MFG Algebraic Expressions and Problem Solving

Section A.3 Algebraic Expressions and Problem Solving

You are familiar with the use of letters, or variables, to stand for unknown numbers in equations or formulas. Variables are also used to represent numerical quantities that change over time or in different situations. For example, $p$ might stand for the atmospheric pressure at different heights above the Earth's surface. Or $N$ might represent the number of people infected with cholera $t$ days after the start of an epidemic.

An algebraic expression is any meaningful combination of numbers, variables, and symbols of operation. Algebraic expressions are used to express relationships between variable quantities.

Example A.17.

Loren makes $$6$ an hour working at the campus bookstore.

Choose a variable for the number of hours Loren works per week.
Write an algebraic expression for the amount of Loren's weekly earnings.

Solution

Let $h$ stand for the number of hours Loren works per week.
The amount Loren earns is given by

\begin{equation*} \blert{6\times (\text{number of hours Loren worked})} \end{equation*}

or $6\cdot h\text{.}$ Loren's weekly earnings can be expressed as $6h\text{.}$

The algebraic expression $6h$ represents the amount of money Loren earns in terms of the number of hours she works. If we substitute a specific value for the variable in an expression, we find a numerical value for the expression. This is called evaluating the expression.

Example A.18.

If Loren from Example A.17 works for $16$ hours in the bookstore this week, how much will she earn?

Solution

Evaluate the expression $6h$ for $h=\alert{16}\text{.}$

\begin{equation*} 6h = 6(\alert{16}) = 96 \end{equation*}

Loren will make $$96\text{.}$

Example A.19.

April sells environmentally friendly cleaning products. Her income consists of $$200$ per week plus a commission of $9\%$ of her sales.

Choose variables to represent the unknown quantities and write an algebraic expression for April's weekly income in terms of her sales.
Find April's income for a week in which she sells $$350$ worth of cleaning products.

Solution

Let $I$ represent April's total income for the week, and let $S$ represent the total amount of her sales. We translate the information from the problem into mathematical language as follows:

\begin{gather*} \blert{\text{Her income consists of }$200 . . .\text{ plus }. . . 9\% \text{ of her sales}} \\ I \hphantom{consists of}= \hphantom{of}200 \hphantom{plus+}+ \hphantom{....}0.09 \hphantom{of her}S \end{gather*}

Thus, $I = 200 + 0.09S\text{.}$
We want to evaluate our expression from part (a) with $S = 350\text{.}$ We substitute $\alert{350}$ for $S$ to find

\begin{equation*} I = 200 + 0.09(\alert{350}) \end{equation*}

Following the order of operations, we perform the multiplication before the addition. Thus, we begin by computing $0.09(350)\text{.}$

\begin{align*} I \amp = 200 + 0.09(350)\amp\amp\blert{\text{Multiply }0.09 (350) \text{ first.}}\\ \amp = 200 + 31.5\\ \amp = 231.50 \end{align*}

April's income for the week is $$231.50\text{.}$

Remark A.20. Calculator Tip.

On a scientific or a graphing calculator, we can enter the expression from Example A.19 just as it is written:

\begin{equation*} 200 + 0.09 ~ \boxed{\times}~ 350 ~\boxed{\text{ENTER}} \end{equation*}

The calculator will perform the operations in the correct order—multiplication first.

Example A.21.

Economy Parcel Service charges $$2.80$ per pound to deliver a package from Pasadena to Cedar Rapids. Andrew wants to mail a painting that weighs $8.3$ pounds, plus whatever packing material he uses.

Choose variables to represent the unknown quantities and write an expression for the cost of shipping Andrew's painting.
Find the shipping cost if Andrew uses $2.9$ pounds of packing material.

Solution

Let $C$ stand for the shipping cost and let $w$ stand for the weight of the packing material. Andrew must find the total weight of his package first, then multiply by the shipping charge.

