Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Cathedral Catholic High School
Algebra 1A
Problem Solving Workbook
Table of Contents
Book Section
3.1
3.2, 3.3
3.3
3.4
3.4
3.6, 3.7
3.8
4.6
7.0
7.1
7.5
7.5
9.6 to 9.8
10.6
12.7
Title
Page
Introduction to Problem Solving………………………………2
Introduction to Problem Solving………………………………5
Consecutive Integer Problems…………………………………7
Distance, Rate, Time Problems………………………………..9
Mixture Problems…………………………………………….13
Percent Problems and Simple Interest………………………..17
Problem Solving Review of Chapter 3……………………….21
Modeling Direct Variation……………………………………23
Introduction to Systems of Equations………………………...26
Solving Systems of Equations with Graphing Calculators…...27
Problem Solving Using Systems of Equations……………….28
Wind and Current Problems………………………………….31
Problem Solving Using Factoring……………………………34
Vertical Motion Problems……………………………………37
Work Problems……………………………………………….40
1
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Introduction to Problem Solving
(To Be Used With Section 3.1)
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. _____________
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. _____________
2
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Introduction to Problem Solving
(To Be Used With Section 3.1)
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 draught. 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. _______________
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. _______________
3
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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 hours.
How long was the bulb left burning if it consumed 3.3 kWh of energy?
11. _____________
12. One kilogram of sea water contains, on average, 35 grams of salt. How many grams of sea
water 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. ______________
14. The perimeter of the United States Pentagon is 1 miles. How long is each side in feet?
14. ______________
4
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Introduction to Problem Solving
(To Be Used With Section 3.2 and 3.3)
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 taxi cab 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. ____________
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. _____________
5
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Introduction to Problem Solving
(To Be Used With Section 3.2 and 3.3)
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 100-point 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. _____________
______________
______________
6. With the major options package and destination charge, a car cost $24, 416. The base price
of the car was 10 times the price of the major options package and fifty times the destination
charge. What was the base price of the car?
6. ______________
6
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Consecutive Integer Problems
(To Be Used With Section 3.3)
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. ___________________
7
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Consecutive Integer Problems
(To Be Used With Section 3.3)
Exercises
1. Find 3 consecutive integers whose sum is 171.
1. _____________________
2. Find 3 consecutive odd integers who sum is 105.
2. _____________________
3. Find four consecutive even integers who 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. _____________________
8
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Distance, Rate, Time Problems
(To Be Used With Section 3.4)
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
(rate) x (time) = (distance)
80
t
80t
72
t
72t
Heidi
Spencer
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 catch 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.
(rate) x (time) = (distance)
35
t
35t
50
t – 0.75 50(t – 0.75)
Felicia
Jose
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)
9
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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
Note: t = 2240/280 = 8 h
Plane 2
r
7
2240
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
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?
10
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Distance, Rate, Time Problems
(To Be Used With Section 3.4)
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?
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?
11
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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?
12
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Mixture Problems
(To Be Used With Section 3.4)
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
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
13
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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 were 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
Mixture Problems
14
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
(To Be Used With Section 3.4)
Exercises
1. A liter of cream has 9.2% butterfat. How much skim mile 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 2 coffees,
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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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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. _____________
16
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Percent Problems and Simple Interest
(To Be Used With Section 3.6 or 3.7)
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
.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.
17
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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. _____________
18
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Percent Problems and Simple Interest
(To Be Used With Section 3.6 or 3.7)
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. _______________________
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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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. __________________
20
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Problem Solving Review of Chapter 3
(To Be Used Following Chapter 3)
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. ___________________
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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
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?
11. _________________
22
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Modeling Direct Variation
(To Be Used With Section 4.6)
Goal: Solve real world problems that involve variables that vary directly with each other.
Skills: Find the constant of variation to write the direct variation model, and/or
Utilize proportions to set up and solve problems with direct variation.
Example 1: The weight of an object on the moon varies directly as its weight on Earth. With
all his gear on, Neil Armstrong weighed 360 pounds on Earth, but on the moon he only weighed
60 pounds. If Kristina weighs 108 pounds on Earth, what would she weigh on the moon?
