Solving One-Step Equation Word Problems | Grade 7 Math | LevelKite

A word problem is a short story about quantities, which are amounts you can count or measure. Some numbers are given, one number is unknown, and the story tells you how they are related. Your job is to find the unknown number.

An equation is a statement that two amounts are equal, written with an equals sign. A one-step equation is an equation you can solve with one operation: adding, subtracting, multiplying or dividing. You already know how to solve one, even with fractions and decimals. In this lesson you build one from a story, solve it, and check that your answer fits the story.

Start with what a story means. Read this one:

Story. A tank holds some water. After $12.5$ liters are added, the tank holds $40$ liters.

The story describes the final amount of water in two ways:

the starting amount plus $12.5$ liters
$40$ liters

Both describe the same amount, so they are equal.

To write this in symbols, use a variable, a letter that stands for a number. Let $w$ be the number of liters in the tank at the start. Then the first description is the expression $w + 12.5$ . An expression is made of numbers, variables and operations, with no equals sign. Setting the two descriptions equal gives this equation:

w + 12.5 = 40

Key concept

The equation retells the story

In each word problem in this lesson, the story describes one quantity in two ways. Setting the two descriptions equal gives a one-step equation that says the same thing as the story. To solve the problem:

Define the variable. Say what it counts or measures, and name its unit.
Write the equation that matches the story.
Solve the equation. Do the same operation to both sides, and never divide by zero. Two equal amounts stay equal when you do this.
Check your value. It is a solution only if it makes the original equation, the one from step 2, true. It must also make sense in the story.

Then answer the question in a full sentence.

Worked example

Finish the tank story

A tank holds some water. After $12.5$ liters are added, the tank holds $40$ liters. How many liters of water did the tank hold at the start?

Define the variable and its unit. Let w be the number of liters of water in the tank at the start. An amount of water cannot be negative, so w must be 0 or more.
Write the equation. The final amount is the starting amount plus 12.5 liters, and the story says it is 40 liters. Both describe the same amount, so they are equal.
$w + 12.5 = 40$
Undo the addition by subtracting 12.5 from both sides. Subtracting the same number from both sides keeps the two sides equal.
$w + 12.5 - 12.5 = 40 - 12.5$
Simplify each side. On the left, w + 12.5 - 12.5 equals w for every value of w. On the right, 40 - 12.5 = 27.5.
$w = 27.5$
Check the value in the original equation. Replace w with 27.5. The left side is 27.5 + 12.5 = 40, the same as the right side, so 27.5 makes the original equation true and is a solution. It is not negative, so it also makes sense in the story.
$27.5 + 12.5 = 40$

Answer

The tank held $27.5$ liters of water at the start.

A common mistake: letting one word choose the operation. In the tank story, the word "added" can tempt you to add the two numbers you see: $40 + 12.5 = 52.5$ . This fails because the story says the starting amount plus $12.5$ equals $40$ . The $12.5$ liters are already part of the $40$ liters. Test $52.5$ in the original equation $w + 12.5 = 40$ . The left side is $52.5 + 12.5 = 65$ , not $40$ , so $52.5$ does not make the original equation true and is not a solution. It also makes no sense in the story: the tank cannot hold more water before the water is added than after. In this story, "added" tells you what to write in the equation. Solving then undoes that addition by subtracting $12.5$ from both sides.

One-step stories often follow one of the four patterns in the table below. The tank story follows the first one. In the table, $x$ is the unknown number, and $a$ and $b$ are numbers the story gives you. Every row does one operation to both sides. The table divides only by a number that is not zero, because division by zero is not defined. Whichever pattern you use, check your result: it is a solution only if it makes the original equation true.

Four one-step patterns

What the story says	Equation	Do this to both sides	Result
An unknown amount $x$ goes up by $a$ and becomes $b$ .	$x + a = b$	Subtract $a$	$x = b - a$
An unknown amount $x$ goes down by $a$ and becomes $b$ .	$x - a = b$	Add $a$	$x = b + a$
$a$ times an unknown amount $x$ is $b$ , and $a \neq 0$ .	$ax = b$	Divide by $a$ , which is allowed because $a \neq 0$	$x = b \div a$
An unknown amount $x$ divided by $a$ is $b$ , and $a \neq 0$ .	$\dfrac{x}{a} = b$	Multiply by $a$	$x = b \cdot a$

Worked example

An elevation below zero

Elevation is height measured in meters from the surface of the water, so a point below the surface has a negative elevation. A diver swims straight down $7.5$ meters and reaches an elevation of $-20$ meters. What was the diver's elevation before swimming down?

Define the variable and its unit. Let e be the diver's elevation, in meters, before swimming down. Here e may be negative, because a negative elevation means a point below the surface.
Write the equation. Swimming down 7.5 meters lowers the elevation by 7.5, so the final elevation is the starting elevation minus 7.5. The story says the final elevation is -20 meters. Both describe the final elevation, so they are equal.
$e - 7.5 = -20$
Undo the subtraction by adding 7.5 to both sides. Adding the same number to both sides keeps the two sides equal.
$e - 7.5 + 7.5 = -20 + 7.5$
Simplify each side. On the left, e - 7.5 + 7.5 equals e for every value of e. On the right, -20 is farther from zero than 7.5, so the sum is negative, and its distance from zero is 20 - 7.5 = 12.5.
$e = -20 + 7.5 = -(20 - 7.5) = -12.5$
Check in the original equation. Replace e with -12.5. The left side is -12.5 - 7.5 = -20, the same as the right side, so -12.5 makes the original equation true. It also makes sense in the story: before swimming down, the diver was higher than -20 meters.
$-12.5 - 7.5 = -20$

Answer

Before swimming down, the diver's elevation was $-12.5$ meters, which is $12.5$ meters below the surface.

Worked example

A fraction of an unknown total

Ravi has read $\frac{2}{3}$ of the pages in a book. That is $84$ pages. How many pages are in the whole book?

Define the variable and its unit. Let p be the number of pages in the whole book. A number of pages must be a positive whole number.
Write the equation. The pages Ravi has read are 2/3 of p, which means 2/3 times p. The story says this is 84 pages, so the two are equal.
$\frac{2}{3}p = 84$
Undo the multiplication by dividing both sides by 2/3. This is allowed because 2/3 is not zero. Dividing by 2/3 gives the same result as multiplying by 3/2, so multiply both sides by 3/2.
$\frac{3}{2} \cdot \frac{2}{3}p = \frac{3}{2} \cdot 84$
Simplify each side. On the left, 3/2 times 2/3 is 1, and 1 times p equals p for every value of p. On the right, 3 times 84 is 252, and 252 divided by 2 is 126.
$p = \frac{3 \cdot 84}{2} = \frac{252}{2} = 126$
Check in the original equation. Replace p with 126. Then 2 times 126 is 252, and 252 divided by 3 is 84, the same as the right side, so 126 makes the original equation true. It is a positive whole number, so it makes sense as a number of pages.
$\frac{2}{3} \cdot 126 = \frac{252}{3} = 84$

Answer

The whole book has $126$ pages.

Here is the idea to keep: an equation that matches a story says the same thing as the story. Define the variable and its unit, write the equation, and solve it by doing the same operation to both sides, never dividing by zero. Then check that your value makes the original equation true and makes sense in the story. A value that passes both checks answers the question, so state it in a full sentence with its unit.