The total weight of the package is $8.3 + w$ pounds. We use parentheses around this expression to show that it should be computed first, and the sum should be multiplied by the shipping charge of $$2.80$ per pound. Thus,

\begin{equation*} C = 2.80(8.3 + w) \end{equation*}
Evaluate the formula from part (a) with $w = \alert{2.9}\text{.}$

\begin{align*} C \amp = 2.80(8.3 + \alert{2.9})\amp\amp\blert{\text{Add inside parentheses.}}\\ \amp = 2.80(11.2)\amp\amp\blert{\text{Multiply.}}\\ \amp = 31.36 \end{align*}

The cost of shipping the painting is $$31.36\text{.}$

Remark A.22. Calculator Tip.

On a calculator, we enter the expression for $C$ in the order it appears, including the parentheses. (Experiment to see whether your calculator requires you to enter the $\times$ symbol after 2.80.) The keying sequence

\begin{equation*} 2.80 \times ( 8.3 + 2.9 )~~ \boxed{\text{ENTER}} \end{equation*}

gives the correct result, $31.36\text{.}$

Caution A.23.

If we omit the parentheses, the calculator will perform the multiplication before the addition. Thus, the keying sequence

\begin{equation*} 2.80 \times 8.3 + 2.9 \end{equation*}

gives an incorrect result for Example A.21. (The sequence

\begin{equation*} 8.3 + 2.9 \times 2.80 \end{equation*}

does not work either!)

Subsection Problem Solving

Problem solving often involves translating a real-life problem into a computer programming language, or, in our case, into algebraic expressions. We can then use algebra to solve the mathematical problem and interpret the solution in the context of the original problem. Here are some guidelines for problem solving with algebraic equations.

Guidelines for Problem Solving.

Identify the unknown quantity and assign a variable to represent it.
Find some quantity that can be expressed in two different ways and write an equation.
Solve the equation.
Interpret your solution to answer the question in the problem.

In step 1, begin by writing an English phrase to describe the quantity you are looking for. Be as specific as possible—if you are going to write an equation about this quantity, you must understand its properties! Remember that your variable must represent a numerical quantity. For example, $x$ can represent the speed of a train, but not just “the train.”

Writing an equation is the hardest part of the problem. Note that the quantity mentioned in step 2 will probably not be the same unknown quantity you are looking for, but the algebraic expressions you write will involve your variable. For example, if your variable represents the speed of a train, your equation might be about the distance the train traveled.

Subsection Supply and Demand

The law of supply and demand is fundamental in economics. If you increase the price of a product, the supply increases because its manufacturers are willing to provide more of the product, but the demand decreases because consumers are not willing to buy as much at a higher price. The price at which the demand for a product equals the supply is called the equilibrium price.

Example A.24.

The Coffee Connection finds that when it charges $p$ dollars for a pound of coffee, it can sell $800 - 60p$ pounds per month. On the other hand, at a price of $p$ dollars a pound, International Food and Beverage will supply the Connection with $175 + 40p$ pounds of coffee per month. What price should the Coffee Connection charge for a pound of coffee so that its monthly inventory will sell out?

Solution

We are looking for the equilibrium price, $p\text{.}$
The Coffee Connection would like the demand for its coffee to equal its supply. We equate the expressions for supply and for demand to obtain the equation

\begin{equation*} 800 - 60p = 175 + 40p \end{equation*}
Solve the equation. To get all terms containing the variable, $p\text{,}$ on one side of the equation, we add $60p$ to both sides and subtract $175$ from both sides to obtain

\begin{align*} 800 - 60p + \alert{60p - 175} \amp= 175 + 40p + \alert{60p - 175}\\ 625 \amp = 100p \amp\amp\blert{\text{Divide both sides by }100.}\\ 6.25 \amp = p \end{align*}
The Coffee Connection should charge $$6.25$ per pound for its coffee.

Subsection Percent Problems

Recall the basic formula for computing percents.

