Introduction to Problem Solving
(To Be Used With Section 1.1)
Goal: Introduce students to setting up and solving introductory word problems.
Skills: Define a variable to represent the unknown quantity.
Use this variable to write the equation described by the word problem.
Example 1: A lion can run 18 mph faster than a giraffe. If a lion can run 50 mph, how
fast can a giraffe run?
Solution 1: Let g = giraffe’s running speed
(Giraffe’s running speed) + 18 = (lion’s running speed), so g + 18 = 50, so g =
32 mph
Example 2: Wylie Coyote hiked into the Grand Canyon in search of the Roadrunner
from its South Rim, which is 6876 ft. above sea level. Walking along the 7.8-mile Bright
Angel Trail, he reached the Colorado River in 4 hours. At that point, he was 2460 feet
lower in the Grand Canyon than at his starting point. How far above sea level is the
Colorado River at this point?
Solution 2: Hint: Draw a picture below. Let c = Colorado River’s elevation at this
point
(Starting elevation) – (Colorado River elevation) = 2460 ft.
6876 – c = 2460
- c = -4416
C = 4416 ft.
Guided Practice
Attempt the following problems on your own. Check your answers with the instructor.
1. Thirty-seven less than a number is -19. Find the number.
1. ___________
2. Coreen ran the 400-meter dash in 56.8 seconds. This was 1.3 seconds less than her
previous personal record. What was her previous personal record?
2. ____________
3. The temperature in Palm Springs rose 400 F between 8:00 A.M. and noon. At noon,
the temperature was 1050 F. What was the temperature at 8:00 A.M.?
3. _____________
1
4. Rico paid $4.75 for a sandwich, a drink, and frozen yogurt. He remembered that the
drink and the yogurt were each $1.15 and that the sandwich had too much mustard, but he
forgot how much the sandwich cost. How much did the sandwich cost?
4. _____________
Introduction to Problem Solving
(To Be Used With Section 1.1)
Exercises
Directions: Define a variable to represent the unknown quantity then write and equation
and solve. Although many of these can be solved mentally, it is important to practice
defining the unknown. Please show this work.
1. A number is increased by 15 and is now equal to 34. Find the number.
1. _____________
2. A number decreased by 14 is -46. Find the number.
2. _____________
3. Lisa skied down the slalom run in 139.8 seconds. This was 13.7 seconds slower than
her best time. What was her best time?
3._____________
4. Farmer John lost 47 cattle because of the summer drought. His herd now numbers
396. How large was the herd before the drought?
4. _____________
5. Joyce bought 5 tickets for the CCHS vs. Saints football game for a total of $47.50.
How much did each ticket cost?
5. ______________
6. The temperature on the top of Mt. Soledad dropped 170 F between 4 P.M. and 11 P.M.
If the temperature is 11 P.M. was 460 F, what was the temperature at 4 P.M.?
6. _______________
2
7. A factory hired 130 new workers during a year in which 27 workers retired and 59 left
for other reasons. If there were 498 workers at the end of the year, how many were there
at the beginning of the year?
7. _______________
8. During one day of trading in the stock market, an investor lost $2500 on one stock, but
gained $1700 on another. At the end of the trading day, the investor’s holdings in those
two stocks were worth $52,400. What were they worth when the market opened?
8. ______________
9. Gino paid $3.23 for two tubes of toothpaste. He paid the regular price of $1.79 for
one tube. However, he bought the second tube for less because he used a coupon. How
much the coupon worth?
9. ______________
10. The Dons girl’s lacrosse team won 3 times as many games as it lost. If they won 21
games, how many did they lose?
10. _____________
11. A 75-watt bulb consumer 0.075 kWh (kilowatt-hours) of energy when it burns for 1
hour. How long was the bulb left burning if it consumed 3.3 kWh of energy?
11. _____________
12. One kilogram of seawater contains, on average, 35 grams of salt. How many grams
of seawater contain 4.2 grams of salt?
12. ______________
13. One hundred twenty seniors are on the honor roll. This represents one-third of the
senior class. How big is the senior class?
13. ______________
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14. The perimeter of the United States Pentagon is 1 mile. How long is each side in
feet?
