Worded Problems

Writing Word Problems as Algebraic Expressions

In algebra, the variables that we write represent unknown values. It is these unknown values that we usually try to find.

Each part of an algebraic expression (number sentence) can be expressed using words and vice versa. (This is why we call it a number sentence!)

Any operation can be used in algebraic expressions - in fact, any combination of operations can be used. To be able to turn written expressions into algebra, we need to be aware of all the words that can be used to describe different operations.

• For the operation "$+$+" we might say "more than", "add", "plus" or "increased by", as well as other such words.
• For the operation "$\times$×" we might say "groups of", "multiply", "double" (for $\times2$×2), "triple" (for $\times3$×3) or "times".
• Write down all the words you can think of for subtraction "$-$−" and division "$\div$÷​".

Example

Write an algebraic expression for the following:

A number (call it $n$n) minus four equals ten.

Think:  What operation, number or symbol does each word in the question represent?

Do: In this case, "minus" is the operation "$-$−". So the sentence can be written using algebra as $n-4=10$n−4=10.

Finding unknown values

Once we can write these worded problems as algebraic equations, we can solve them by finding the unknown values. Let's look at an example of this:

Example

In the equation $n-4=10$n−4=10, what is the value of $n$n?

Think: One way of writing this equation in words is "take four away from a number and the result is ten". What number would we need to start with for this to be true?

Do: If we take $4$4 away from a number to get $10$10, then the number we started with must have been $4$4 larger than $10$10. Since $10+4=14$10+4=14, we can see that $n=14$n=14.

More examples

Lets watch the worked solutions to these different types of problems.

Question 1

Write an equation in simplest form to represent "$y$y is $18$18 times $x$x".

Question 2

A number (call it $n$n) plus ten equals eighteen.

A) Write this sentence using mathematical symbols.

B) Find the number.

Question 3

A number (call it $n$n) multiplied by five gives you fifteen.

A) Write this sentence using mathematical symbols.

B) Find the number.