WORD PROBLEMS
Word problems are mathematical problems or equations expressed in words rather than numbers. The
main idea of word problems is to try to understand and translate what is being said in words into
numbers and variables, which are the unknowns. Solving word problems begins with a good
translation. If you can successfully understand and are able to translate words into numerical
relations, word problems will not be difficult. As with any mathematical problem, the key to success
is practice.
Translating Words into Mathematical Problems
When converting word problems to equations, certain "key" words tell you what kind of operations to
use: addition, multiplication, subtraction, and division. The table below shows some common phrases
and the operation to use.
Word
Sum
Operation
Addition
Total
Addition
More Than
Addition
Difference*
Subtraction
Less Than*
Times
Product
Subtraction
Multiplication
Multiplication
Example
The sum of my age and 10 equals 27.
The total of my pocket change and 20 dollars
is $22.43.
Eleven more than my age equals 43.
The difference between my age and my
younger sister's age, which is 11 years old, is
5 years.
Seven less than my age equals 32.
Three times my age is 60.
The product of my age and 14 is 168.
Equation
x + 10 = 27
x + 20 = 22.43
11 + x = 43
x – 11 = 5
x – 7 = 32
3 × y = 60
y × 14 = 168
*Note: When the problem asks for the difference of two numbers, the greater number should always go first to ensure you
obtain a positive number.
Algebra Dictionary
Words
is, are, was, has, cost, is equal to
is not equal to
which, what
and, more, sum, older than
less, difference, younger than
times, product
quotient of y and z,
or y divided by z
greater than
less than
the square of x
c times as old as John
Mathematical Way to Say It
=
≠
y (or any other letter)
+ (addition)
− (subtraction)
× (multiplication)
y
or y ÷ z
z
>
<
x2
( c ) × ( John's age )
TRANSLATING WORDS INTO MATHEMATICAL EXPRESSIONS – EXERCISES
Translate these phrases into correct mathematical expressions:
1. y increased by four
2. the sum of m and n
3. the product of k and w
4. s divided by t
5. the square of x
6. seven multiplied by p
7. two-thirds of y
8. a less b
9. the sum of four y’s
10. the difference between x and y
11. the quotient of 3a and 4b
12. five decreased by e
13. three times the product of m and n
14. twice x divided by four
15. one-half increased by the product of three and x
16. three times the square of y
17. the square of (three multiplied by a)
18. the product of h and the square of k
19. a number increased by 8
20. eight less than some number t
21. the difference between a number y and a number x
22. seven diminished by a number x
23. the square of the sum of x and four
24. the sum of the squares of x and four
25. four added to twice an unknown quantity x
26. twice an unknown quantity, x, diminished by seven
27. the square of r minus the product of x and y
28. twice the product of the square of x and the square of y
29. the quotient of five times the square of w and four times the cube of y
30. the difference between the cube of x and the square of z
TRANSLATING WORDS INTO MATHEMATICAL EXPRESSIONS –
ANSWERS TO EXERCISES
1. y + 4
2. m + n
3. k ⋅ w
4.
5. x 2
6. 7 p
7.
2
y
3
8. a − b
9. y + y + y + y =
4y
11.
3a
4b
14.
1
+ 3x
2
17. ( 3a )
10. x − y
12. 5 − e
13. 3 ⋅ m ⋅ n
15.
s
t
2x
4
16. 3y 2
2
18. h ⋅ k 2
19. x + 8
20. t − 8
21. y − x
22. 7 − x
23. ( x + 4 )
2
24. x 2 + 42
25. 2 x + 4
26. 2 x − 7
27. r 2 − xy
28. 2x 2 y 2
5w2
29.
4 y3
30. x 3 − z 2