14. ______________
Introduction to Problem Solving
(To Be Used With Section 1.2 and 1.3)
Goal: To solve multi-step equations by defining a variable to represent the unknown
quantity, write and equation to represent the given information, and solve said equation.
Skills: Define a variable to represent the unknown.
Utilize said variable to write an equation to represent the described scenario.
Solve said equation using algebraic equation solving techniques.
Example 1: Lynn had to take a taxicab from her office to the airport to catch a flight.
The taxi charged Lynn a flat fee of $2.05 plus $.90 per mile. The total cost of the trip
was $5.65. How many miles long was the taxi ride?
Solution 1: Let m = number of miles long the taxi ride was
Total cost = Flat Fee + .90(Number of miles driven)
$5.65 = 2.05 + .90m
3. 60 = .9m
4=m
Example 2: Bonnie sold some stock for $42 per share. This was $10 per share more
than twice what she paid for it. What was the price when she bought the stock?
Solution 1: Let p = price of stock when bought
42 = 2p + 10
32 = 2p
16 = p
Guided Practice
Attempt the following problems on your own. Check with your instructor for the
solutions.
1. The sum of 38 and twice a number if 124. Find the number.
1. ____________
2. Karen has 6 more than twice as many newspaper customers as when she started
selling newspapers. She now has 98 customers. How many did she have when she
started?
2. ____________
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3. One season, Rickey Henderson scored 9 more than twice the number of runs he batted
in. He scored 117 runs that season. How many runs did he bat in?
3. _____________
Introduction to Problem Solving
(To Be Used With Section 1.2 and 1.3)
Exercises
1. Four more than two thirds of a number is 22. Find the number.
1. ___________
2. Hodad’s sold 495 hamburgers today. The number sold with cheese was half the
number sold without cheese. How many of each type of burger was sold?
2. ____________
3. A company added a new oil tank that holds 350 barrels of oil more than its old oil
tank. Together they hold 3650 barrels of oil. How much does each tank hold?
3. ____________
4. Carl has an average of 76 on four tests. What score does he have to get on the 100point final exam if it counts double and he wants to have an average of 80 or better?
4. ____________
5. Theo has $5 more than Denise and Denise has $11 more than Rudy. Together they
have $45. How much money does Rudy have? How much money do Denise and Theo
have?
5. _____________
5
Consecutive Integer Problems
(To Be Used With Section 1.4)
Goal: To solve equations involving consecutive integers.
Skills: Be able to define a series of consecutive integers with only defining one variable.
Example 1: Find 3 consecutive integers who sum is 87.
Solution 1: Let n = first number, so logically…
n + 1 = 2nd number, so logically…
n + 2 = 3rd number
n + (n + 1) + (n + 2) = 87
3n + 3 = 87
3n = 84
n = 28
so, the 3 consecutive integers are 28, 29, and 30
Example 2: Find two consecutive even integers whose sum is 118.
Solution 2: Let n = first even integer, so logically….
n +2 = next even integer
n + (n + 2) = 118
2n + 2 = 118
2n = 116
n = 58
so, the two consecutive even integers are 58 and 60
Example 3: Find 3 consecutive odd integers who sum is 42.
Solution 3: Let n = first odd integer, so logically….
n + 2 = second odd integer
n + 4 = third odd integer
n + (n + 2) + (n + 4) = 42
3n + 6 = 42
3n = 36
n = 12
so, there is actually no solution since n is not odd
Guided Practice
Attempt the following problem on your own. Check with your instructor for the solution.
1. The lengths of the sides of a triangle are consecutive odd integers. If the triangle’s
perimeter is 27 meters, what are the lengths of the sides?
1. ______________
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Consecutive Integer Problems
(To Be Used With Section 1.4)
Exercises
1. Find 3 consecutive integers whose sum is 171.
1.________________
2. Find 3 consecutive odd integers whose sum is 105.
2. _____________
3. Find four consecutive even integers whose sum is 244.
3. _____________
4. Find four consecutive even integers such that twice the least increased by the greatest
is 96.
