Problem Solving

Learning Outcomes

Define a process for problem solving
- Translate words into algebraic expressions and equations
- Define a process for solving word problems
Solve problems containing rates
- Apply the steps for solving word problems to distance, rate, and time problems
- Apply the steps for solving word problems to interest rate problems
- Evaluate a formula using substitution
- Rearrange formulas to isolate specific variables
- Identify an unknown given a formula
Solve additional applications of linear equations
- Apply the steps for solving word problems to geometry problems
- Use the formula for converting between Fahrenheit and Celsius
- Evaluate a formula using substitution
- Rearrange formulas to isolate specific variables
- Identify an unknown given a formula

Define a Process for Problem Solving

Word problems can be tricky. Often it takes a bit of practice to convert an English sentence into a mathematical sentence, which is one of the first steps to solving word problems. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it’s good to see what mathematical operators are associated with them.

Addition [latex]+[/latex]	Subtraction [latex]-[/latex]	Multiplication [latex]\times[/latex]	Variable ?	Equals [latex]=[/latex]
More than	Less than	Double	A number	Is
Together	In the past	Product	Often, a value for which no information is given.	The same as
Sum	slower than	times	After how many hours?
Total	the remainder of		How much will it cost?
In the future	difference
faster than

Some examples follow:

[latex]x\text{ is }5[/latex] becomes [latex]x=5[/latex]
Three more than a number becomes [latex]x+3[/latex]
Four less than a number becomes [latex]x-4[/latex]
Double the cost becomes [latex]2\cdot\text{ cost }[/latex]
Groceries and gas together for the week cost $250 means [latex]\text{ groceries }+\text{ gas }=250[/latex]
The difference of 9 and a number becomes [latex]9-x[/latex]. Notice how 9 is first in the sentence and the expression

Let’s practice translating a few more English phrases into algebraic expressions.

Example

Translate the table into algebraic expressions:

some number	the sum of the number and 3	twice the sum of the number and 3
a length	double the length	double the length, decreased by 6
a cost	the difference of the cost and 20	2 times the difference of the cost and 20
some quantity	the difference of 5 and the quantity	the difference of 5 and the quantity, divided by 2
an amount of time	triple the amount of time	triple the amount of time, increased by 5
a distance	the sum of [latex]-4[/latex] and the distance	the sum of [latex]-4[/latex] and the twice the distance

Show Solution

In this example video, we show how to translate more words into mathematical expressions.

The power of algebra is how it can help you model real situations in order to answer questions about them.

Here are some steps to translate problem situations into algebraic equations you can solve. Not every word problem fits perfectly into these steps, but they will help you get started.

Read and understand the problem.
Determine the constants and variables in the problem.
Translate words into algebraic expressions and equations.
Write an equation to represent the problem.
Solve the equation.
Check and interpret your answer. Sometimes writing a sentence helps.

Example

Twenty-eight less than five times a certain number is 232. What is the number?

Show Solution

In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation.

For example, let’s say I want to know the next consecutive integer after 4. In mathematical terms, we would add 1 to 4 to get 5. We can generalize this idea as follows: the consecutive integer of any number, x, is [latex]x+1[/latex]. If we continue this pattern we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.

First	[latex]x[/latex]
Second	[latex]x+1[/latex]
Third	[latex]x+2[/latex]
Fourth	[latex]x+3[/latex]

We apply the idea of consecutive integers to solving a word problem in the following example.

Example

The sum of three consecutive integers is 93. What are the integers?

Show Solution

In the following video we show another example of a consecutive integer problem.

Rates

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. [latex]d=rt[/latex] 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 [latex]d=rt[/latex], 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 [latex]30\frac{\text{ miles }}{\text{ hour }}[/latex] for 2 hours, you can find the distance you would travel by multiplying rate times time or [latex]\left(30\frac{\text{ miles }}{\text{ hour }}\right)\left(2\text{ hours }\right)=60\text{ miles }[/latex].

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 [latex]d=rt[/latex] equation for t using division:

[latex]d=rt\\\frac{d}{r}=t[/latex]

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

[latex]d=rt\\\frac{d}{t}=r[/latex]

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.^[1] 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.^[2]

In the next examples we will use the [latex]d=rt[/latex] formula to answer the following questions about the two runners.