Solution 1a) Finding the constant of variation
Let x = weight on Earth and let y = weight on the moon, then y = kx
y
60 1
1
k  , so k 
 , therefore y  x , so to find the Kristina’s weight
x
360 6
6
1
on the moon, y  108  =18, so Kristina would weigh 18 pounds on the
6
moon.
Solution 1b) If two variables vary directly, then their values are proportional
Let x = weight on Earth and let y = weight on the moon, then
x1 x2
360 108
60 108

 , therefore,
, so y2   360
 18 (same answer as above)
60
y2
y1 y2
Guided Practice
Attemp the following problems, consult with your teacher for the solutions.
1. In an electrical transformer, voltage is directly proportional to the number of turns on the coil.
If 110 volts comes from 55 turns, what would be the voltage produced by 66 turns.
a) Solve by finding the constant of variation k first.
1 a) k = ________________
Voltage = ___________
b) Solve by setting up and solving a proportion.
1 b) ___________________
2. An employee’s wages are directly proportional to the time worked. If an employee earns
$100 for 5 hours, how much will the employee earn for 18 hours worked?
(note: If you find the constant of variation, k, what does it represent in this scenario?)
23
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Modeling Direct Variation
(To Be Used With Section 4.6)
Exercises
1. A certain car used 15 gallons of gasoline in hours. If the rate of gas consumption varies
directly with the time of use, how much gasoline will the car use on a 35-hour trip?
1. _______________
2. The distance traveled by a truck at a constant rate of speed varies directly with the amount of
time driving. If the travels 168 miles in 4 hours of driving, how far will is travel in 7 hours of
driving at the same constant rate of speed?
2. _______________
3. A restaurant buys 20 pounds of ground beef to prepare 110 serving of chili. At this rate, how
many serving can be made with 30 pounds of ground beef?
3. _______________
4. A mass of 25 grams stretches a spring 10 cm. If the distance a spring is stretched is directly
proportional to the mass, what mass will stretch the spring 22 cm?
4
_______________
5. In Chemistry, Charles’ Law says that the volume of a gas in directly proportional to its
temperature. If the volume of a gas is 2.5 cubic feet at 150 degrees (absolute temperature), what
is the volume of the same gas at 200 degrees (absolute temperature)?
5. _______________
24
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
6. The thermometer F is marked off into 180 equal units. Thermometer C is marked off into 100
equal units. A reading of 66.6% on thermometer F is equal to a reading of how many degrees on
thermometer C?
6. _______________
7. The odometer of the Feldman’s car was not measuring distance correctly. For a 220-mile trip
the odometer registered only 216.7 miles. On the return trip, the Feldman’s had to detour due to
road repairs. If the odometer registered 453.1 miles for the round trip, how many actual miles
was the detour?
7. _______________
8. On a map, 1 cm represents an actual distance of 75 m. Find the area of a piece of land that is
represented on the map by a rectangle measuring 11.5 cm by 18.5 cm.
8. _______________
9. In a scale model of a sailboat, an object that is 6 ft. tall is represented by a figure 8 in. high.
How many feet tall should the mast of the sailboat be in the model if the actual mast of the
sailboat is 38 ft. tall?
9. _______________
10. If the circumference of a circle varies directly as the diameter, and the diameter varies
directly as the radius, show that the circumference varies directly as the radius.
25
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Introduction to Systems of Equations
(To Be Used as an Introduction to Chapter 7)
Competing Cell Phones
Grandma is deciding between two competing cell phone plans. Alpha Wireless offers a plan for
$20 per month plus $.10 per minute of use. Beta Air’s plan costs $25 per month plus $.05 per
minute of use. Both companies have equivalent coverage and sell exact same phones at the exact
same prices.
1. Write an equation to represent the cost of Alpha Wireless’s plan. Be sure to define any
variables used.
1. ___________________________
2. Write an equation to represent the cost of Beta Air’s plan. Be sure to define any variables
used.
2. ___________________________
3. What plan would you choose if
you planned to use the phone
for 30 minutes per month? Why?
4. What plan would you choose if
you planned to use the phone for
3 hours each month? Why?