4. _____________
5. In cross-country, a team’s score is the sum of the place numbers of the first five
finishers on the team. The captain of the tam placed second in a meet. The next four
finishers on the team placed in consecutive order. The team score was 40. In what
places did the other members finish?
5._____________
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Distance, Rate, Time Problems
(To Be Used With Section 1.5)
Goal: Construct a table to help solve word problems involving distance, rate, and time
(aka uniform motion)
Skills: Organize given information into a chart to assist with solving complex word
problems
Apply the formula (Distance) = (Rate) x (Time)
Apply dimensional analysis skills to ensure answer has correct units of measure.
Example 1: Heidi and Spencer leave their home at the same time, traveling in opposite
directions. Heidi travels at 80 km/h and Spencer travels at 72 km/h. In how many hours
will they be 760 km apart?
Solution 1: Draw a diagram to represent their travel.
Organize the information in a chart. Let t = number of hours
Heidi
Spencer
(rate) x (time) = (distance)
80
t
80t
72
t
72t
Write an equation to represent the total distance between them is 760 km:
(Heidi’s Distance) + (Spencer’s Distance) = 760
80t
+
72t
= 760
152t
= 760
t = 5 hr.
Example 2: At 8:00 A.M. Felicia leaves home on a business trip driving 35 mph. A half
hour later, Jose discovers that Felicia forgot her briefcase and her cell phone. He drives
50 mph to catch up with her. If Jose is delayed 15 minutes with a flat tire, when he
catches up with Felicia?
Solution 2: Draw a diagram to represent the relationship between their distances
traveled.
Organize the information into a (rate) x (time) = (distance) chart.
Felicia
Jose
(rate) x (time) = (distance)
35
t
35t
50
t – 0.75 50(t – 0.75)
8
Write an equation to represent that when Jose catches up to Felicia, the
distances traveled by each person will be equal.
(Felicia’s Distance) = (Jose’s Distance)
35t = 50(t – 0.75)
35t = 50t – 37.5
-15t = -37.5
t = 2.5 hours
So, since Felicia time is represented by t (see table) and she left at 8:00
A.M., then Jose catches her at 10:30 A.M.
Guided Practice
Attempt the following problems on your own. Check with your instructor for the
solutions.
1. At 1:30 P.M., an airplane leaves Tucson for Baltimore, a distance of 2240 miles. The
plane flies at 280 mph. A second airplane leaves Tucson at 2:15 P.M., and is scheduled
to land in Baltimore 15 minutes before the first airplane. At what rate must the second
airplane travel to arrive on schedule.
(rate) x (time) = (distance)
Plane 1
280
t
2240
Plane 2
r
7
2240
Note: t =
2240/280 = 8 h
Note: plane 2
must make the flight in 7 hours
2. Two trains leave New York at the same time, on traveling north, the other south. The
first train travels at 40 mph and the second at 30 mph. In how many hours will the trains
be 245 miles apart?
(rate) x (time) = (distance)
Train #1
Train #2
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3. Two bicyclists are traveling in the same direction on the same bike path. One travels
at 20 mph and the other at 14 mph. After how many hours will they be 15 miles apart?
Distance, Rate, Time Problems
(To Be Used With Section 1.5)
Exercises
1. At the same time Kris leaves Washington, D.C. for Detroit, Amy leaves Detroit for
Washington, D.C. The distance between the cities is 510 miles. Amy’s average speed is
5 mph faster than Kris’s. How fast is Kris driving if they pass each other in 6 hours?
2. The Hornblower leaves the pier at 9:00 A.M. at 8 knots (nautical mph). A half hour
later, The Nymph leaves the same pier in the same direction traveling at 10 knots. At
what time will the Nymph overtake the Hornblower?
3. Art leaves at 10:00 A.M., traveling at 50 mph. At 11:30 A.M., Jennifer starts in the
same direction at 45 mph. When will they be 100 miles apart?
4. Guillermo is driving 40 mph. After he has driven 30 miles, his brother Jorge starts
driving in the same direction. At what rate must Jorge drive to catch up with Guillermo
in 5 hours?
10
5. Two airplanes leave Dallas at the same time and fly in opposite directions. One
airplane travels 80 mph faster than the other. After 3 hours, they are 2940 miles apart.