Percent Formula.

\begin{equation*} P = rW \end{equation*}

the Part (or percent) = the percentage rate $\times$ the Whole Amount

A percent increase or percent decrease is calculated as a fraction of the original amount. For example, suppose you make $$16.00$ an hour now, but next month you are expecting a $5\%$ raise. Your new salary should be

\begin{equation*} \stackrel{\text{Original salary}}{\$16.00} + \stackrel{\text{Increase}}{0.05(\$16.00)} = \stackrel{\text{New Salary}}{\$16.80} \end{equation*}

Example A.25.

The price of housing in urban areas increased $4\%$ over the past year. If a certain house costs $$100,000$ today, what was its price last year?

Solution

Let $c$ represent the cost of the house last year.
Express the current price of the house in two different ways. During the past year, the price of the house increased by $4\%\text{,}$ or $0.04c\text{.}$ Its current price is thus

\begin{equation*} \stackrel{\text{Original cost}}{(1)c} + \stackrel{\text{Price increase}}{0.04c} = c(1 + 0.04) = 1.04c \end{equation*}

This expression is equal to the value given for current price of the house:

\begin{equation*} 1.04c = 100,000 \end{equation*}
To solve this equation, we divide both sides by $1.04$ to find

\begin{equation*} c = \frac{100,000}{1.04}= 96,153.846 \end{equation*}
To the nearest cent, the cost of the house last year was $$96,153.85\text{.}$

Caution A.26.

In Example A.25, it would be incorrect to calculate last year's price by subtracting $4\%$ of $$100,000$ from $$100,000$ to get $$96,000\text{.}$ (Do you see why?)

Subsection Weighted Averages

We find the average, or mean, of a set of values by adding up the values and dividing the sum by the number of values. Thus, the average, $\overline{x}\text{,}$ of the numbers $x_1, x_2, \ldots , x_n$ is given by

\begin{equation*} \overline{x} = \frac{x_1 + x_2 + \cdot \cdot \cdot +x_n}{n} \end{equation*}

In a weighted average, the numbers being averaged occur with different frequencies or are weighted differently in their contribution to the average value. For instance, suppose a biology class of 12 students takes a 10-point quiz. Of the 12 students, 2 receive 10s, three receive 9s, 5 receive 8s, and 2 receive scores of 6. The average score earned on the quiz is then

\begin{equation*} \overline{x} = \frac{\alert{2}(10) + \alert{3}(9) + \alert{5}(8) + \alert{2}(6)}{12}=8.25 \end{equation*}

The numbers in color are called the weights—in this example they represent the number of times each score was counted. Note that $n\text{,}$ the total number of scores, is equal to the sum of the weights:

\begin{equation*} 12 = 2 + 3 + 5 + 2 \end{equation*}

Example A.27.

Kwan's grade in his accounting class will be computed as follows: Tests count for $50\%$ of the grade, homework counts for $20\%\text{,}$ and the final exam counts for $30\%\text{.}$ If Kwan has an average of $84$ on tests and $92$ on homework, what score does he need on the final exam to earn a grade of $90\text{?}$

Solution

Let $x$ represent the final exam score Kwan needs.
Kwan's grade is the weighted average of his test, homework, and final exam scores.

\begin{equation*} \frac{\alert{0.50}(84) + \alert{0.20}(92) + \alert{0.30}x}{1.00}=90 \end{equation*}

(The sum of the weights is 1.00, or 100% of Kwan’s grade.) Multiply both sides of the equation by $1.00$ to get

\begin{equation*} 0.50(84) + 0.20(92) + 0.30x = 1.00(90) \end{equation*}
Solve the equation. Simplify the left side first.

\begin{align*} 60.4 + 0.30x \amp = 90\amp\amp\blert{\text{Subtract 60.4 from both sides.}}\\ 0.30x \amp = 29.6\amp\amp\blert{\text{Divide both sides by 0.30.}}\\ x \amp = 98.7 \end{align*}
Kwan needs a score of $98.7$ on the final exam to earn a grade of $90\text{.}$

In step 2 of Example A.27, we rewrote the formula for a weighted average in a simpler form.

