Deck 1: Equations and Inequalities

Question 1

Five people decide to share the cost of a yacht. By bringing in an additional partner, they can reduce the cost to each person by $7000. What is the total cost of the yacht?

A)$112,000
B)$252,000
C)$210,000
D)$140,000
E)$175,000

Answer 1

$210,000

Question 2

The SMOG (Simplified Measure of Gobbledygook)readability formula estimates the reading grade level required by a person if he or she is to fully understand the written material assessed. The SMOG formula is defined as where w is the number of words that have three or more syllables that should appear in a sample of 30 sentences. A writer of children s books has decided to write a book that can be fully understood by children at the 7th-grade level. What is the maximum number of words with three or more syllables that should appear in any sample of 30 sentences of the book?

A)20
B)17
C)16
D)18
E)19

Answer 2

16

Question 3

Jordan is driving along a highway that passes through Moyock. His distance, d , in miles, from Moyock is given by the equation where t is the time, in hours, since the start of his trip and . Determine when Jordan will be exactly 85 miles from Moyock. Round to the nearest minute.

A)8 hrs 7.5 min and 4 hrs 22.5 min
B)4 hrs 7.5 min and 4 hrs 22.5 min
C)8 hrs 22.5 min
D)4 hrs 7.5 min and 8 hrs 22.5 min
E)4 hrs 22.5 min

Answer 3

4 hrs 7.5 min and 8 hrs 22.5 min

Question 4

Solve the following formula for the specified variable.

A)
B)
C)
D)
E)

Question 5

The National Football League uses the following formula to rate quarterbacks: In this formula, C is the percentage of pass completions, Y is the average number of yards gained per pass attempt, T is the percentage of touchdown passes, and I is the percentage of interceptions. During the 2006 season, Brett Farve, the quarterback for the Green Bay Packers, completed 55.95% of his passes. He averaged 6.34 yards per pass attempt, 2.94% of his passes were for touchdowns, and 2.94% of his passes were intercepted. Determine Brett Farve s quarterback rating for the 2006 season. Round to the nearest tenth.

A)70.2
B)72.7
C)75.1
D)77.6
E)69.9

Question 6

Solve the absolute value equation below for x .

A)3 and 24
B)-10 and 24
C)-2 and 3
D)-10 and -2
E)-2 and 24

Question 7

Solve the absolute value equation below for x .

A) and
B)
C) and
D) and
E) and

Question 8

The gas mileage m , in miles per gallon, obtained during a long trip is given by where s is the speed of the automobile in miles per hour and . At what constant speed can the automobile drive to obtain a gas mileage of exactly 25 miles per gallon?

A)25 mph and 40 mph
B)45 mph and 49 mph
C)25 mph and 45 mph
D)40 mph and 49 mph
E)40 mph and 45 mph

Question 9

An investment adviser invested $100,000 in two accounts. One investment earned 9% annual simple interest, and the other investment earned 8.5% annual simple interest. The amount of interest earned for 1 year was $8700 How much was invested in each account?

A)$49,000 → 9% ; $51,000 → 8.5%
B)$40,000 → 9% ; $60,000 → 8.5%
C)$73,000 → 9% ; $27,000 → 8.5%
D)$36,000 → 9% ; $64,000 → 8.5%
E)$34,000 → 9% ; $66,000 → 8.5%

Question 10

The length of a rectangle is 12 feet less than 4 times the width of a rectangle. If the perimeter of the rectangle is 466 feet, find the width and the length.

A)
B)
C)
D)
E)

Question 11

Solve the following absolute value equation for p.

A)-9, 3
B)-9, 7
C)3
D)-9, 9
E)7

Question 12

Solve the equation below.

A)8
B)-1
C)-6
D)-10
E)6

Question 13

Solve the following equation.

A)10
B)
C)
D)
E)

Question 14

The SMOG (Simplified Measure of Gobbledygook)readability formula is often used to estimate the reading grade level required by a person if he or she is to fully understand the written material being assessed. The formula is given by where w is the number of words that have three or more syllables in a sample of 30 sentences. A sample of 30 sentences from a given book shows a total of 75 words that have three or more syllables. Use the SMOG reading grade level formula to estimate the reading grade level required to fully understand this novel. Round your answer to the nearest tenth.

A)17.7 reading grade level
B)5.7 reading grade level
C)8.7 reading grade level
D)11.7 reading grade level
E)2.7 reading grade level

Question 15

The percent of a file that remains to be downloaded using an Internet connection is given by the equation Percent Remaining where N is the size of the file in bytes and t is the number of seconds since the download began. In how many minutes will 20% of a 264,000-byte file remain to be downloaded? Round your answer to the nearest minute.

A)13 minutes
B)8 minutes
C)18 minutes
D)12 minutes
E)10 minutes

Question 16

Solve the following formula for the specified variable.

A)
B)
C)
D)
E)

Question 17

The price of a computer fell 30% this year. If the computer now costs $525, how much did it cost last year?

