Essential modeling

Essential modeling is the skill of taking a real-world scenario (usually written as a word problem) and turning it into a mathematical problem you can solve. This is a step up from simply writing an equation from a sentence. In modeling problems, you may need to draw a diagram, set up one or more equations, and use several algebra skills together.

Words to algebra

Here’s a quick summary of how to turn a word problem into an algebraic equation.

Look for:

• Numbers (given values)
• Variables (unknown values you need to find)
• Keywords like more, less, times, total, difference, and each

Keywords suggest which operations to use (addition, subtraction, multiplication, division). The numbers tell you the amounts involved, and the variables represent what you’re solving for.

Remember: a variable might be given as a letter like $a$, or it might simply be “the missing value” the question asks you to find.

Modeling practice

Let’s work through a few examples to practice identifying the important information and turning it into an equation or a drawing.

Example 1

Pat is a bus driver who drives to school twice a day, Monday through Friday. How many times will Pat have driven to school in $4$ weeks?

Step 1 is to identify the key information:

• Pat drives twice a day.
• Pat drives Monday through Friday, which is 5 days per week.

So Pat drives $2\times 5=10$ times per week. In $4$ weeks, that’s:

$$10\ast 4=40$$

Pat drove to school $40$ times in $4$ weeks.

In this example, you didn’t really need a formal equation because you could solve it step by step. You could write an equation, but it isn’t necessary here.

Example 2

Ryan wants to buy a subscription to a streaming service that costs $\$9$ per month. The service allows him to stream the first month for free, but each following month will cost $\$9$. How many months can Ryan keep this subscription if he only has $\$39$?

This problem has more pieces, so it helps to write an equation.

Key information:

• The subscription costs $\$9$ per month.
• The first month is free.
• Ryan has $\$39$ total.
• The number of months is the unknown.

Let $m$ be the total number of months Ryan has the subscription.

If the first month is free, then Ryan pays for only $m-1$ months. Each paid month costs $9$, so the total cost is $9(m-1)$. Set that equal to $39$:

$$9(m-1)=39$$

This is equivalent to:

$$9m-9=39$$

Solving for $m$, we get:

$$\begin{array}{rl}9m& =39+9\\ m& =48/9\\ m& =5.33\end{array}$$

The result $5.33$ means Ryan can afford 5 full months, but not 6. Since the question asks how many months he can keep the subscription, the final answer is $5$ months.

Example 3

A rectangular dining room has a length that is twice as long as its width. The tables in the room have a radius of $1$ meter. If the width of the dining room is $10$ meters and there are $30$ tables, how much of the room in square meters is not taken up by the tables?

There’s a lot of information here, so start by listing what you know and what formulas you’ll use:

• Area of room $=$ length $\ast$ width
• Length $=2\ast$ width
• Width $=10$ meters
• Area of table $=pi\ast r^2$
• Radius ($r$) $=1$ meter
• Space not taken up by tables $=$ area of dining room $-$ (area of a table) $\ast 30$

Start with the room.

• Width is $10$ meters.
• Length is twice the width, so $L=20$ meters.
• Area of the room is $A=20\times 10=200$ square meters.

Now find the area of the tables.

• Area of one table is $A=pi\ast (1{)}^2=pi$ square meters.
• Area of 30 tables is $30\ast pi\approx 94.3$ square meters.

Now subtract to find the space not taken up by tables:

$$200-94.3=105.7\text{square meters}$$