What is the rate of each airplane?
6. An express train travels 80 kph from Wheaton to Ward. A local train, traveling at 48
kph, takes 2 hours longer for the same trip. How far apart are Wheaton and Ward?
7. Mark runs a 440-yard dash in 55 seconds and Al runs it in 88 seconds. To have Mark
and Al finish at the same time, how much of a head start should Mark give Al? State
your answer in yards.
8. If it takes a plane 40 minutes longer to fly from Boston to Los Angeles at 525 mph
than it does to return at 600 mph. How far apart are the cities?
9. A bus traveled 387 km in 5 hours. One hour of the trip was in city traffic. The bus’s
city speed was just half of its speed on open highway. The rest of the trip was on open
highway. Find the bus’s city speed.
10. It took Cindy 2 hours to bike from Abbot to Benson at a constant speed. The return
trip took only 90 minutes because she increased her speed by 6 km/h. How far apart are
Abbot and Benson?
11
Mixture Problems
(To Be Used With Section 1.5)
Goal: To use tables to set-up and solve complex mixture problems including Chemistry
problems.
Skills: Apply knowledge from previous section to solve problems involving the mixture
of ingredients, coins, and chemical solutions.
Example 1: The CCHS cafeteria makes 2 kinds of cookies daily: chocolate chip at
$6.50 per dozen and peanut butter at $9.00 per dozen. On Thursday, the cafeteria sold 85
dozen more chocolate chip than peanut butter cookies. The total sales for both were
$4055.50. How many dozen of each were sold?
Solution 1: Define the variable, so let x = number of dozen peanut butter cookies sold.
Create a table to organize the information
Peanut Butter
Chocolate Chip
(# of dozens) x (price per dozen) = (Revenue)
x
9
9x
x + 85
6.5
6.5(x + 85)
Write an equation to represent the total amount of revenue was $4055.50.
(Revenue for peanut butter cookies) + (revenue for chocolate chip) =
4055.50
9x + 6.5(x + 85) = 4055.50
9x + 6.5x + 552.5 = 4055.5
15.5 x + 552.5 = 4055.5
15.5x = 3503
x = 226 So, the cafeteria sold 226 dozen peanut butter and 311 dozen
chocolate chip cookies
Example 2: Rudy has $2.55 in dimes and quarters. He has eight more dimes than
quarters. How many quarters does he have?
Solution 2: Define a variable to represent the unknown number of quarters.
Let q = number of quarters
Use this variable to also represent the number of dimes.
q + 8 = number of dimes
Write an equation representing how much this coin mixture is worth
.25q + .10(q + 8) = 2.55
.25q + .1q + .8 = 2.55
.35q + .8 = 2.55
.35q = 1.75
q=5
So, Rudy has 5 quarters and 13 dimes
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Example 3: Kendra is doing a chemistry experiment that calls for a 30% solution of
copper sulfate. She has 40 mL of 25% solution. How many milliliters of 60% solution
should Kendra add to obtain the required 30% solution.
Solution 3: Define a variable for the unknown, so let x = mL of the 60% solution needed
Create a table to organize the information:
Amount of
Amount of
Solution (mL) Copper Sulfate (mL)
25% Solution
40
0.25(40)
60% Solution
x
0.60x
30% Solution
40 + x
0.30 (40 + x)
The second column will give you the information for the necessary
equation.
(Amt of cop. sulf. in 25% sol.) + (Amt of cop sulf in 60% sol) = (Amt of cop sulf in the
mixture)
0.25(40) + 0.60x = 0.30 (40 + x)
10 + 0.6x = 12 + .3x
10 + .3x = 12
.3x = 2
x = 6.67 mL So, Kendra needs to add 6.67 mL of the 60% solution
Guided Practice
Attempt the following problems. Consult your instructor for the solutions.
1. The CCHS Athletic Office is selling tickets for Friday’s football game. Tickets for
adults cost $5.50 and tickets for students cost $3.50. How many of each king of ticket
was purchased at break yesterday if 21 tickets were sold and the office brought in $83.50
in revenue.
2. Peanuts sell for $3.00 per pound and cashews sell for $6.00 per pound. How many
pounds of cashews should be mixed with 12 pounds of peanuts to obtain a mixture that
sells for $4.20 per pound?
