Math Core

Solving $15m + 25 = 130$ is only half the job. In real life, nobody hands you the equation. You get a story about a gym membership or a taxi ride, and you have to build the equation yourself. This lesson shows you how.

From story to equation

Most two-step stories have the same shape: a starting amount plus a rate repeated some number of times.

Four steps for word problems

Name the unknown. Choose a letter and say exactly what it stands for.
Write an equation that says the same thing as the story.
Solve the equation.
Answer the question in a sentence with units, and check that it makes sense.

Look for clue words. "Per," "each" and "every" usually point to the number that multiplies the variable. A one-time fee, a starting amount, or a head start is usually the number added on.

Worked example: A membership fee

A gym charges a $25 sign-up fee plus $15 per month. Ana has paid $130 in all. How many months has she been a member?

Let $m$ be the number of months. Each month costs $15$ dollars, so $m$ months cost $15m$ . Add the one-time fee:

\begin{aligned} 15m + 25 &= 130 \\ 15m &= 105 \\ m &= 7 \end{aligned}

Ana has been a member for 7 months. Check: $15 \times 7 = 105$ dollars, plus $25$ is $130$ . ✓

Arithmetic or algebra?

You could also solve the gym problem with arithmetic: take away the $25 fee to get $105, then divide by $15 to get 7. Notice those are the same two steps as solving the equation, in the same order. The equation is just a tidy record of your thinking. It becomes much more helpful when the story gets harder, because the steps are written down and easy to check.

Stories with parentheses

Sometimes a number is added before everything gets multiplied. That gives an equation of the form $p(x + q) = r$ .

Worked example: Tickets and popcorn

Six friends go to a movie. Each one buys a ticket and a $4 popcorn. Together they spend $72. How much is one ticket?

Let $t$ be the price of one ticket. Each person spends $t + 4$ dollars, and there are $6$ people:

\begin{aligned} 6(t + 4) &= 72 \\ t + 4 &= 12 \\ t &= 8 \end{aligned}

One ticket costs $8. Check: each friend spends $8 + 4 = 12$ dollars, and $6 \times 12 = 72$ . ✓

Perimeter problems often look like this too. A rectangle's perimeter is $2(\ell + w)$ : add the length and width, then double.

Worked example: A garden fence

A rectangular garden is $11$ meters long. Its perimeter is $36$ meters. How wide is it?

Let $w$ be the width in meters.

\begin{aligned} 2(w + 11) &= 36 \\ w + 11 &= 18 \\ w &= 7 \end{aligned}

The garden is 7 meters wide. Check: $2(7 + 11) = 2(18) = 36$ . ✓

Stories with negative numbers

Temperatures, elevations and bank balances can go below zero. A decrease each hour or each minute becomes a negative rate.

Worked example: A falling temperature

At 6 p.m. the temperature is $5^\circ\text{F}$ . It drops $3$ degrees every hour. After how many hours will it be $-13^\circ\text{F}$ ?

Let $h$ be the number of hours. The temperature goes down $3$ degrees each hour, so it changes by $-3h$ :

\begin{aligned} 5 - 3h &= -13 \\ -3h &= -18 \\ h &= 6 \end{aligned}

It will be $-13^\circ\text{F}$ after 6 hours, at midnight. Check: $5 - 3(6) = 5 - 18 = -13$ . ✓

Common mistake

Don't just grab the numbers in the order they appear. In the gym story, "$25 sign-up fee plus $15 per month" does not mean $25m + 15$ . Ask yourself which amount happens each month. That one multiplies the variable.

Tip

Before you solve, estimate. If Ana paid $130 at $15 a month, she can't have been a member for 20 months, since that alone would be $300. An estimate helps you spot an answer that makes no sense.