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

Show Solution

Runner	Distance =	(Rate )	(Time)
Trason	50 miles	r	6 hours
Johnson	50 miles	r	5.5 hours

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?

Show Solution

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 as long as Trason?

Show Solution

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?

Show Solution

Here is the table we created for reference:

Runner	Distance =	(Rate )	(Time)
Trason	50 miles	8.33 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]	6 hours
Johnson	50 miles	9.1 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]	5.5 hours

Read and Understand: we are looking for time, and this time our distance has changed from 50 miles to 1 mile, so we can use

[latex]d=rt\\\frac{d}{r}=t[/latex]

Define and Translate: we can use the formula [latex]d=rt[/latex] again. This time the unknown is t, the distance is 1 mile, and we know each runner’s rate. It may help to create a new table:

Runner	Distance =	(Rate )	(Time)
Trason	1 mile	8.33 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]	t hours
Johnson	1 mile	9.1 [latex]\frac{\text{ miles }}{\text{ hour }}[/latex]	t hours

Write and Solve:

Trason:

We will need to divide to isolate time.

[latex]\begin{array}{c}d=rt\\\\1\text{ mile }=8.33\frac{\text{ miles }}{\text{ hour }}\left(t\text{ hours }\right)\\\\\frac{1\text{ mile }}{\frac{8.33\text{ miles }}{\text{ hour }}}=t\text{ hours }\\\\0.12\text{ hours }=t\end{array}[/latex].

0.12 hours is about 7.2 minutes, so Trason’s time for running one mile was about 7.2 minutes. WOW! She did that for 6 hours!

Johnson:

We will need to divide to isolate time.

[latex]\begin{array}{c}d=rt\\\\1\text{ mile }=9.1\frac{\text{ miles }}{\text{ hour }}\left(t\text{ hours }\right)\\\\\frac{1\text{ mile }}{\frac{9.1\text{ miles }}{\text{ hour }}}=t\text{ hours }\\\\0.11\text{ hours }=t\end{array}[/latex].

0.11 hours is about 6.6 minutes, so Johnson’s time for running one mile was about 6.6 minutes. WOW! He did that for 5.5 hours!

Check and Interpret:

Have we answered the question? We were asked to find how long it took each runner to run one mile given the rate at which they ran the whole 50-mile course. Yes, we answered our question.

Trason’s mile time was [latex]7.2\frac{\text{minutes}}{\text{mile}}[/latex] and Johnsons’ mile time was [latex]6.6\frac{\text{minutes}}{\text{mile}}[/latex]

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 [latex]\displaystyle I=P\,\cdot \,R\,\cdot \,T[/latex].

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:

[latex]\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}[/latex]

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

Solve For	Result
I	[latex]I=P\,\cdot \,R\,\cdot \,T[/latex]
P	[latex]{P}=\frac{I}{{R}\,\cdot \,T}[/latex]
R	[latex]{R}=\frac{I}{{P}\,\cdot \,T}[/latex]
T	[latex]{T}=\frac{I}{{P}\,\cdot \,R}[/latex]

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?

Show Solution

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 3.25% monthly interest for 3 years. What amount of interest has Alex earned?

Show Solution

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?

Show Solution

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

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

Further Applications of Linear Equations

Formulas come up in many different areas of life. We have seen the formula that relates distance, rate, and time and the formula for simple interest on an investment. In this section we will look further at formulas and see examples of formulas for dimensions of geometric shapes as well as the formula for converting temperature between Fahrenheit and Celsius.

Geometry

There are many geometric shapes that have been well studied over the years. We know quite a bit about circles, rectangles, and triangles. Mathematicians have proven many formulas that describe the dimensions of geometric shapes including area, perimeter, surface area, and volume.

Perimeter

Perimeter is the distance around an object. For example, consider a rectangle with a length of 8 and a width of 3. There are two lengths and two widths in a rectangle (opposite sides), so we add [latex]8+8+3+3=22[/latex]. Since there are two lengths and two widths in a rectangle, you may find the perimeter of a rectangle using the formula [latex]{P}=2\left({L}\right)+2\left({W}\right)[/latex] where

L = Length

W = Width

In the following example, we will use the problem-solving method we developed to find an unknown width using the formula for the perimeter of a rectangle. By substituting the dimensions we know into the formula, we will be able to isolate the unknown width and find our solution.

