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Translating Words Into Mathematical Symbols

Before we begin solving word problems, we need to be very comfortable with translating words into mathematical symbols. Following is a partial list of words and their mathematical equivalents.
 

Concept

Symbol

Words

Example

Translation

equality

=

is

2 plus 2 is 4

2 + 2 = 4

 

 

equals

x minus 5 equals 2

x – 5 = 2

 

 

is the same as

multiplying x by 2 is the same as dividing x by 7

2x = x/7

addition

+

sum

the sum of y and π is 20

y + π = 20

 

 

plus

x plus y equals 5

x + y = 5

 

 

add

how many marbles must John add to collection P so that he has 13 marbles

x + P = 13

 

 

increase

a number is increased by 10%

x + 10%x

 

 

more

the perimeter of the square is 3 more than the area

P = 3 + A

subtraction

minus

x minus y

x – y

 

 

difference

the difference of x and y is 8

|x – y| = 8

 

 

subtracted

x subtracted from y

y – x

 

 

less than

the circumference is 5 less than the area

C = A – 5

multiplication

× or •

times

the acceleration is 5 times the velocity

a = 5v

 

 

product

the product of two consecutive integers

x(x + 1)

 

 

of

x is 125% of y

x = 125%y

division

÷

quotient

the quotient of x and y is 9

x ÷ y = 9

 

 

divided

if x  is divided by y, the result is 4

x ÷ y = 4

 
Although exact steps for solving word problems cannot be given, the following guidelines will help:
  1. First, choose a variable to stand for the least unknown quantity, and then try to write the other unknown quantities in terms of that variable.
     
    For example, suppose we are given that Sue’s age is 5 years less than twice Jane’s and the sum of their ages is 16. Then Jane’s age would be the least unknown, and we let x = Jane's age. Expressing Sue’s age in terms of x gives Sue's age = 2x – 5.
  2. Second, write an equation that involves the expressions in Step 1. Most (though not all) word problems pivot on the fact that two quantities in the problem are equal. Deciding which two quantities should be set equal is usually the hardest part in solving a word problem since it can require considerable ingenuity to discover which expressions are equal.
     
    For the example above, we would get (2x – 5) + x = 16.
  3. Third, solve the equation in Step 2 and interpret the result.
For the example above, we would get by adding the x’s:
3x – 5 = 16
Then adding 5 to both sides gives
3x  = 21
Finally, dividing by 3 gives
x = 7
 
Hence, Jane is 7 years old and Sue is 2x – 5 = 2 7 – 5 = 9 years old.
 
*Notice that with “minus” and “difference” the terms are subtracted in the same order as they are written, from left to right (x minus y —> x – y). However, with “subtracted” and “less than,” the order of subtraction is reversed (x subtracted from y —> y – x). Many students translate “subtracted from” in the wrong order.




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