to
10 times the quantity n + 8. Whenever you have multiplication its always between 2 numbers.
In this case one number is 10 and the other was the sum of a number and 8. So we need
to use grouping symbols to group the expression formed by the sum which is then multiplied
by 10. Alright let’s try the next one:
“The difference between three times a number and one”
So here we have a subtraction because of the key word difference, the two expressions that
are being subtracted are 3n and 1 so the final answer is 3n - 1
Let’s try the following phrase: “Twice a number, increased by 3”
Here we have a sum because of the key word increased, so this phrase translates to 2n
+ 3 or 3 + 2n, whenever you have phrases such as twice, tripled, half, a third,
these phrases denote a multiplication. Alright so now let’s go over examples that
involve translating phrases that are slightly more practical.
Answer the following with a variable expression. Bob is 2 ft taller than Joe. If Joe’s height
is J ft, how tall is Bob? It’s important to read the question as many
times as it takes you to understand it completely. The question is describing the heights of
Bob and Joe. The question is asking how tall Bob is given the fact that Bob is 2 ft taller
than Joe, we are also told that Joe’s height is represented by the variable J. So Bob’s
height is equal to Joe’s height + 2 or written algebraically j + 2 ft and this is our final answer.
Let’s try the next one:
Maria is 10 in. shorter than Claudia. If Claudia’s height is C inches, how tall is Maria?
Once again re-read the problem over and over until you understand it. The question is asking
how tall Maria is given the fact that Maria is 10 inches shorter than Claudia whose height
is represented by the variable C. So since Maria is 10 inches shorter than Claudia we
take Claudia’s height C and subtract 10 from that. This results in a final expression
C – 10 inches. Which represents Maria’s height.
Let’s try a slightly more challenging one: Sam is 3 feet taller than Dan.
If Dan’s height is D ft, how tall is Sam? Also, If Sam’s height is S ft, How tall is Dan?
Let’s answer the first question; we know that Sam is taller than Dan by 3 feet. We
also know that Dan’s height is D feet so to find Sam’s height we take Dan’s height
and add 3. This results in a final expression of D + 3 feet. Notice that this expression
is from the reference point of Sam, imagine that you are Sam and you glance at Dan you
see that you are taller than Dan. So this expression makes sense. The second question
tells us that Sam’s height is S feet and is asking us how tall is Dan. So if you were
Dan and look at Sam you would notice that you were shorter than Sam so your height would
be equal to Sam’s height minus 3 this results in a final expression of S - 3 feet. The first
variable expression we obtained is from the reference point of Sam and the second variable
expression is from the reference point of Dan. Both contain the same constrain the fact
that Sam is 3 feet taller than Dan. The variable expressions are different depending on whose
point of view we focus on. At times it’s useful to put yourself on the characters shoes
and think about their perspectives relative to the other.
Alright let’s work on questions that require the use of formulas
Answer the following with a variable expression. Matthew drove for H + 6 hours at a constant
rate of R + 5 mph. How far did he go? This problem is a typical distance equals
rate times time problem so we use the distance traveled formula at a constant rate. D = rt
In this problem the rate is equal to the quantity R + 5 so we use grouping symbols and substitute
this expression for r on our formula and in the same manner the time interval traveled
is equal to the quantity h + 6 so we substitute this expression using grouping symbols for
the variable t our final variable expression is equal to the quantity R + 5 times the quantity
H + 6 Miles. Alright lets try the next one:
Pencils cost W cents each and note books cost N cents each. How much will 3 pencils and
2 notebooks cost? This problem is an application of the cost
formula C = n times P, but first notice that we have 2 distinct items in this case pencils
and notebooks so to find the total cost we need to add the total cost of the pencils
and the total cost of the notebooks. The total cost of the pencils is equal to 3w and total
cost of the note books is equal to 2N so the total cost is equal to 3W + 2N cents. This
expression represents the total cost of 3 pencils and 2 notebooks. Notice that the formula
we used was a slightly modified version of the cost formula because of the fact that
we had 2 different types of items in this case pencils and notebooks. Make sure that
you understand the problem before you blindly use a formula.
Alright let’s work on the final example. A rectangle has a length of 5R yd and width
of 2S yd. What is the perimeter and area of the rectangle?
To find the perimeter of the rectangle we use the formula P = 2L plus 2W or just add
all the sides of the rectangle so using the formula and simplifying the expression we
have that the perimeter is equal to 10R + 4s. In the same manner the area of the rectangle
can be found by using the formula length times width so substituting the expressions into
the formula and simplifying we have that the final variable expression for the area of
the rectangle is equal to 10 times R times S or 10RS squared yards, and that's our final answer.
Alright it’s important that you develop this important skill of translating phrases
into variable expressions and start thinking algebraically by working solely with variable
expressions as oppose to numerical expressions. Okay in our next video we will expand on this
skill of translating phrases into variable expressions by taking sentences and translating
them into equations.