5. In the space to the right, graph
the equations from #1 and #2 on
the same graph. Think about
your scales and label your axes.
6. What are the coordinates for the point of intersection? __________ What does this point
represent?
7. How would you advise grandma on which cell phone plan to purchase?
26
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Solving Systems of Equations by Graphing
Graphing Calculator Techniques
(To Be Used With Section 7.1)
Exercises
Directions: Find the solution to the following systems of equations by utilizing a graphing
calculator.
1. y = -x + 2
y = 2x + 5
1. ________________
2. x – y = 6
2x + y = 0
2. ________________
3. y = 3x + 1
y = 3x – 8
3. ________________
4. 6x + 4y = 2
3x + 2y = 1
4. ________________
5. 9x = 10 – 4y
y = 3x – 8
5. _________________
27
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Problem Solving Using Systems of Equations
(To Be Used After Section 7.5)
Goal: When introducing two variables (or when there are two unknowns), being able to write
two equations to solve a system.
Skills: Setting up and solving a system of equations by substitution or elimination.
Determining which solution method would be more efficient.
Example 1: Audry and Rusty each had a birthday party at Wally World last weekend. The cost
of admission to Wally World was $137.50 for the 13 children and 2 adults at Audry’s party, and
the cost of admission was $103.50 for the 9 children and 2 adults at Rusty’s party. What was the
admission price to Wally World for a child and for an adult?
Solution 1: There are 2 unknown quantities: let c = admission price for a child
let a = admission price for an adult
Write an equation representing the amount spent on admission for Audry’s party:
13c + 2a = 137.50
Write an equation representing the amount spent on admission for Rusty’s party:
9c + 2a = 103.50
The following system of 2 equations and 2 unknowns has been defined:
13c + 2a = 137.50
9c + 2a = 103.50
Solve the system by elimination:
13c + 2a = 137.50
13c + 2a = 137.50
(-1)(9c + 2a = 103.50)
-9c – 2a = -103.50
4c
= 34
c = 8.5, so a child’s ticket is $8.50
Determine the cost of an adult ticket by using either of the two original equations.
9c + 2a = 103.50, so 9(8.5) + 2a = 103.50, so 76.50 + 2a = 103.50
2a = 27
a = 13.50
so an adult ticket is $13.50
Guided Practice
Attempt the following problem. Check with your teacher for the correct answer.
1. A1 Car Rental rents compact cars for a fixed amount per day plus a fixed amount for each
mile driven. Benito rented a car from A-1 for 6 days, drove it 550 miles, and spent $337. Lisa
rented the same car for 3 days, drove it 350 miles, and spent $185. What are the charge per day
and charge per mile driven?
1. __________________________
28
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Problem Solving Using Systems of Equations
(To Be Used After Section 7.5)
Exercises
1. The Beach Resort is offering two weekend specials. One includes a 2-night stay with 3 meals
and costs $195. The other includes a 3-night stay with 5 meals and costs $300. What is the
nightly rate and what is the cost per meal?
1. _________________________________
2. Ted’s bill for 6 cans of grape juice and 4 cans of orange juice was $13.20. When he got
home, he found that he should have bought 4 cans of grape juice and 6 cans of orange juice.
Although he screwed up, he did save 60 cents. How much does each can cost?
2. _________________________________
3. Last season, the CCHS place kicker kicked 38 times and never missed. Each field goal is
worth 3 points and each point-after-touchdown is worth 1 point. If this kicker scored a total of
70 points last season, how many field goals and how many PATs did he make?
3. _________________________________
4. Rebecca has 45 coins, all nickels and dimes. The total value of the coins is $3.60. How many
of each type of coin does she have?
4. _________________________________
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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
5. A car traveled at a steady speed for 130 km. Due to a mechanical problem, it returned at half
that speed. If the total time for the round trip was 4 hours 30 minutes, find the two speeds?
5. _________________________________
6. At an amusement park you get 5 points for each bull’s eye you hit, but you lose 10 points for
every miss. After 30 tries, Yolie had lost 90 points. How many bull’s eyes did she have?
6. _________________________________
7. My strange uncle Harry has farm that appeared to be overrun by chickens and dogs. Being a
bit of nutjob, Harry told me that his dogs and chickens had 148 legs and 60 heads combined.