Pounds Total Cost
$3.00 peanuts
12
$6.00 cashews
$4.20 mix
13
Mixture Problems
(To Be Used With Section 1.5)
Exercises
1. A liter of cream has 9.2% butterfat. How much skim milk containing 2% butterfat
should be added to the cream to obtain a mixture with 6.4% butterfat?
1. ______________
2. Java Joe, owner of Java Joe’s Coffee, wants to create a special home brew using two
types of coffee, one priced at $6.40 per pound and the other at $7.28 per pound. How
many pounds of the $7.28 coffee should he mix with 9 pounds of the $6.40 coffee to sell
the mixture for $6.95 per pound?
2. _______________
3. A pharmacist has 150 dL of a 25% solution of peroxide in water. How many
deciliters of pure peroxide should be added to obtain a 40% solution?
3. _______________
4. The Martins are going to Wally World (a great amusement park). The total cost of
tickets for a family of 2 adults and three children is $79.50. If an adult ticket costs $6.00
more than a child’s ticket, find the cost of each.
4. ______________
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5. A teacher received her Christmas bonus and it consisted of nickels, dimes, and
quarters and totaled $1.45. If the number of nickels was twice the number of quarters and
there was one more dime than quarter, how many of each type of coin was there?
5. _____________
6. Ground chuck (a meat not a man) sells for $1.75 per pound. How many pounds of
ground round selling for $2.45 per pound should be mixed with 20 pounds of ground
chuck to obtain a mixture that sells for $2.05 per pound?
6. ______________
7. A car radiator has a capacity of 16 quarts and is filled with a 25% antifreeze solution.
How much must be drained off and replaced with pure antifreeze to obtain a 40%
solution?
7. _____________
8. Jane has a collection of nickels and quarters worth $3.05. She has 7 more nickels than
quarters. How many coins of each type does she have?
8. _____________
15
Percent Problems and Simple Interest
(To Be Used With Section 1.5 or 1.5)
Goal: Solve problems using the formula for simple interest and the concept of percent.
Skills: To solve finance problems involving simple interest.
To solve problems involving percentage increases or decreases including
discounts.
Example 1: During a sale, a sporting goods store gave a 40% discount on sleeping bags.
How much did Ross pay for a sleeping bag with an original price of $75.
Solution 1: Find the amount of the discount (different than the percentage discount).
Amount of discount is 40% of original price = .40 x 75 = $30
Find the sale price by subtracting the discount from the original price.
Sale price = original price – discount = 75 – 30 = $45
Example 2: Rachel opened a brokerage account that earned 7% annual interest. After 6
months, she received $52.50 in interest. How much money was originally invested?
Solution 2: Utilize the simple interest formula: Interest = Principal x Rate x Time (I =
prt)
Principal = Amount initially invested
Rate = annual interest rate (expressed as a decimal)
Time = number of years the money has been invested
52.50 = p (.07)(0.5)
52.50 = .035p
1500 = p
So, Rachel invested $1500 into this account.
Example 3: Monica invested $30,000, part at 7% annual interest and the rest at 7.5%
annual interest. Last year, she earned $1995 in interest. How much money was invested
into each account?
Solution 3: Define a variable to represent how much money was invested into each
account.
Let a = amount invested into account 1, so 30,000 – a = amount in account
2.
Construct a table to organize information.
Principal x rate x time = Interest
Account 1
a
.06 1
.06a
Account 2
30000 – a
.075
1 .075(30000 – a)
Write an equation to represent the total amount of interest earned is $1995
.06a + .075(30000 – a) = 1995
16
.06a + 2250 - .075a = 1995
-.015a + 2250 = 1995
-.015a = -255
a = 17000
So, Monica put $17,000 into account 1 and $13,000
into
account 2.
Guided Practice
Attempt the following problems, consult with your teacher for the solutions.
1. A sporting goods dealer estimates that an $85 tennis racket will increase in price by
6% next year. What will the tennis racket cost next year?
1. ____________
2. A record store is selling a $50 Led Zeppelin box set at an 8% discount. If sales tax is
then added on (and sales tax is 8%), what is the final price of the box set?
