## Problems Involving Formulas I

### Learning Outcomes

• Solve a formula for a specific variable
• Use the distance, rate, and time formula
• Apply for the steps for solving word problems to interest rate problems

There is often a well-known formula or relationship that applies to a word problem. For example, if you were to plan a road trip, you would want to know how long it would take you to reach your destination. $d=rt$ is a well-known relationship that associates distance traveled, the rate at which you travel, and how long the travel takes.

## Distance, Rate, and Time

If you know two of the quantities in the relationship $d=rt$, you can easily find the third using methods for solving linear equations. For example, if you know that you will be traveling on a road with a speed limit of $30\frac{\text{ miles }}{\text{ hour }}$ for $2$ hours, you can find the distance you would travel by multiplying rate times time or $\left(30\frac{\text{ miles }}{\text{ hour }}\right)\left(2\text{ hours }\right)=60\text{ miles }$.

### Try It

We can generalize this idea depending on what information we are given and what we are looking for. For example, if we need to find time, we could solve the $d=rt$ equation for $t$ using division:

$d=rt$

$\frac{d}{r}=t$

Likewise, if we want to find rate, we can isolate $r$ using division:

$d=rt$

$\frac{d}{t}=r$

In the following examples, you will see how this formula is applied to answer questions about ultra marathon running. Ann Trason

Ultra marathon running (defined as anything longer than $26.2$ miles) is becoming very popular among women even though it remains a male-dominated niche sport. Ann Trason has broken twenty world records in her career. One such record was the American River $50$-mile Endurance Run, which begins in Sacramento, California, and ends in Auburn, California. In 1993, Trason finished the run with a time of $6:09:08$.  The men’s record for the same course was set in 1994 by Tom Johnson, who finished the course with a time of  $5:33:21$.

In the next examples, we will use the $d=rt$ formula to answer the following questions about the two runners.

1. What was each runner’s rate for their record-setting runs?
2. By the time Johnson had finished, how many more miles did Trason have to run?
3. How much further could Johnson have run if he had run as long as Trason?
4. What was each runner’s time for running one mile?

To make answering the questions easier, we will round the two runners’ times to $6$ hours and $5.5$ hours.

### Example

What was each runner’s rate for their record-setting runs? Round to two decimal places.

Now that we know each runner’s rate, we can answer the second question.

### Example

By the time Johnson had finished, how many more miles did Trason have to run?

The third question is similar to the second. Now that we know each runner’s rate, we can answer questions about individual distances or times.

### Examples

How much further could Johnson have run if he had run for the same amount of time as Trason?

Now we will tackle the last question, where we are asked to find a time for each runner.

### Example

What was each runner’s time for running one mile?

In the following video, we show another example of answering many rate questions given distance and time.

## Simple Interest

In order to entice customers to invest their money, many banks will offer interest-bearing accounts. The accounts work like this: a customer deposits a certain amount of money (called the Principal, or $P$), which then grows slowly according to the interest rate ($r$, measured in percent) and the length of time ($t$, usually measured in months) that the money stays in the account. The amount earned over time is called the interest ($I$), which is then given to the customer. Beware! Interest rates are commonly given as yearly rates, but can also be monthly, quarterly, bimonthly, or even some custom amount of time. It is important that the units of time and the units of the interest rate match. You will see why this matters in a later example.

The simplest way to calculate interest earned on an account is through the formula $\displaystyle I=P\,\cdot \,r\,\cdot \,t$

If we know any of the three amounts related to this equation, we can find the fourth. For example, if we want to find the time it will take to accrue a specific amount of interest, we can solve for $t$ using division:

$\displaystyle\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,I=P\,\cdot \,r\,\cdot \,t\\\\ \frac{I}{{P}\,\cdot \,r}=\frac{P\cdot\,r\,\cdot \,t}{\,P\,\cdot \,r}\\\\\,\,\,\,\,\,\,\,\,\,\,{t}=\frac{I}{\,r\,\cdot \,t}\end{array}$

Below is a table showing the result of solving for each individual variable in the formula.

Solve For Result
$I$ $I=P\,\cdot \,r\,\cdot \,t$
$P$ ${P}=\frac{I}{{r}\,\cdot \,t}$
$r$ ${r}=\frac{I}{{P}\,\cdot \,t}$
$t$  ${t}=\frac{I}{{P}\,\cdot \,r}$

In the next examples, we will show how to substitute given values into the simple interest formula, and decipher which variable to solve for.

### Example

If a customer deposits a principal of $2000$ at a monthly rate of  $0.7\%$, what is the total amount that she has after $24$ months?

The following video shows another example of finding an account balance after a given amount of time, principal invested, and a rate.

In the following example, you will see why it is important to make sure the units of the interest rate match the units of time when using the simple interest formula.

### Example

Alex invests $600$ at $3.5\%$ monthly interest for $3$ years. What amount of interest has Alex earned?

In the following video, we show another example of how to find the amount of interest earned after an investment has been sitting for a given monthly interest.

### Example

After  $10$ years, Jodi’s account balance has earned  $1080$ in interest. The rate on the account is  $0.09\%$ monthly. What was the original amount she invested in the account?

The last video shows another example of finding the principal amount invested based on simple interest.

### Try it

In the next section, we will apply our problem-solving method to problems involving dimensions of geometric shapes.

## Contribute!

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1. "Ann Trason." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/Ann_Trason.
2.  "American River $50$ Mile Endurance Run." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/American_River_50_Mile_Endurance_Run.