How many dogs and how many chickens does Harry have?
7. ________________________________
8. If Tom given Maria 30 cents, they will have equal amounts of money. But if Maria then
gives Tom 50 cents he will have twice as much money as she does. How much money does each
have now?
8. ________________________________
30
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Wind and Current Problems
(To Be Used After Section 7.5)
Goal: To solve uniform motion problems involving wind or current factors that either speeds up
or slows down the vessel.
Skills: Utilize the d = rt formula for uniform motion.
Set up a system of equations to solve for two unknowns (the rate of the vehicle and the
rate of the wind/current).
Example 1: Traveling against the wind, a plane flies 2100 miles from Chicago to San Diego in
4 hours and 45 minutes. The return trip, traveling with the wind that is twice as fast, takes 4
hours. Find the rate of plane in still air. Round velocities to the nearest tenth.
Solution 1: There are two unknowns: let r = rate of plane in still air and let w = wind speed
Set up a (rate) x (time) = (distance) table
(rate) x (time) = (distance)
Chicago to S.D.
r –w
4.75
4.75(r – w)
S.D. to Chicago
r + 2w
4
4(r + 2w)
The distance between the cities is 2100 miles, so set up the system of equations:
4.75(r – w) = 2100
4.75r – 4.75w = 2100
4(r + 2w) = 2100
4r + 8w = 2100, so solve for r, r = -2w + 525
Solve with substitution: 4.75(-2w + 525) – 4.75w = 2100
-9.5w + 2493.75 – 4.75w = 2100
-14.25w = - 393.75
w = 27.6 mph, so the wind speed is 27.6 mph
Solving for r, r = -2(27.6) + 525 = -55.2 + 525 = 469.8 mph (fast plane!)
Guided Practice
Attempt the following problem. Check with your teacher for the correct answer.
1. A riverboat traveled 48 miles downstream in 2 hours. The return trip took 2 hours and 40
minutes. Find the rate of the riverboat in still water and the rate of the current.
1. ___________________________
31
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Wind and Current Problems
(To Be Used After Section 7.5)
Exercises
1. A plane travels 8400 km against the wind in 7 hours. With the wind, the plane makes the
return trip in 6 hours. Find the speed of the plane in still air and the speed of the wind.
1. ___________________________
2. In a canoe race, a team paddles downstream 480 m in 60 seconds. The same team makes the
trip upstream in 80 seconds. Find the team’s rate in still water and the rate of the current.
2. ___________________________
3. It takes an airplane 1 h 30 min to fly 600 km against the wind. The return trip with the wind
takes only 1 h. Find the total flying time for the round trip if there was no wind.
3. ___________________________
4. Len is planning a three-hour trip down the Colorado River and back to his starting point. He
knows that he can paddle in still water at 3 mi/h and that the rate of the current is 2 mi/h. How
much time can he spend going downstream? How far downstream can he travel?
4. __________________________
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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
5. A motorboat has a 4-hour supply of gasoline. How far from the marina can it travel if the rate
going out against the current is 20 mi/h and the rate coming back with the current is 30 mi/h?
5. __________________________
6. A motorboat goes 36 km downstream in the same amount of time that it takes to go 24 km
upstream. If the current is flowing at 3 km/h, what is the rate of the boat in still water?
6. ___________________________
7. The rate of the current in the Susanna River is 4 km/h. If a canoeist can paddle 5 km
downstream in the same amount of time that she can paddle 1 km upstream, how fast can she
paddle in still water?
7. __________________________
8. The 1080 km trip from Madrid to Paris takes 2 h flying against the wind and 1.5 h flying with
the wind. Find the speed of the plane in still air and the speed of the wind.
8. __________________________
33
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Solving Problems Using Factoring
(To Be Used With Sections 9.6 to 9.8)
Goal: Solve real-world quadratics by utilizing factoring techniques learned in Chapter 9.
Skills: Set-up and solve a quadric equation using various factoring techniques.
Determine if any answers are not reasonable.
Example 1: Find two consecutive integers who product is 72.