Weighted Average.

The sum of the weighted values equals the sum of the weights times the average value. In symbols,

\begin{equation*} w_1x_1 + w_2x_2 + \cdots + w_nx_n = W\, \overline{x} \end{equation*}

where $W$ is the sum of the weights.

This form is particularly useful for solving problems involving mixtures.

Example A.28.

The vet advised Delbert to feed his dog Rollo with kibble that is no more than $8\%$ fat. Rollo likes JuicyBits, which are $15\%$ fat. LeanMeal is more expensive, but it is only $5\%$ fat. How much LeanMeal should Delbert mix with $50$ pounds of JuicyBits to make amixture that is $8\%$ fat?

Solution

Let $p$ represent the number of pounds of LeanMeal needed.
In this problem, we want the weighted average of the fat contents in the two kibbles to be $8\%\text{.}$ The weights are the number of pounds of each kibble we use. It is often useful to summarize the given information in a table.

 $\%$ fat Total pounds Pounds of fat

Juicy Bits $15\%$ $50$ $0.15(50)$

LeanMeal $5\%$ $p$ $0.05p$

Mixture $8\%$ $50+p$ $0.08(50+p)$

The amount of fat in the mixture must come from adding the amounts of fat in the two ingredients. This gives us an equation,

\begin{equation*} 0.15(50) + 0.05p = 0.08(50 + p) \end{equation*}

This equation is an example of the formula for weighted averages.
Simplify each side of the equation, using the distributive law on the right side, then solve.

\begin{align*} 7.5 + 0.05p \amp = 4 + 0.08p \amp\amp\blert{\text{Subtract}~4+0.05p~\text{from both sides.}}\\ 3.5 \amp = 0.03p \amp\amp \blert{\text{Divid both sides by}~ 0.03.}\\ p \amp = 116.\overline{6} \end{align*}
Delbert should mix $116\frac{2}{3}$ pounds of LeanMeal with $50$ pounds of JuicyBits to make a mixture that is $8\%$ fat.

Subsection Section Summary

Subsubsection Vocabulary

Look up the definitions of new terms in the Glossary.

Variable
Weighted average
Demand
Equilibrium price
Supply
Algebraic expression
Evaluate an expression

Subsubsection SKILLS

Practice each skill in the exercises listed.

Write an algebraic expression: #1–12
Evaluate an algebraic expression: #1–12
Write and solve an equation to solve a problem: #13–28

Exercises Exercises A.3

For Problems 1-12, write algebraic expressions to describe the situation and then evaluate for the given values.

1.

Jim was 27 years old when Ana was born.

Write an expression for Jim's age in terms of Ana's age.
Use your expression to find Jim's age when ana is 22 years old.

2.

Rani wants to replace the wheels of her in-line skates. New wheels cost $6.59 each.

Write an expression for the total cost of new wheels in terms of the number of wheels Rani must replace.
Use your expression to find the total cost if Rani must replace 8 wheels.

3.

Helen decides to drive to visit her father. The trip is a distance of 1260 miles.

Write an expression for the total number of hours Helen must drive in terms of her average driving speed.
Use your expression to find how long Helen must drive if she averages 45 miles per hour.

4.

Ben will inherit one million dollars on his twenty-first birthday.

Write an expression for the number of years before Ben gets his inheritance in terms of his present age.
Use your expression to find how many more years Ben must wait after he turns 13 years old.

5.

The area of a circle is equal to $\pi$ times the square of its radius.

Write an expression for the area of a circle in terms of its radius.
Find the area of a circle whose radius is 5 centimeters.

6.

The volume of a sphere is equal to $\frac{4}{3} \pi$ times the cube of the radius.

