This chapter focuses on converting words into numbers. That means taking a word problem and rewriting it in numerical terms (an equation). This skill is essential for solving word problems because the wording and details can lead to mistakes even when you understand the math. In this chapter, you’ll practice spotting key words and translating them into mathematical operations.

The following list shows common mathematical operators and the words that often signal them in word problems:

Definitions

Addition: More, increase, sum, plus
Subtraction: Less, decrease, difference, minus
Multiplication: Times, product, each
Division: Out of
Equals: Is, same, equal to

Most of this chapter is practice applying these terms. In the examples below, you’ll be given a word problem and asked to turn it into an equation. It’s most helpful to try each one yourself before revealing the answer.

Practice: Rewrite the following problems as equations

Example 1

We’ll do the first one together:

A number is three less than four times a second number.

“A number” is an unknown value, so we’ll represent it with a variable. Use $x$ .
“Is” translates to $=$ .
“Three less than” means subtract $3$ .
“Four times” means multiply by $4$ .
“A second number” is a different unknown, so use a new variable, $y$ .

Writing it as an equation:

$x = 4 * y - 3$

Example 2

$5$ times the original number is $2$ more than the new number.

(spoiler)

“ $5$ times” translates to $5 *$ .
“The original number” is $x$ .
“Is” translates to $=$ .
“ $2$ more than” means add $2$ .
“The new number” is $y$ .

All together:

$5 * x = y + 2$

Example 3

A second book is worth four more than two out of seven of the original book.

(spoiler)

“Second book” is $x$ .
“Is worth” translates to $=$ .
“Four more than” means add $4$ .
“Two out of seven” translates to $2/7$ .
“Original book” is $y$ .

All together:

$x = (2/7) * y + 4$

Example 4

When a car is driven off its dealership parking lot, its value becomes half its original value

(spoiler)

“Value” is $x$ .
“Becomes” translates to $=$ .
“Half its original value” means multiply the original value by $\frac{1}{2}$ . Let the original value be $y$ , so this is $\frac{1}{2} * y$ .

All together:

$x = \frac{1}{2} * y$

Example 5

There is an apartment in town that charges $$400$ as a down payment and $$600$ more for each month rented out. How much would it cost to rent the apartment for $6$ months?

(spoiler)

$$400$ is a fixed starting cost (a base payment). This part does not depend on the number of months, so it stays as $+ 400$ .
$$600$ is charged for each month, so you multiply $600$ by the number of months. For $6$ months, this is $600 * 6$ .

All together:

$x = 400 + 600 * 6 = 4, 000$

Key word operators

Converting words to numbers

The following list shows common mathematical operators and the words that often signal them in word problems:

Definitions

Addition: More, increase, sum, plus
Subtraction: Less, decrease, difference, minus
Multiplication: Times, product, each
Division: Out of
Equals: Is, same, equal to

Practice: Rewrite the following problems as equations

Example 1

We’ll do the first one together:

A number is three less than four times a second number.

“A number” is an unknown value, so we’ll represent it with a variable. Use $x$ .
“Is” translates to $=$ .
“Three less than” means subtract $3$ .
“Four times” means multiply by $4$ .
“A second number” is a different unknown, so use a new variable, $y$ .

Writing it as an equation:

$x = 4 * y - 3$

Example 2

$5$ times the original number is $2$ more than the new number.

(spoiler)

“ $5$ times” translates to $5 *$ .
“The original number” is $x$ .
“Is” translates to $=$ .
“ $2$ more than” means add $2$ .
“The new number” is $y$ .

All together:

$5 * x = y + 2$

Example 3

A second book is worth four more than two out of seven of the original book.

(spoiler)

“Second book” is $x$ .
“Is worth” translates to $=$ .
“Four more than” means add $4$ .
“Two out of seven” translates to $2/7$ .
“Original book” is $y$ .

All together:

$x = (2/7) * y + 4$

Example 4

When a car is driven off its dealership parking lot, its value becomes half its original value

(spoiler)

“Value” is $x$ .
“Becomes” translates to $=$ .
“Half its original value” means multiply the original value by $\frac{1}{2}$ . Let the original value be $y$ , so this is $\frac{1}{2} * y$ .

All together:

$x = \frac{1}{2} * y$

Example 5

There is an apartment in town that charges $$400$ as a down payment and $$600$ more for each month rented out. How much would it cost to rent the apartment for $6$ months?