Solution 1: Define a variable to represent the unknown: let n = first integer
so, n + 1 = 2nd integer
Write an equation to represent what is described in the problem:
n(n + 1) = 72
Solve the quadratic:
n2 + n = 72
n2 + n – 72 = 0
(n + 9)(n – 8) = 0
n + 9 = 0 or n – 8 = 0
n = -9 or n = 8
Two answers: If n = -9, then the two consecutive integers are -9 and -8
If n = 8, then the two consecutive integers are 8 and 9
Example 2: As a professional photographer, Patty often needs to make prints of her photos to
show to clients. Usually, she is asked to make different sized prints of the same photo. She has
just finished making a print that is 8 cm long by 6 cm wide. Now, Patty wants to reduce the
length and width by the same amount so that the are of the new print is one-half the area of the
original print. By what amount should Patty reduce the length and width of the original print?
Solution 2: Note: It may help to draw of sketch of the original and new photo.
Let x = amount to reduce the length and width of the original photo
The length of the new print: 8 – x
The width of the new print: 6 – x
Equation to represent the described situation: (8 – x)(6 – x) = ½ (48)
Solve: 48 – 8x – 6x + x2 = 24
x2 – 14x + 24 = 0
(x – 12)(x – 2) = 0, So x = 12 or 2.
The only reasonable answer is 2 cm because she cannot reduce the original by 12cm.
Guided Practice
Attempt the following problem. Check with your teacher for the correct answer.
1. The length of Rachel’s rectangular garden is 5 yards more than its width. The area of the
garden is 234 square yards. What are its dimensions?
1. __________________________
34
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Solving Problems Using Factoring
(To Be Used With Sections 9.6 to 9.8)
Exercises
1. A rectangular photograph is 8 cm by 12 cm. The photo is enlarged by increasing the length
and width by the same amount. If the are of the new photo is 69 square centimeters greater than
the area of the original photo, what are the dimensions of the new photo?
1. __________________________
2. A strip of uniform width is plowed along all four sides of a 12-km by 9-km rectangular
cornfield. How wide is the plowed strip if the cornfield is half plowed?
2. __________________________
3. If a number is added to its square, the result is 56. Find the number.
3. ___________________________
4. Find two consecutive negative integers who product is 90.
4. ___________________________
5. The sum of the squares of two consecutive negative even integers is 100. Find the integers.
5. ___________________________
35
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
6. The length of a rectangle is 8 cm greater than its width. Find the dimensions of the rectangle
if its area is 105 cm2?
6. _________________________
7. Find the dimensions of rectangle whose perimeter is 42 m and whose area is 104 m2.
7. _________________________
8. Originally the dimensions of a rectangle were 20 cm by 23 cm. When both dimensions were
decreased by the same amount, the area of the rectangle decreased by 120 cm2. Find the
dimensions of the new rectangle.
8. ________________________
9. Vanessa built a rectangular pen for her dogs. She used an outside wall of the garage for one
side of the pen. She used 20 m of fencing in order to build the other 3 sides to complete the
rectangle. Find the dimensions of the pen if the area of the resulting pen was 48 m2.
9. ________________________
10. A 50 m by 120 m park consists of a rectangular lawn surrounded by a path of uniform width.
Find the dimensions of the lawn if its area is the same as the are of the path. (Hint: let x = width
of the path)
10. _________________________
36
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Vertical Motion Problems
(To Be Used With Section 10.6)
1
Goal: Solve vertical motion problems defined by formula h   gt 2  vt  ho
2
Skills: Understand the different parameter of the formula.
When solving for t, determine which solution method is most effective.
Determine the reasonableness of the solutions.
Note: The acceleration due to gravity, g, is 32 feet/sec2 or 9.8 m/sec2
Example 1: A model rocket is shot directly upward with an initial speed of 34.3 m/s. When
will it be at a height of 49 m?
Solution 1: The unknown quantity to solve for is t (time).