Write an expression for the volume of a sphere in terms of its radius.
Find the volume of a sphere whose radius is 5 centimeters.

7.

The sales tax in the city of Preston is 7.9%.

Write an expression for the total bill for an item (price plus tax) in terms of the price of the item.
Find the total bill for an item whose price is $490.

8.

A savings account pays 6.4% annual interest on the amount deposited.

Write an expression for the balance (initial deposit plus interest) in the account after one year in terms of the amount deposited.
Find the total amount in the account after one year if $350 was deposited.

9.

Your best friend moves to another state. To call her, a long-distance phone call costs $1.97 plus $0.39 for each minute.

Write an expression for the cost of a long-distance phone call in terms of the number of minutes of the call.
Find the cost of a 27-minute phone call.

10.

Arenac Airplines charges 47 cents per pound on its flight from Omer to Pinconning, both for passengers and for luggage. Mr. Owsley wants to take the flight with 15 pounds of luggage.

Write an expression for the cost of the flight in terms of Mr. Owlsley's weight.
Find the cost if Mr. Owsley weights 162 pounds.

11.

Juan buys a 50-pound bag of rice and consumes about 0.4 pound per week.

Write an expression for the amount of rice Juan has consumed in terms of the number of weeks since he bought the bag.
Write an expression for the amount of rice Juan has left in terms of the number of weeks since he bought the bag.
Find the amount of rice Juan has left after 6 weeks.

12.

Trinh is bicycling down a mountain road that loses 500 feet in elevation for each 1 mile of road. She started at an elevation of 6300 feet.

Write an expression for the elevation that Trinh has lost in terms of the distance she has cycled.
Write an expression for Trinh's elevation in terms of the number of miles she has cycled.
Find Trinh's elevation after she has cycled 9 miles.

For Problems 13-28, write and solve an equation to answer the question.

13.

Celine's boutique carries a line of jewelry made by a local artists' co-op. If Celine charges $p$ dollars for a pair of earrings, she finds that she can sell $200 - 5p$ pairs per month. On the other hand, the co-op will provide her with $56 + 3p$ pairs of earrings when she charges $p$ dollars per pair. What price should Celine charge so that the demand for earrings will equal her supply?

14.

Curio Electronics sells garage door openers. If it charges $p$ dollars per unit, it sells $120 - p$ openers per month. The manufacturer will supply $20 + 2p$ openers at a price of $p$ dollars each. What price should Curio Electronics charge so that its monthly supply will meet its demand?

15.

Roger sets out on a bicycle trip at an average speed of 16 miles per hour. Six hours later, his wife finds his patch kit on the dining room table. If she heads after him in the car at 45 miles per hour, how long will it be before she catches him?

What are we asked to find in this problem? Assign a variable to represent it.
Write an expression in terms of your variable for the distance Roger's wife drives.
Write an expression in terms of your variable for the distance Roger has cycled.
Write an equation and solve it.

16.

Kate and Julie set out in their sailboat on a straight course at 9 miles per hour. Two hours later, their mother becomes worried and sends their father after them in the speedboat. If their father travels at 24 miles per hour, how long will it be before he catches them?

What are we asked to find in this problem? Assign a variable to represent it.
Write an expression in terms of your variable for the distance Kate and Julie sailed.
Write an expression in terms of your variable for the distance their father traveled.
Write an equation and solve it.

17.

The reprographics department has a choice of 2 new copying machines. One sells for $20,000 and costs $0.02 per copy to operate. The other sells for $17,500, but its operating costs are $0.025 per copy. The repro department decides to buy the more expensive machine. How many copies must the repro department make before the higher price is justified?

What are we asked to find in this problem? Assign a variable to represent it.
Write an expression in terms of your variable for the total cost incurred by each machine.
Write an equation and solve it.

18.

Annie needs a new refrigerator and can choose between two models of the same size. One model sells for $525 and costs $0.08 per hour to run. A more energy-efficient model sells for $700 but runs for $0.05 per hour. If Annie buys the more expensive model, how long will it be before she starts saving money?