(spoiler)

$$400$ is a fixed starting cost (a base payment). This part does not depend on the number of months, so it stays as $+ 400$ .
$$600$ is charged for each month, so you multiply $600$ by the number of months. For $6$ months, this is $600 * 6$ .

All together:

$x = 400 + 600 * 6 = 4, 000$

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Key word operators

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Converting words to numbers

The following list shows common mathematical operators and the words that often signal them in word problems:

Definitions

Addition: More, increase, sum, plus
Subtraction: Less, decrease, difference, minus
Multiplication: Times, product, each
Division: Out of
Equals: Is, same, equal to

Practice: Rewrite the following problems as equations

Example 1

We’ll do the first one together:

A number is three less than four times a second number.

“A number” is an unknown value, so we’ll represent it with a variable. Use $x$ .
“Is” translates to $=$ .
“Three less than” means subtract $3$ .
“Four times” means multiply by $4$ .
“A second number” is a different unknown, so use a new variable, $y$ .

Writing it as an equation:

$x = 4 * y - 3$

Example 2

$5$ times the original number is $2$ more than the new number.

(spoiler)

“ $5$ times” translates to $5 *$ .
“The original number” is $x$ .
“Is” translates to $=$ .
“ $2$ more than” means add $2$ .
“The new number” is $y$ .

All together:

$5 * x = y + 2$

Example 3

A second book is worth four more than two out of seven of the original book.

(spoiler)

“Second book” is $x$ .
“Is worth” translates to $=$ .
“Four more than” means add $4$ .
“Two out of seven” translates to $2/7$ .
“Original book” is $y$ .

All together:

$x = (2/7) * y + 4$

Example 4

When a car is driven off its dealership parking lot, its value becomes half its original value

(spoiler)

“Value” is $x$ .
“Becomes” translates to $=$ .
“Half its original value” means multiply the original value by $\frac{1}{2}$ . Let the original value be $y$ , so this is $\frac{1}{2} * y$ .

All together:

$x = \frac{1}{2} * y$

Example 5

There is an apartment in town that charges $$400$ as a down payment and $$600$ more for each month rented out. How much would it cost to rent the apartment for $6$ months?

(spoiler)

$$400$ is a fixed starting cost (a base payment). This part does not depend on the number of months, so it stays as $+ 400$ .
$$600$ is charged for each month, so you multiply $600$ by the number of months. For $6$ months, this is $600 * 6$ .

All together:

$x = 400 + 600 * 6 = 4, 000$

Key word operators

Converting words to numbers

The following list shows common mathematical operators and the words that often signal them in word problems:

Definitions

Addition: More, increase, sum, plus
Subtraction: Less, decrease, difference, minus
Multiplication: Times, product, each
Division: Out of
Equals: Is, same, equal to

Practice: Rewrite the following problems as equations

Example 1

We’ll do the first one together:

A number is three less than four times a second number.

“A number” is an unknown value, so we’ll represent it with a variable. Use $x$ .
“Is” translates to $=$ .
“Three less than” means subtract $3$ .
“Four times” means multiply by $4$ .
“A second number” is a different unknown, so use a new variable, $y$ .

Writing it as an equation:

$x = 4 * y - 3$

Example 2

$5$ times the original number is $2$ more than the new number.

(spoiler)

“ $5$ times” translates to $5 *$ .
“The original number” is $x$ .
“Is” translates to $=$ .
“ $2$ more than” means add $2$ .
“The new number” is $y$ .

All together:

$5 * x = y + 2$

Example 3

A second book is worth four more than two out of seven of the original book.

(spoiler)

“Second book” is $x$ .
“Is worth” translates to $=$ .
“Four more than” means add $4$ .
“Two out of seven” translates to $2/7$ .
“Original book” is $y$ .

All together:

$x = (2/7) * y + 4$

Example 4

When a car is driven off its dealership parking lot, its value becomes half its original value

(spoiler)

“Value” is $x$ .
“Becomes” translates to $=$ .
“Half its original value” means multiply the original value by $\frac{1}{2}$ . Let the original value be $y$ , so this is $\frac{1}{2} * y$ .

All together:

$x = \frac{1}{2} * y$

Example 5

There is an apartment in town that charges $$400$ as a down payment and $$600$ more for each month rented out. How much would it cost to rent the apartment for $6$ months?

(spoiler)

$$400$ is a fixed starting cost (a base payment). This part does not depend on the number of months, so it stays as $+ 400$ .
$$600$ is charged for each month, so you multiply $600$ by the number of months. For $6$ months, this is $600 * 6$ .

All together:

$x = 400 + 600 * 6 = 4, 000$