Math Core

Solving equations is only half the skill. In real life, nobody hands you $8c + 25 = 89$ . They hand you a situation, and you have to turn it into an equation yourself. This lesson shows you how to translate words into math, one careful step at a time.

A four-step plan

From words to an answer

Name the unknown. Choose a variable and write down what it stands for, with units.
Write the equation. Translate the words, piece by piece.
Solve the equation.
Answer the question in a sentence, with units, and check that it makes sense in the story.

Step 1 matters more than it looks. Writing "Let $c$ = the number of classes" keeps you from mixing up what you're finding.

Translating key phrases

Many word problems use the same phrases over and over.

Words	Math
sum, more than, increased by, total	$+$
difference, less than, decreased by, fewer	$-$
product, times, twice, of	$\times$
quotient, divided by, per, split equally	$\div$
is, equals, gives, results in, will be	$=$

For example, "five more than three times a number is $26$ " becomes $3n + 5 = 26$ .

Common mistake

"Less than" flips the order. " $7$ less than a number" means you start with the number and take away $7$ : $n - 7$ , not $7 - n$ . Think of it this way: $7$ less than your age is your age minus $7$ .

Worked example: A membership plus a fee

A climbing gym charges a $25 membership fee plus $8 for each class. Jordan spent $89 in total. How many classes did Jordan take?

Name the unknown. Let $c$ = the number of classes.

Write the equation. Each class costs $8$ dollars, so $c$ classes cost $8c$ . Add the fee:

8c + 25 = 89

Solve. Subtract $25$ : $8c = 64$ . Divide by $8$ : $c = 8$ .

Answer. Jordan took $8$ classes. Check: $8 \times 8 = 64$ dollars, plus $25$ is $89$ dollars. ✓

Unknowns that depend on each other

Sometimes a problem has two or more unknown amounts, but one is described using another. Let the variable stand for the basic amount and write the others in terms of it.

Worked example: Consecutive integers

The sum of three consecutive integers is $87$ . What are the integers?

Name the unknown. Let $n$ = the smallest integer. Consecutive integers go up by $1$ , so the next two are $n + 1$ and $n + 2$ .

Write and solve.

\begin{aligned} n + (n + 1) + (n + 2) &= 87 \\ 3n + 3 &= 87 \\ 3n &= 84 \\ n &= 28 \end{aligned}

Answer. The integers are $28$ , $29$ and $30$ . Check: $28 + 29 + 30 = 87$ . ✓

Worked example: A rectangle

A rectangle's length is $4$ cm more than its width. Its perimeter is $44$ cm. Find the length and the width.

Name the unknown. Let $w$ = the width in cm. Then the length is $w + 4$ .

Write the equation. Perimeter is two widths plus two lengths:

\begin{aligned} 2w + 2(w + 4) &= 44 \\ 2w + 2w + 8 &= 44 \\ 4w + 8 &= 44 \\ 4w &= 36 \\ w &= 9 \end{aligned}

Answer. The width is $9$ cm and the length is $9 + 4 = 13$ cm. Check: $2(9) + 2(13) = 18 + 26 = 44$ . ✓

When two options cost the same

A common question compares two plans: "When do they cost the same?" Write an expression for each plan and set them equal. That gives an equation with the variable on both sides.

Worked example: Comparing phone plans

Plan A costs $30 per month plus $0.10 per minute. Plan B costs $20 per month plus $0.15 per minute. For how many minutes do the plans cost the same?

Name the unknown. Let $m$ = the number of minutes.

Write the equation. Cost of A equals cost of B:

\begin{aligned} 30 + 0.10m &= 20 + 0.15m \\ 10 + 0.10m &= 0.15m \\ 10 &= 0.05m \\ m &= 200 \end{aligned}

Answer. The plans cost the same for $200$ minutes. Check: A is $30 + 0.10(200) = 50$ dollars and B is $20 + 0.15(200) = 50$ dollars. ✓

Tip

Ask whether your answer makes sense. A number of classes or people should be a whole number. A length can't be negative. If you get $c = 7.5$ classes, go back and look for a mistake.