In the formula, g = 9.8, v = 34.3, and ho = 0 (launched from ground level)
Set up and solve an equation where the height is equal to 49m:
1
  9.8  t 2  34.3t  49
2
- 4.9t2 + 34.3t – 49 = 0
Solve with the quadratic formula:
34.3  34.32  4  4.9  49 
34.3  216.09 34.3  14.7 49 19.6

or
 5 or 2
=
9.8
9.8
9.8
2  4.9 
9.8
Therefore, the rocket is at a height of 49 m after 2 seconds (on the way up) and after 5 seconds
(on the way down). For this problem, both answers make sense.
t

Guided Practice
Answer each of the following questions. Check with your teacher for the correct answers.
1. A flare is launched from a life raft (at sea level) with an initial upward velocity of 192 feet per
second. How many seconds will it take for the flare to return to sea level?
1. ____________________
2. A punter punts a football straight up into the air at 120 feet per second. If the ball was kicked
from a height of 3 feet above the ground, how long will it take for the ball to hit the ground?
2. _____________________
37
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Vertical Motion Problems
(To Be Used With Section 10.6)
Exercises
1. A certain firework rocket is set off at an initial vertical velocity of 440 feet per second. This
firework is designed to explode at a height of 3000 feet. How many seconds after it is set off
will the rocket reach 3000 feet and explode?
1. __________________________
2. A ball is thrown upward with an initial speed of 24.5 m/s. When is it 19.6 m high? (Assume
the initial height is 0).
2. __________________________
3. A batter hit a ball from a height 3 feet above ground straight up with a velocity of 120 ft/s. If
the catcher caught it 5 feet above the ground, how long was the ball in the air?
3. __________________________
4. The CCHS physics students dropped eggs from the 2nd level of the gym to see if the eggs
would break. If the eggs are dropped from a height of 24 feet, how long are they in the air before
they hit the ground?
4. __________________________
5. Mitch tossed an apple to Kathy, who was on a balcony 40 ft above him, with an initial speed
of 56 ft/s. Kathy missed the apple on its way, but caught it on its way down. How long was the
apple in the air?
5. _________________________
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Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
6. A signal flare is fired upward from an initial height of 3 meters at a speed of 245 m/s. A
stationary balloonist at a height of 1966 meters sees the flare pass on its way up. How long after
this will the flare pass the balloonist on the way down?
6. ________________________
7. A ball is thrown upward from the top of a 98 m tower with an initial speed of 29.2 m/s. How
much later will it hit the ground?
7. ________________________
8. A rocket is fired upward with an initial velocity of 160 ft/s.
a) When is the rocket 400 ft. high?
8a. __________________
b) How do you know that 400 ft. is the maximum height the rocket reaches?
8b. __________________
9. The Charger’s punter hits a punt with an initial vertical velocity of 80ft/s from a height of 4 ft
above the ground.
a) What is the punt’s hang time?
9a. __________________
b) What is the maximum height reached by the ball?
9b. __________________
10. Prince Charming tossed a rose to Princess Guinevere from an initial height of 1 meter above
the ground with a vertical velocity of 3 m/s. If Princess Guinevere is sticking her head out of a
tower that is 10 meters above the ground, will the rose reach her? Why or why not?
10. ___________________
39
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Work Problems
(To Be Used With Section 12.7)
Goal: Solve equations involving work times to complete a job.
Skills: Understand how to represent the proportion of the job completed each time period.
Understand how to set up an equation to represent more than one person working
together.
Understand how to solve equations that contain fractions by multiplying both sides by the
LCD.
Example 1: Josh can paint a barn in 4 days. His father can do the same job in 2 days. How
long will it them to paint the barn if they work together?
Solution 1: Josh completes ¼ of the job each day (So, ¼ is Josh’s work rate)
1
Josh’s work time on the barn is unknown, x, so let x = amount of job done by Josh
4
Josh’s father completes ½ of the job each day (So, ½ is dad’s work rate)
1
Dad’s work time is the same as Josh’s, x, so let x  amount of job done by dad.
2
The BIG Idea: (Josh’s part of the job) + (Dad’s part of the job) = The whole job
1
1
x  x  1 (The equation is equal to 1 to mean 1 whole job)
4
2
x x
  1 (Now multiply both sides by the LCD, 4)
4 2
x
 x
x x
4     1 so, 4    4    4 1 so, x + 2x = 4
4
2
4 2
3x = 4
4
x  hours
3
1
So, working together it takes them 4/3 of an hour or 1 hours, or 1 hour, 20 min.
