Writing and Solving Equations from Word Problems | Grade 8 Math | LevelKite

An equation is a statement that two amounts are equal, and you already know how to solve two-step equations. In a word problem, nobody hands you the equation, so the new skill is writing it yourself.

Suppose a story says that 5 tickets at the same price cost 40 dollars. Five times the ticket price and 40 dollars describe one amount, the total cost, in two different ways. Writing an equation means finding two descriptions like these and stating that they are equal.

The price is the number you don't know, so you name it with a variable, a letter that stands for an unknown number. Let $p$ be the price of one ticket in dollars. A price can't be negative, so the permitted values of $p$ , the values that make sense in the story, are the numbers with $p \geq 0$ .

\underbrace{5p}_{\text{how the cost is built}} = \underbrace{40}_{\text{what the cost comes to}}

Each side of that equation is an expression, which is made of numbers, variables and operations and has no equals sign. A solution is a value of the variable that makes the equation true. Here, dividing both sides by 5, which is not zero, gives $p = 8$ . That is the solution, since $5(8) = 40$ , and $8$ is a permitted value, so each ticket costs 8 dollars.

Key concept

Translate, solve, check

When a word problem leads to an equation, the story describes one quantity in two ways, and the equation says the two descriptions are equal.

Translate. Name the unknown with a variable, its unit and its permitted values. Write an expression for how the story builds a quantity, and set it equal to what the story says that quantity is.
Solve. Do the same valid operation to both sides. Adding or subtracting the same number, or multiplying or dividing by the same nonzero number, keeps exactly the same solutions. Never divide by zero.
Check. Put your value in place of the variable in the original equation, the one you first wrote. Then test it against the story: it must be a permitted value, and the story's numbers must work out.

Worked example

One step: distance and time

Rosa cycles at a steady 18 kilometers per hour. How many hours does she need to cycle 45 kilometers?

Name the unknown before you write anything else. Let h be the number of hours Rosa cycles. A time cannot be negative, but it can be a decimal or a fraction, so the permitted values are all numbers that are 0 or greater.
$h \geq 0$
Translate. Each hour adds 18 kilometers, so h hours build a distance of 18 times h kilometers. The story states that this same distance is 45 kilometers, so the two descriptions are equal.
$18h = 45$
Solve. Divide both sides by 18 to undo the multiplication. Equal amounts stay equal when you divide both by the same nonzero number, and multiplying by 18 would reverse the step, so no solution is lost or added. Here 18 is not zero.
$\frac{18h}{18} = \frac{45}{18} \quad \text{so} \quad h = \frac{5}{2} = 2.5$
Check in the original equation, then in the story. Replacing h with 2.5 gives 18 times 2.5, which is 45, so the equation is true. In the story, 2 hours cover 36 kilometers and the extra half hour covers 9 more, which makes 45. And 2.5 is not negative, so it is a permitted value.
$18(2.5) = 36 + 9 = 45$

Answer

Rosa needs exactly $2.5$ hours, which is 2 hours 30 minutes, to cycle 45 kilometers.

Many stories mix an amount that happens once with an amount that repeats. When the equation isn't obvious, try a few specific numbers first.

A climbing gym charges 25 dollars once to join and then 15 dollars for each month. Let $m$ be the number of months paid for. You pay for whole months, so the permitted values of $m$ are the whole numbers $0$ , $1$ , $2$ and so on.

The table tries 1, 2 and 3 months. Watch the middle column: the 15 and the 25 stay the same, and only the number of months changes. The changing number is where the variable goes, which gives the last row.

Cost of the climbing gym for different numbers of months

Months paid	How the story builds the cost	Cost in dollars
$1$	$15(1) + 25$	$40$
$2$	$15(2) + 25$	$55$
$3$	$15(3) + 25$	$70$
$m$	$15(m) + 25$	$15m + 25$

Worked example

Two steps: a one-time fee plus a monthly charge

At the climbing gym, joining costs 25 dollars once and each month costs 15 dollars. Maya has paid 115 dollars in total. For how many months has she paid?

Name the unknown. As in the table, let m be the number of months Maya has paid for. She pays for whole months, so the permitted values are the whole numbers 0, 1, 2 and so on.
Translate. The table shows that m months cost 15 times m, plus the 25 that Maya pays once. The story states that this cost is 115 dollars, so set the two equal. Writing 25 + 15m on the left is just as correct. The two expressions are equivalent, meaning they have the same value for every permitted m, so both equations have the same solutions.
$15m + 25 = 115$
Solve by undoing the addition first. Subtract 25 from both sides. Equal amounts stay equal when you subtract the same number from each, and adding 25 back would reverse the step, so the solutions stay the same.
$15m + 25 - 25 = 115 - 25 \quad \text{so} \quad 15m = 90$
Divide both sides by 15 to undo the multiplication. Dividing both sides by the same nonzero number keeps the same solutions, and 15 is not zero.
$\frac{15m}{15} = \frac{90}{15} \quad \text{so} \quad m = 6$
Check in the original equation, then in the story. Replacing m with 6 gives 90 + 25, which is 115, so the equation is true. In the story, six monthly charges of 15 dollars make 90 dollars, and the one joining fee of 25 dollars brings the total to 115 dollars. And 6 is a whole number, so it is a permitted value.
$15(6) + 25 = 90 + 25 = 115$

Answer

Maya has paid for $6$ months.

Worked example

Two steps: an amount that shrinks

A full tank holds 180 liters of water. It then drains at a steady 12 liters per minute. After how many minutes are 48 liters left?

Name the unknown. Let t be the number of minutes the tank has been draining. The tank is empty after 180 divided by 12, which is 15 minutes, so the permitted values run from 0 to 15. They do not have to be whole numbers.
$0 \leq t \leq 15$
Translate. The water left is the starting amount minus what has drained. Each minute removes 12 liters, so t minutes remove 12 times t liters. The story states that 48 liters are left.
$180 - 12t = 48$
Subtract 180 from both sides, which keeps the same solutions. The 12t keeps its minus sign, because the left side subtracts it.
$180 - 12t - 180 = 48 - 180 \quad \text{so} \quad -12t = -132$
Divide both sides by negative 12. It is not zero, so the solutions stay the same. A negative number divided by a negative number is positive.
$\frac{-12t}{-12} = \frac{-132}{-12} \quad \text{so} \quad t = 11$
Check in the original equation, then in the story. Replacing t with 11 gives 180 minus 132, which is 48, so the equation is true. In the story, 11 minutes of draining remove 132 liters and leave 48. And 11 lies between 0 and 15, so it is a permitted value.
$180 - 12(11) = 180 - 132 = 48$

Answer

The tank has 48 liters left after $11$ minutes of draining.

The idea to keep: to write an equation from a word problem, find one quantity that the story describes in two ways and state that the two are equal. Name the variable with its unit and permitted values first. Solve by doing the same valid operation to both sides, never dividing by zero. Then check twice: in the original equation to test your solving, and against the story to test your translation.