Because this statement is true, 7 is indeed the solution.
Example5.3.1.
How long will it take Rudi to swim one mile if she can swim at a rate of 50 yards per minute?
Solution.
We will use the formula \(d=rt\text{.}\) We are asked to find the time it takes Rudi to swim a mile, so \(t\) will be the variable in our equation.
Rudi’s swimming rate is given in yards per minute, so we must first convert the distance, one mile, into yards: one mile is equal to 1760 yards.
Next we substitute \(\alert{50}\) for \(r\) and \(\alert{1760}\) for \(d\) into the formula.
\begin{align*}
d \amp = rt\\
\alert{1760} \amp = \alert{50}t
\end{align*}
Finally, we solve the equation for \(t\text{.}\) Because \(t\) has been multiplied by 50, we divide both sides of the equation by 50.
\begin{align*}
\dfrac{1760}{\blert{50}} \amp = \dfrac{50t}{\blert{50}} \hphantom {0000} \blert{\text{Divide both sides by 50.}}\\
35.2 = t
\end{align*}
It takes Rudi 35.2 minutes to swim one mile. (How long is 0.2 minutes in seconds?)
Note5.3.2.
To finish solving a problem, we write a sentence that answers the question in the problem.
Checkpoint5.3.3.
Logicorp calculates that its costs next month will be $9100. What revenue do they need to make a profit of $1100?
\(\blert{\text{Choose the appropriate formula.}}\)
\(\blert{\text{Solve an equation.}}\)
\(\blert{\text{Write a sentence to answer the question.}}\)
Subsection5.3.2Writing an Equation
Sometimes we can write an equation to solve a problem. Here is a simple example.
Example5.3.4.
The sum of 17 and a number is 45. What is the number?
Solution.
We choose a variable, say \(n\text{,}\) to represent the unknown number. Then we write an equation about the number \(n\text{.}\) Look for mathematical words to help you: "sum" indicates addition, and "is" means equals.
\begin{align*}
\text{The sum of 17 and a number} \amp = 45.\\
\hphantom{00000000} 17 ~~ + ~~ n \amp = 45
\end{align*}
Finally, we solve the equation. Because 17 is added to \(n\text{,}\) we subtract 17 from both sides of the equation.
\begin{align*}
17 + n \amp = ~~~45 \hphantom{000} \blert{\text{Subtract 17 from both sides.}}\\
\underline{-17}~~~~~~~ \amp = \underline{-17}\\
n \amp = 28
\end{align*}
The number is 28.
Checkpoint5.3.5.
The product of 13 and a number is 598.
Write an equation that describes this problem.
Solve your equation. What is the number?
\(\blert{\large{\text{PATIENCE!}}}\).
Yes, we know that you can solve these problems in your head!
But the point is not the answers to the problems, the point is to
which can help us solve much harder (and more realistic) problems once we have developed the necessary skills.
Subsection5.3.3Modeling a Problem
For many problems we must write an equation by putting together the information given in the problem.
Example5.3.6.
For healthy weight loss, the ratio of calories you get from carbohydrates to calories from protein should be 3 to 2. On a diet of 1500 calories per day, you should consume 450 calories of protein. How many calories of carbohydrates should you eat?
Solution.
Follow the steps to write an equation.
Step 1: What are we asked to find in the problem? Choose a variable to represent it.
\begin{equation*}
\blert{\text{Calories from carbohydrates:} ~C}
\end{equation*}
Step 2: Find something in the problem that we can express in two different ways.
\begin{equation*}
\blert{\text{Ratio of carb calories to protein calories}}
\end{equation*}
Write an equation using the two expressions.
\begin{equation*}
\dfrac{\blert{\text{Carb calories}}}{\blert{\text{Protein calories}}}:~~~~~~\dfrac{C}{450} = \dfrac{3}{2} ~~~~~~\blert{\text{Ratio of 3 to 2.}}
\end{equation*}
Step 3: Solve the equation.
\begin{align*}
\dfrac{C}{450} \amp = \dfrac{3}{2}~~~~~~~~~~~~~~{\blert{\text{Multiply both sides by 450.}}}\\
450 \left(\dfrac{C}{450}\right) \amp = \left(\dfrac{3}{2}\right)450~~~~~~~~{\blert{\text{Multiply.}}}\\
C \amp = 675
\end{align*}
Answer the question in the problem.
\begin{equation*}
\blert{\text{You should eat 675 calories of carbohydrates.}}
\end{equation*}
Note5.3.7.
Observe some things about our solution:
Our equation did not start with "\(C = \)" ! The variable is part of the equation, but the equation itself is about something else, in this case, a ratio.
We did not use the 1500 calories per day in our solution. We used only the information we needed to write an equation.
We summarize our method in three steps.
Steps for Modeling a Problem.
Identify the unknown quantity and choose a variable to represent it.
Find some quantity that can be expressed in two different ways, and write an equation.