3
1
Example 2: Robot A takes 6 minutes to weld a fender. Robot B takes on 5 minutes. If they
2
work together for 2 minutes, how long will it take Robot B to finish welding the fender by itself?
Solution 2: This question is different than examp1e #1 because we know robot A will work for
exactly 2 minutes, but robot B will work 2 + x minutes where x represents the additional time it
will take robot B to finish the job itself.
1
1
Robot A’s Work Rate:
(Completes
of the job each minute)
6
6
2
1
1
2
Robot B’s Work Rate:
(Completes
of the job each minute)


1
11
11
11
5
2
2
(Robot A’s part of the job) + (Robot B’s part of the job) = Whole Job
1
2
1 4 2x
 2    2  x   1 , so    1 , so multiply both sides by the LCD (33)
6
11
3 11 11
For this problem:
40
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
1
4
 2x 
 1 4 2x 
33      33(1) , so 33    33    33    33 ,
3
 11 
 11 
 3 11 11 
2
11 + 12 + 6x = 33, so 6x = 10, so x  5  1 min.
3
3
2
So, it will take Robot B 1 minutes or 1 minute, 40 seconds to finish the job itself.
3
Guided Practice
Try each of the following problems. Check with your teacher for the solutions.
1. Using a new mower, Abby can mow the lawn in 2 hours. Her sister Carla uses an older
mower and takes 3 hours to mow the lawn. How long will it take the sisters to mow the lawn if
they work together?
1. ____________________
2. Brett usually takes 50 minutes to groom the horses. After working for 10 minutes, he was
joined by Angela and they finished the grooming in 15 minutes. How long would it have taken
Angela working alone.
Hints: 1. Let 1 by Brett’s work rate and let 1 by Angela’s work rate. The time Brett
50
x
worked was 25 minutes and the time Angela worked was 15 minutes. Set up an equation and
solve for x. This will be how long it will take Angela to do the work alone.
2. _____________________
41
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
Work Problems
(To Be Used With Section 12.7)
Exercises
1. It takes Sally 15 minutes to pick the apples from the tree in her backyard. Lisa can do it in 25
minutes. How long will it take them working together?
1. _______________
2. Phil can paint a garage in 12 hours and Rick can do it in 10 hours. They work together for 3
hours. How long will it take Rick to finish the job alone?
2. _______________
3. Chuck can shovel the snow off his driveway in 40 minutes. He shovels snow for 20 minutes
and then is joined by his wife, Joan. If they shovel the remaining snow in 10 minutes, how long
would it have taken Joan to shovel the driveway by herself?
3. _______________
4. A roofing contractor estimates that he can shingle a house in 20 hours and that his assistant
can do it in 30 hours. How long will it take them to shingle the house working together?
4. _______________
5. Stan can load his truck in 24 minutes. If his brother helps him, it takes them 15 minutes to
load the truck. How long does it take Stan’s brother alone?
5. _______________
42
Cathedral Catholic High School
Algebra 1A Problem Solving Workbook
Name:____________________________
Date: ________________Period:_______
6. One painting machine works twice as fast as another. When both machines are used, they can
print a magazine in 3 hours. How many hours would each machine require to do the job alone?
6. _____________________
7. Art can do a job in 30 minutes, Bonnie can do it in 40 minutes, and Claire can do it in 60
minutes. How long will it take them to complete the job if they work together?
7. _______________
8. It takes my father 3 hours to plow our cornfield with his new tractor. Using the old tractor it
takes me 5 hours. If we plow together for 1 hour before I go to school, how long will it take him
to finish the plowing?
8. _______________
9. One pump can fill a water tank in 3 hours, and another pump takes 5 hours. When the tank
was empty, both pumps were turned on for 30 minutes and them the faster pump was turned off.
How much longer did the slower pump have to run before the tank was filled?
9. ______________
10. The fill pipe for a tank can fill the tank in 4 hours, and the drain pipe can drain it in 2 hours.
If both pipes are accidentally opened, how long will it take to empty a half filled tank?
10. ______________
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