Example

You want to make another garden box the same size as the one you already have. You write down the dimensions of the box and go to the lumber store to buy some boards. When you get there you realize you didn’t write down the width dimension—only the perimeter and length. You want the exact dimensions so you can have the store cut the lumber for you.

Here is what you have written down:

Perimeter = 16.4 feet
Length = 4.7 feet

Can you find the dimensions you need to have your boards cut at the lumber store? If so, how many boards do you need and what lengths should they be?

Show Solution

This video shows a similar garden box problem.

We could have isolated the w in the formula for perimeter before we solved the equation, and if we were going to use the formula many times, it could save a lot of time. The next example shows how to isolate a variable in a formula before substituting known dimensions or values into the formula.

Example

Isolate the term containing the variable, w, from the formula for the perimeter of a rectangle:

[latex]{P}=2\left({L}\right)+2\left({W}\right)[/latex].

Show Solution

Area

The area of a triangle is given by [latex] A=\frac{1}{2}bh[/latex] where

A = area
b = the length of the base
h = the height of the triangle

Remember that when two variables or a number and a variable are sitting next to each other without a mathematical operator between them, you can assume they are being multiplied. This can seem frustrating, but you can think of it like mathematical slang. Over the years, people who use math frequently have just made that shortcut enough that it has been adopted as convention.

In the next example we will use the formula for area of a triangle to find a missing dimension, as well as use substitution to solve for the base of a triangle given the area and height.

Example

Find the base (b) of a triangle with an area (A) of 20 square feet and a height (h) of 8 feet.

Show Solution

We can rewrite the formula in terms of b or h as we did with perimeter previously. This probably seems abstract, but it can help you develop your equation-solving skills, as well as help you get more comfortable with working with all kinds of variables, not just x.

Example

Use the multiplication and division properties of equality to isolate the variable b.

Show Solution

Use the multiplication and division properties of equality to isolate the variable h.

Show Solution

The following video shows another example of finding the base of a triangle given area and height.

Temperature

Let’s look at another formula that includes parentheses and fractions, the formula for converting from the Fahrenheit temperature scale to the Celsius scale.

[latex]C=\left(F-32\right)\cdot \frac{5}{9}[/latex]

Example

Given a temperature of [latex]12^{\circ}{C}[/latex], find the equivalent in [latex]{}^{\circ}{F}[/latex].

Show Solution

As with the other formulas we have worked with, we could have isolated the variable F first, then substituted in the given temperature in Celsius.

Example

Solve the formula shown below for converting from the Fahrenheit scale to the Celsius scale for F.

[latex]C=\left(F-32\right)\cdot \frac{5}{9}[/latex]

Show Solution

Think About It

Express the formula for the surface area of a cylinder, [latex]s=2\pi rh+2\pi r^{2}[/latex], in terms of the height, h.

In this example, the variable h is buried pretty deeply in the formula for surface area of a cylinder. Using the order of operations, it can be isolated. Before you look at the solution, use the box below to write down what you think is the best first step to take to isolate h.

Show Solution

In the last video, we show how to convert from celsius to fahrenheit.

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

[latex]a[/latex]	[latex]a+3[/latex]	[latex]2\left(x+3\right)[/latex]
[latex]l[/latex]	[latex]2l[/latex]	[latex]2l-6[/latex]
[latex]c[/latex]	[latex]c-20[/latex]	[latex]2\left(c-20\right)[/latex]
[latex]q[/latex]	[latex]5-q[/latex]	[latex]\frac{5-q}{2}[/latex]
[latex]t[/latex]	[latex]3t[/latex]	[latex]3t+5[/latex]
[latex]d[/latex]	[latex]-4+d[/latex]	[latex]-4+2d[/latex]

Module 1: Solving Equations and Inequalities

Learning Outcomes

Define a Process for Problem Solving

Example

Example

Example

Rates

Distance, Rate, and Time

Example

Example

Examples

Example

Simple Interest

Example

Example

Example

Further Applications of Linear Equations

Geometry

Perimeter

Example

Example

Area

Example

Answer

Example

Temperature

Example

Example

Answer

Think About It