Solve the equation, and answer the question in the problem.
Checkpoint5.3.8.
Claudia and her alpaca Lima together weigh 173 pounds. Claudia alone weighs 94 pounds. How much does Lima weigh?
Step 1: Let stand for Lima’s weight in pounds.
Step 2: Find two different expressions for the sum of Claudia’s and Lima’s weights, and write an equation.
Step 3: Solve your equation and answer the question.
You can probably find Lima’s weight using arithmetic alone, but can you write an equation for the problem?!
Here are two things to remember about writing equations to solve problems.
Problem-Solving Hints.
It is very important to specify precisely what the variable represents. You will have to write an equation about this variable, so you must have a clear idea of what it stands for.
In particular, the variable must stand for a number, so in the previous exercise, \(w\) stands for "Lima’s weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity.
In the previous Example, the variable \(C\) stands for calories, but the equation gives two ways to express a ratio.
Activity5.3.1.Problem Solving.
Follow the three steps to solve the problems. Remember that writing an equation is more important than finding the answer!
Toshiro must reduce his daily calorie intake by 260 calories. If his goal is 1350 calories per day, what is his current calorie intake?
Step 1: What are we asked to find? Choose a variable for that quantity.
Step 2: Find two different ways to express Toshiro’s dieting goal, and write an equation.
Step 3: Solve the equation and answer the question.
The ratio of calories from saturated fat to calories from polyunsaturated fat in most margarine is 0.4. A tablespoon of margarine contains 45 calories from polyunsaturated fat. How many calories from saturated fat does a tablespoon of margarine contain?
Step 1: What are we asked to find? Choose a variable for that quantity.
Step 2: Write the ratio of calories from saturated fat to calories from polyunsaturated fat in two different ways.
Step 3: Solve the equation and answer the question.
How much money must Francisco deposit in a savings account that pays 3% interest if he wants to earn $240 interest in 2 years?
Step 1: What are we asked to find? Choose a variable for that quantity.
Step 2: Find two different ways to express Francisco’s interest, and write an equation.
Step 3: Solve the equation and answer the question.
Exercises5.3.4Practice 5.3
Exercise Group.
For Problems 1-12,
Choose the appropriate formula and write an equation.
Solve the equation and answer the question.
1.
The Camp for Kids Foundation hopes to make $1500 profit at its next rummage sale. If the cost of putting on the sale is $215, how much revenue will they need?
2.
A gallon of paint will cover 350 square feet. The wooden fence around the public gardens is 7 feet high. What length of fence will one gallon of paint cover?
3.
36% of the freshman class at Hollins College is from out-of-state. If there are 162 freshmen from out-of-state, how large is the freshman class?
4.
A telephone poll of 800 voters showed 496 in favor of a school bond. What percent of those polled favored the bond?
5.
Millie invested $1300 in a certificate of deposit for one year and earned $98.80 interest. What interest rate did the certificate earn?
6.
Scott earned $14.70 in interest from his credit union last year. He made no deposits or withdrawals from the account. If the interest rate is 4.9%, how much is in the account?
7.
A fruit punch contains 36% ginger ale. If you have 9 quarts of ginger ale, how much fruit punch can you make?
8.
45% of the voters voted for Senator Fogbank. If the Senator received 540 votes, how many people voted?
9.
Staci invested some money in a T-bill account that pays 912% interest, and one year later the account had earned $171 interest. How much did Staci deposit in the T-bill?
10.
Clive loaned his brother some money to buy a new truck, and his brother agreed to repay the loan in one year with 3% interest. Clive earned $75 interest on the loan. How much did Clive loan his brother?
11.
Garden snails travel at about 2.6 feet per minute. At that rate, how long will it take a snail to cross a 10-foot patio?
12.
The peregrine falcon is the fastest flying creature. At a 45° angle of descent, it can cover 312 yards in 3 seconds. What is the falcon’s speed in yards per second? In miles per hour? (Hint: There are 1720 yards in a mile.)
Exercise Group.
For Problems 13-20, write an equation. Let \(n\) stand for the unknown number.
13.
The product of a number and 6 is 162. What is the number?
14.
The quotient of a number and 8 is 13. What is the number?
15.
A number decreased by 24 is 38. What is the number?
16.
Eleven more than a number is 30. What is the number?
17.
A number divided by 2.5 is equal to 6.6. What is the number?
18.
Fifty-eight times a number is 34.8. What is the number?
19.
Twenty is 15.3 more than a number. What is the number?
20.
Eighteen is the difference of a number and 7.1. What is the number?
Exercise Group.
For Problems 21-26, follow the steps to write an equation and solve the problem. (Remember: writing the equation is the important part!)
21.
Lupe spent $24.50 at the Craft Fair. She now has $39.75 left. How much did she have before the Craft Fair?
What are we asked to find? Choose a variable to represent it.
Find two ways to express the amount of money Lupe had after the Craft Fair, and write an equation.
