Escrevendo literalmente uma equação com uma variável aleatória

We're going to talk about what literal equations are and how to represent them using variables (which are usually letters, of course). To do so, we're going to use a type of brainteaser that has been swarming over social media for the past few years.

This innocuous looking blend of fruits and mathematical symbols has appeared in many different instances, on Twitter, Facebook, and Instagram challenging people to solve equations where the variables are represented with fruits instead of letters. We call any equation with more than one different variable a literal equation, so the above is a literal equation because each piece of fruit is a different variable.

This equation, however, is NOT a literal equation because there is only one variable, the kiwi, even though it appears multiple times in the equation.

We can turn both of these equations into equations with variables, by first turning them into equations with words.

The first picture says "oranges plus kiwis plus bananas equals $12,$" or the total number of all three fruits is $12.$ To write this as an equation with variables, we need variable names. Let $o$ be the number of oranges, $k$ the number of kiwis, $b$ the number of bananas, then we get

$$o + k + b = 12,$$

replacing every name with its variable. The second equation reads, "kiwis plus five times kiwis equals $66,$" which becomes

$$k + 5 \times k = 66$$

by replacing the word "kiwis" with a $k$ every time.

Then solve this two-step equation like usual, following the Addition and Subtraction Properties of Equality & the Multiplication and Division Properties of Equality.

$$k + 5k = 66$$ $$6k = 66$$ $$k = 11$$

Therefore, the kiwi, $k$ represents $11$.

Writing literal equations using variables is a lot like translating word problems into mathematical expressions. The general approach is to take the information given to you in words and turn it into an equation.

Sometimes, you may already know the actual equation you should be using. For example, say you want to find the amount of simple interest, and you know the principal, rate, and time. Then, you can plug that information into the simple interest literal equation, $I=PRT,$ to find the answer!

Let's look at another example:

Simon is interested in comparing temperatures in China (in Celsius) and temperatures in the US (in Fahrenheit). He finds out that the temperature in Fahrenheit is $32\text{ degrees}$ more than $\frac{9}{5}$ of the temperature in Celsius. Write an equation to model this information.

We've got two quantities whose values we don't actually know—the temperature in Celsius and the temperature in Fahrenheit. We need to represent those as variables.

You can certainly use $x$ and $y$, but we're going to use $C$ and $F$ to help us keep track of which variable represents which temperature scale.

We simply need to translate the information given to us in the word problem into an equation with the variables $C$ and $F$. Try writing down an equation on your own, then click continue to reveal the answer.

$$F = \frac{9}{5}C + 32$$

This equation tells us what measurements in Fahrenheit ($F$) are in terms of measurements in Celsius ($C$). To find it, we knew we needed to have an equation that compared $F$ to $C$, so $F$ and $C$ have to be on opposite sides of an equals sign.

Let's start by writing that down. $$F = C$$

Next, we know that Fahrenheit is $32\text{ degrees}$ more than Celsius multiplied by $\frac{9}{5}$. So to find a measurement in $F$, we'll have to multiply $C$ by $\frac{9}{5}$, then add $32$: $$F = \frac{9}{5}C + 32$$

Done!

Example: Word Problem

Liz is a Writing major in college and she always have a bunch of papers to write. Like the rest of us, Liz procrastinates from time to time and leaves her papers until the last minute.

She decided to figure out a way to calculate how much time she needs to finish any paper. So when she does want to procrastinate, she will still be able to finish on time.

Let $y$ be the amount of time in minutes she needs to finish, let $x$ be the number of pages she needs to finish for any given paper. On average, it takes Liz $45$ minutes to write a page.

With the units included, the relationship between $x$ and $y$ is: $$y \text{ (min)} = 45\left(\frac{\text{min}}{\text{page}}\right)x\text{ (page)}$$

To be more realistic, Liz wants to take into account of rest time. Liz thinks she needs a $20\text{ minute}$ break every $3\text{ pages}$ she writes. Can you write a new equation?

Every $3\text{ pages}$, Liz rests $20\text{ minutes}$. Therefore for $x\text{ pages}$, she spends $20\times\frac{x}{3}=\frac{20x}{3}\text{ minutes}$ for a break.

(You can also think this way: on average, Liz spends $\frac{20}{3}\text{ minutes}$ for a break every page. If she writes $x\text{ pages}$, she spends $20\frac{x}{3}=\frac{20x}{3}\text{ minutes}$ for a break.

With the units included, the relationship between $x$ and $y$ is: $$y \text{ (min)} = 45\left(\frac{\text{min}}{\text{page}}\right)x\text{ (page)}+\frac{20}{3}\left(\frac{\text{min}}{\text{page}}\right)x\text{ (page)}$$

(Video) Solving Literal Equations Made Easy

by Brian McLogan

This video by Brian McLogan works through the steps of solving a literal equation.

$\large{\text{Summary}}$

A literal equation is an equation where every variable represents a quantity in the real-world. In other words, literal equations model real-world scenarios. These scenarios can be whatever you want, and we can pick any scenario to demonstrate HOW to build a literal equation. The key is to read the description very carefully to answer two important questions:

1. What variables should be in the equation?
2. How are the variables in the equation related?

To answer the first question, we keep track of what objects are mentioned in the description given. To answer the second equation, we look for words and phrases that compare quantities, such as "more," "less," "twice as many," and so on. This video walks through an example involving an area formula.

$\large{\text{Bonus Example}}$

Let's work through an example that wasn't in the video.

Here is the scenario:

"Tim and Janie planted tulips, roses, sunflowers, and watermelons in their garden. The product of the number of tulips and sunflowers in the garden is $2$ less than the total number of roses and watermelons in the garden."

Our task is to write a literal equation describing this relationship. As we said earlier, the first step is to identify the variables. Each plant will be a variable, since the scenario talks about numbers of different plants. We assign each plant a letter:

• $T$: number of tulips
• $R$: number of roses
• $S$: number of sunflowers
• $W$: number of watermelons

Now that we have the variables, we need to consider how they are related. The first phrase to consider is "the product of the number of tulips and sunflowers in the garden." This is a cue that we will be multiplying $T$ and $S$, so beginning to build our equation looks like $$T \times S$$

The next phrase is "two less than," which means when we build our equation, the left side of the equation will be $T \times S,$ and the right side will be something minus $2.$

Now we need to figure out what the $?$ is. The final phrase in our scenario is "the total number of roses and watermelons." Here the variables involved are $R$ and $W,$ and we see the word "total," which means we have a sum. Thus our final equation is

$$T \times S = R + W - 2.$$

That's all there is to it. You just need to be careful about where the equals sign goes. Usually, a scenario will start talking about one side of the equation, and then switch over to talk about the other side. If you see the words "as many," "equals," "more," "less," or "the same as," these are good indicators that we've switched which side of the equation we're talking about.