Translating Words Into Expressions

In order to use algebra to solve word problems, we need to learn how to express English phrases in the symbols and operations of algebra. We should also be able to recognize in an algebraic equation the words from which it might have come.
 
Here is how to translate an English phrase into algebra:
 

1. Identify quantities and express them as symbols (variables).
2. Identify the operation.
3. Connect the symbols with the operation sign.

We also need to learn the phrases that indicate the basic arithmetic operations of addition, subtraction, multiplication, and division.
 

Different phrases can be used for addition. Here are some phrases that are equivalent to the algebraic expression "b + 5".
 

• b increased by 5
• 5 more than b
• b added to 5
• the sum of b and 5
• 5 plus b

Subtraction can be expressed with different phrases. The following phrases are equivalent to the algebraic expression "y - 4".
 

• the difference of y and 4
• 4 fewer than y
• 4 subtracted from y
• 4 less than y
• y decreased by 4

Certain phrases can be used for multiplication. These phrases are equivalent to the algebraic expression "3y".
 

• the product of 3 and y
• 3 multiplied by y
• 3 times y

The phrases below are equivalent to the algebraic expression "12 ÷ x"
 

• the quotient of 12 and x
• the ratio of 12 to x
• 12 divided by x
• 12 over x

What algebraic expression is equivalent to the English expression below:

The quantity P increased by the quantity Q plus R

To find an equivalent expression, look at the above expression piece by piece.
 

1. Look at the partial expression "the quantity Q plus R"; an equivalent algebraic expression would be:

   Q + R

2. Look at the partial expression "the quantity P increased by..."; an equivalent algebraic expression would be:

   "P + ..."

3. Our entire expression would be:

   P + Q + R