What are we asked to find in this problem? Assign a variable to represent it.
Write an expression in terms of your variable for the total cost incurred by each refrigerator.
Write an equation and solve it.

19.

The population of Midland has been growing at an annual rate of 8% over the past 5 years. Its present population is 135,000.

Assuming the same rate of growth, what do you predict for the population of Midland next year?
What was the population of Midland last year?

20.

For the past 3 years, the annual inflation rate has been 6%. This year, a steak dinner at Benny's costs $12.

Assuming the same rate of inflation, what do you predict for the price of a steak dinner next year?
What did a steak dinner cost last year?

21.

Virginia took a 7% pay cut when she changed jobs last year. What percent pay increase must she receive this year in order to match her old salary of $24,000?

Hint

What was Virginia's salary after the pay cut?

22.

Clarence W. Networth took a 16% loss in the stock market last year. What percent gain must he realize this year in order to restore his original holdings of $85,000?

Hint

What was the value of Clarence's stock holdings after the loss?

23.

Delbert's test average in algebra is 77. If the final exam counts for 30% of the grade and the test average counts for 70%, what must Delbert score on the final exam to have a term average of 80?

24.

Harold’s batting average for the first 8 weeks of the baseball season is 0.385. What batting average must he maintain over the last 18 weeks so that his season average will be 0.350 (assuming he continues the same number of at-bats per week)?

25.

A horticulturist needs a fertilizer that is 8% potash, but she can find only fertilizers that contain 6% and 15% potash. How much of each should she mix to obtain 10 pounds of 8% potash fertilizer?

Pounds of fertilizer	% potash	Pounds of potash

What are we asked to find in this problem? Assign a variable to represent it.
Write algebraic expressions in terms of your variable for the amounts of each fertilizer the horticulturist uses. Use the table.
Write expressions for the amount of potash in each batch of fertilizer.
Write two different expressions for the amount of potash in the mixture. Now write an equation and solve it.

26.

A sculptor wants to cast a bronze statue from an alloy that is 60% copper. He has 30 pounds of a 45% alloy. How much 80% copper alloy should he mix with it to obtain the 60% copper alloy?

Pounds of alloy	% copper	Pounds of copper

What are we asked to find in this problem? Assign a variable to represent it.
Write algebraic expressions in terms of your variable for the amounts of each alloy the sculptor uses. Use the table.
Write expressions for the amount of copper in each batch of alloy.
Write two different expressions for the amount of copper in the mixture. Now write an equation and solve it.

27.

Lacy's Department Stores wants to keep the average salary of its employees under $19,000 per year. If the downtown store pays its 4 managers $28,000 per year and its 12 department heads $22,000 per year, how much can it pay its 30 clerks?

What are we asked to find in this problem? Assign a variable to represent it.
Write algebraic expressions for the total amounts Lacy’s pays its managers, its department heads, and its clerks.
Write two different expressions for the total amount Lacy's pays in salaries each year.
Write an equation and solve it.

28.

Federal regulations require that 60% of all vehicles manufactured next year comply with new emission standards. Major Motors can bring 85% of its small trucks in line with the standards, but only 40% of its automobiles. If Major Motors plans to manufacture 20,000 automobiles next year, how many trucks will it have to produce in order to comply with the federal regulations?

What are we asked to find in this problem? Assign a variable to represent it.
Write algebraic expressions for the number of trucks and the number of cars that will meet emission standards.
Write two different expressions for the total number of vehicles that will meet the standards.
Write an equation and solve it.

\(\)	\(\%\) fat	Total pounds	Pounds of fat
Juicy Bits	\(15\%\)	\(50\)	\(0.15(50)\)
LeanMeal	\(5\%\)	\(p\)	\(0.05p\)
Mixture	\(8\%\)	\(50+p\)	\(0.08(50+p)\)