2. _____________
3. Michelle invested $10,000 for one year, part at 8% annual interest and the rest at 12%
annual interest. Her total interest for the year was $944. How much money did she
invest at each rate?
3. _____________
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Percent Problems and Simple Interest
(To Be Used With Section 1.5)
Exercises
1. Steve invested $7200 for one year, part at 10% annual interest and the rest at 14%
annual interest. His total interest for the year was $960. How much did money did Steve
invest in each account?
1. _________________________
2. Angie wants to invest $8500, part at 14% annual interest and part at 12% annual
interest. If she wants to earn the same amount of interest from each investment, how
much should she invest at 14%? (Round answer to nearest cent.)
2. _________________________
3. Ken invested $9450, part at 8% annual interest and the rest at 11% annual interest. He
earned twice as much interest at 11% as he did at 8%. How much money did he have
invested at 11%?
3. _________________________
4. Jesse invested $2000 more in stocks than in bonds. The bonds paid 7.2% annual
interest and stock paid 6% annual interest. The income from each investment was the
same. How much interest did he receive in all?
4. _______________________
18
5. Laura is a real estate agent and she earns a 10.5% commission on each house sold.
How much does Laura earn in commission if she sells a house for $515,000 and a house
for $325,000 during a month?
5.__________________
6. The Dons Athletic Club has raised $18,700 towards the fund raising goal for building
a snack bar at the baseball field. This is 22% of the goal. What is the fund raising goal?
6. ___________________
7. Because an item was slightly damaged, the student store reduced the price by $6. This
represents a 15% discount from the original price. What was the original price?
7._____________
8. Last year, Molly was given a performance bonus of 3% of her base salary for
outstanding customer satisfaction. If her bonus was $720, what is Molly’s base salary?
8. ______________
9. A $200 cost is on sale for $166. What is the percent of the discount?
9.______________
10. The readership of the Union Tribune has decreased by 10% each of the last 2 years.
If two years ago the readership was 200,000 subscribers, what is the readership now?
10. _______________
11. The number of students in the freshmen class at USD is now 1120. This is 6% more
than last year. How many students were in the freshmen class last year?
11. ______________
19
Problem Solving Review of Chapter 1
(To Be Used Following Chapter 1)
Exercises
1. The sum of twice a number and -6 is 9 more than the opposite of the number. Find the
number.
1. ____________
2. Roger spent $22 on a wiffle ball and a waffle bat. If the wiffle bat cost $2 less than 5
times the cost of the wiffle ball, find the cost of each.
2. ____________
3. A rectangle has a perimeter of 48 cm. If the width and the length are consecutive odd
integers, find the dimensions of the rectangle.
3. _____________
4. Find three consecutive integers such that three times the smallest is equal to the
middle number increased by the greatest number.
4. ___________________
5. Rudy has $125 in $5 bills and $10 bills. If he has four more $5 bills than $10 bills,
how many of each does he have?
5. _______________
6. Maria invested $8000 for one year, part at 8% annual interest and the rest at 12%
annual interest. Her total interest for the year was $744. How much money did Maria
invest at each rate?
6. ______________
20
7. Alfonso made a purchase at a Sue Mills totaling $179.96. If he pays 7.75% sales tax,
then how much will he owe?
7. ______________
8. Larry is buying a new I-Phone for $399. Since he is an employee of the store, he
receives a 15% discount, but must also pay the 8% sales tax. How much will he owe if
a) The discount is taken off first and then the sales tax is added?
8a. _________________
b) The sales tax is added first and then the discount is taken off?
8b. _________________
9. How much whipping cream (9% butterfat) should be added to 1 gallon of milk (4%
butterfat) to obtain a 6% butterfat mixture?
9. _______________
10. At noon a private jet left Austin for San Diego, 2100 km away, flying at 500 km/h.
One hour later, a commercial jet left San Diego for Austin at 700 km/h. At what time did
they pass each other?
10. _______________
11. A person weights 0.5% less at the Equator than at the North or South Poles. How
much would a person weigh at the Equator if the person weighed 148 pounds at the North
Pole?
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