Solve the equation and answer the question in the problem.
22.
The sale price on a washing machine is $29.80 less than the regular price. The sale price is $258.35 What is the regular price?
What are we asked to find? Choose a variable to represent it.
Find two ways to express the sale price of the washing machine, and write an equation.
Solve the equation and answer the question in the problem.
23.
Miranda worked 22 hours this week and made $149.60. What is Miranda’s hourly wage?
What are we asked to find? Choose a variable to represent it.
Find two ways to express Miranda’s total earnings, and write an equation.
Solve the equation and answer the question in the problem.
24.
Bruce goes jogging on the same course every morning except Sundays, when he rests. His weekly mileage is 57 miles. How long is the course?
What are we asked to find? Choose a variable to represent it.
Find two ways to express Bruce’s weekly mileage, and write an equation.
Solve the equation and answer the question in the problem.
25.
Struggling Students Gardening Service splits their profit equally among their eight members. If each member made $64.35 last week, what was the total profit?
What are we asked to find? Choose a variable to represent it.
Find two ways to express each member’s share, and write an equation.
Solve the equation and answer the question in the problem.
26.
Judy’s average score on 12 homework assignments was 17.5 points. What was the total number of points she earned?
What are we asked to find? Choose a variable to represent it.
Find two ways to express Judy’s average score, and write an equation.
Solve the equation and answer the question in the problem.
Exercise Group.
For Problems 27-36,
Identify the unknown quantity and choose a variable to represent it.
Write an eqution to model the problem.
27.
A cheeseburger and fries together contain 1070 calories. An order of fires contains 342 calories. How many calories are in a cheeseburger?
28.
Mishell weighs 28.5 pounds less than her dog, Bear. If Mishell weighs 96 pounds, how much does Bear weigh?
29.
Caroline Gottrich divided her fortune equally among her three no-good sons and her cat. Each beneficiary received $350,000. How much was Caroline’s fortune?
30.
The ratio of your quiz score to 80 points is 0.85. What is your quiz score?
31.
After writing a check for $2378, Averil’s bank account shows a balance of $1978. How much was in Averil’s account before the check cleared?
32.
Akiko’s great-grandmother says she was 14 years old when the hotel burned down. If Akiko’s great-grandmother is 92 years old now, how long ago did the hotel burn down?
33.
Beth’s monthly income is $1800, and her rent is $576 per month. What percent of her monthlyincome does Beth spend on rent?
34.
Andrea made a down payment of $1776 on her new car. If this was 12% of the price of the car, how much did the car cost?
35.
In the 50 states, the average area devoted to state parks is 222,960 acres of land. How many acres of state park are there in the whole United States?
36.
Latrisha invested $6500 in a certificate of deposit for one year, and earned $308.75 interest. What rate of interest did the certificate earn?
Exercise Group.
For Problems 37-46, use the appropriate formula to write an equation relating two variables.
37.
The distance traveled in \(t\) hours by a small plane flying at 150 miles per hour.
38.
distance traveled in \(t\) seconds by a car moving at 88 feet per second.
39.
The amount of interest earned after \(t\) years by $1600 deposited in an accountthat pays 4% annual interest rate. (Don’t forget to change 4% to a decimal.)
40.
Your credit union loans you $3000 to be repaid with 4% annual interest. Write an equation for the amount of interest you will owe after \(t\) years.
41.
Milton’s great-aunt plans to put $6000 in a trust fund for Milton until he turns twenty-one three years from now. She has a choice of several different accounts. Write an equation in terms of \(r\) for the amount of interest the money will earn in three years.
42.
Write an equation in terms of \(r\) for the amount of interest earned by $1600 deposited in an interest-bearing account for one year.
43.
Write an equation for the amount \(G\) of grape juice in a fruit punch that is 20% grape juice.
44.
Eggnog is 70% milk. Write an equation for the amount of milk \(M\) in a container of eggnog.
45.
Laureen took 12 quizzes in her math class this semester and earned a total of \(S\) points. Write an equation for Laureen’s quiz average.
46.
Errol has saved $1200 for his vacation. If he goes on vacation for \(d\) days, write an equation for the average amount he can spend each day.
Exercise Group.
For Problems 47-54,
Identify two unknown quantities and choose variables to represent them. Specify precisely what your variables represent.
Write an eqution relating the variables.
47.
Danny weighs 32 pounds more than Brenda.
48.
Corey worked for 5 fewer hours than Shant.
49.
Trinh is 14 years younger than his brother Loc.
50.
Mac’s bowling score is 28 points higher than Tyrone’s.
51.
Each winter coat requires three buttons.
52.
For each camp counselor, there are eight children.
53.
Interest rates are twice what they were five years ago.
54.
Many birds eat half their body weight every day.
Exercise Group.
For Problems 55-58,
Look for a pattern relating the numbers in the second column of the table to the numbers in the first column.
Write an equation relating the variables, and fill in the rest of the table.
