# Applications of One-Step Equations

## Real-world problems using single variable equations.

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Applications of One-Step Equations

### Applications of One-Step Equations

Let's use the skills we learned in the last two concepts to solve applications of one-step problems.

#### Real-World Application: Birth Date

In the year 2017, Anne will be 45 years old. In what year was Anne born?

The unknown here is the year Anne was born, so that’s our variable \begin{align*}x\end{align*}. Here’s our equation:

\begin{align*}x + 45 &= 2017\\ x + 45 - 45 &= 2017 - 45\\ x &= 1972\end{align*}

Anne was born in 1972.

#### Real-World Application: DVD Players

A mail order electronics company stocks a new mini DVD player and is using a balance to determine the shipping weight. Using only one-pound weights, the shipping department found that the following arrangement balances:

How much does each DVD player weigh?

Since the system balances, the total weight on each side must be equal. To write our equation, we’ll use \begin{align*}x\end{align*} for the weight of one DVD player, which is unknown. There are two DVD players, weighing a total of \begin{align*}2x\end{align*} pounds, on the left side of the balance, and on the right side are 5 1-pound weights, weighing a total of 5 pounds. So our equation is \begin{align*}2x = 5\end{align*}. Dividing both sides by 2 gives us \begin{align*}x = 2.5\end{align*}.

Each DVD player weighs 2.5 pounds.

#### Real-World Application: Hot Dogs

In 2004, Takeru Kobayashi of Nagano, Japan, ate 53.5 hot dogs in 12 minutes. This was 3 more hot dogs than his own previous world record, set in 2002. Calculate how many minutes it took him to eat one hot dog.

We know that the total time for 53.5 hot dogs is 12 minutes. We want to know the time for one hot dog, so that’s \begin{align*}x\end{align*}. Our equation is \begin{align*}53.5x = 12\end{align*}. Then we divide both sides by 53.5 to get \begin{align*}x = \frac{12}{53.5}\end{align*}, or \begin{align*}x = 0.224 \ minutes\end{align*}.

We can also multiply by 60 to get the time in seconds; 0.224 minutes is about 13.5 seconds. So that’s how long it took Takeru to eat one hot dog.

### Examples

Calculate the following, using the problem from the hot dog example:

#### Example 1

How many hot dogs he ate per minute.

For this questions we’re looking for hot dogs per minute instead of minutes per hot dog. We’ll use the variable \begin{align*}y\end{align*} instead of \begin{align*}x\end{align*} this time so we don’t get the two confused. 12 minutes, times the number of hot dogs per minute, equals the total number of hot dogs, so \begin{align*}12y = 53.5\end{align*}. Dividing both sides by 12 gives us \begin{align*}y = \frac{53.5}{12}\end{align*}, or \begin{align*}y = 4.458\end{align*} hot dogs per minute.

#### Example 2

What his old record was.

We know that his new record is 53.5, and we know that’s three more than his old record. If we call his old record \begin{align*}z\end{align*}, we can write the following equation: \begin{align*}z + 3 = 53.5\end{align*}. Subtracting 3 from both sides gives us \begin{align*}z = 50.5\end{align*}. So Takeru’s old record was 50.5 hot dogs in 12 minutes.

### Review

Peter is collecting tokens on breakfast cereal packets in order to get a model boat. In eight weeks he has collected 10 tokens. He needs 25 tokens for the boat. Write an equation and determine the following information.

1. How many more tokens he needs to collect, \begin{align*}n\end{align*}.
2. How many tokens he collects per week, \begin{align*}w\end{align*}.
3. How many more weeks remain until he can send off for his boat, \begin{align*}r\end{align*}.

Juan has baked a cake and wants to sell it in his bakery. He is going to cut it into 12 slices and sell them individually. He wants to sell it for three times the cost of making it. The ingredients cost him $8.50, and he allowed$1.25 to cover the cost of electricity to bake it. Write equations that describe the following statements.

1. The amount of money that he sells the cake for \begin{align*}(u)\end{align*}.
2. The amount of money he charges for each slice \begin{align*}(c)\end{align*}.
3. The total profit he makes on the cake \begin{align*}(w)\end{align*}.

Jane is baking cookies for a large party. She has a recipe that will make one batch of two dozen cookies, and she decides to make five batches. To make five batches, she finds that she will need 12.5 cups of flour and 15 eggs.

1. How many cookies will she make in all?
2. How many cups of flour go into one batch?
3. How many eggs go into one batch?
4. If Jane only has a dozen eggs on hand, how many more does she need to make five batches?
5. If she doesn’t go out to get more eggs, how many batches can she make? How many cookies will that be?

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