A)$730
B)$760
C)$720
D)$740
E)$750

Question 18

The Gunning-Fog Index is often used to estimate the reading grade level required by a person if he or she is to fully understand the written material being assessed. The formula is given by where A is the average number of words per sentence and P is the percentage of words that have three or more syllables. The Gunning-Fog Index is defined as the minimum grade level required by a person if he or she is to easily understand the text on the first reading. In a large sample of sentences from a given novel, the average number of words per sentence is 12.7, and the percentage of words with three or more syllables is 18.2. Use the Gunning-Fog Index formula to estimate the reading grade level required to easily understand this novel. Round your answer to the nearest tenth.

A)15.7 reading grade level
B)14.6 reading grade level
C)12.4 reading grade level
D)16.8 reading grade level
E)13.5 reading grade level

Question 19

Solve the following equation.

A)
B)
C)
D)
E)

Question 20

Solve the following equation.

A)
B)
C)
D)
E)

Question 21

Find the solution of the following equation containing a rational exponent on the variable.

A)
B)9
C)27
D)
E)

Question 22

A mason can lay the bricks in a sidewalk in 14 hours. The mason's apprentice requires 20 hours to do the same job. After working together for 3 hours, the mason leaves for another job and the apprentice continues working. How long will it take the apprentice to complete the job?

A) hour
B) hours
C) hours
D) hours
E) hours

Question 23

Use the power principle to solve the following radical equation.

A)28
B)96
C)20
D)10
E)266

Question 24

A veterinarian wishes to use 162 feet of chain-link fencing to enclose a rectangular region and subdivide the region into two smaller rectangular regions, as shown below. If the total enclosed area is 1092 square feet, find the dimensions of the enclosed area.

A) or
B) or
C) or
D) or
E) or

Question 25

Solve the rational equation below.

A)3
B)1
C)-6
D)9
E)-1

Question 26

Find all the real solutions of the following equation by first rewriting the equation as a quadratic equation.

A)
B)
C)
D)
E)

Question 27

Solve the following rational equation.

A)
B)
C)4
D)17
E)

Question 28

At the present time a company manufactures rectangular solid candy bars the measure 4.5 inches by 1.5 inches by 0.5 inch. Due to financial times, the company has decided to keep the price of the candy bar fixed and reduce its volume by 15%. What should be the dimensions, to the nearest tenth of an inch, of the new candy bar if it is decided to keep the height at 0.5 inch and to make the length on the candy bar 2.25 times as long as its width?

A)width: 1.3 in; length: 2.9 in
B)width: 1.7 in; length: 3.8 in
C)width: 1.4 in; length: 3.1 in
D)width: 1.8 in; length: 4.0 in
E)width: 1.6 in; length: 3.6 in

Question 29

Jon went to his local electronics store and purchased an HDTV whose screen measured 64 inches diagonally, and whose aspect ratio was 16 to 9. Find the width and height of the screen. Round to the nearest tenth.

A)width: 57.9 in; height: 34.0 in
B)width: 54.5 in; height: 32.9 in
C)width: 55.8 in; height: 31.4 in
D)width: 55.0 in; height: 30.2 in
E)width: 54.0 in; height: 32.8 in

Question 30

The length of each side of an equilateral triangle is 29 centimeters. Find the altitude of the triangle. Round your answer to the nearest tenth of a centimeter.

A)25.1 centimeters
B)12.6 centimeters
C)20.5 centimeters
D)41 centimeters
E)50.2 centimeters

Question 31

Use the square root procedure to solve the quadratic equation below.

A)
B)
C)
D)
E)

Question 32

A company has determined that the profit P , in dollars, it can expect from the manufacture and sale of x tennis racquets is given by . How many racquets should the company manufacture and sell to earn a profit of $425,000?

A)4600 or 8400 racquets
B)4600 or 5000 racquets
C)5000 or 8400 racquets
D)5000 or 5900 racquets
E)4600 or 5900 racquets

Question 33

How many liters of a 42% acid solution should be mixed with 5 liters of a 30% acid solution to produce a 39% solution?

A)15 liters
B)4 liters
C)9 liters
D)6 liters
E)8 liters

Question 34

A professional painter can paint a kitchen in 10 hours. His apprentice can paint the same kitchen in 20 hours. If they worked together, how long would it take for them to paint the kitchen?

A)15
B)
C)
D)
E)10

Question 35

A square piece of cardboard is formed into a box by cutting out 3-inch squares from each of the corners and folding up the sides. If the volume of the box needs to be 126.75 cubic inches, what size square piece of cardboard is needed?

A)3.5 inches by 3.5 inches
B)13.5 inches by 13.5 inches
C)11.5 inches by 11.5 inches
D)9.5 inches by 9.5 inches
E)12.5 inches by 12.5 inches

Question 36

The height, h , in feet, of a baseball above the ground t seconds after it is hit is given by . Use this equation to determine the number of seconds, to the nearest tenth of a second, from the time the ball is hit until the ball hits the ground.

A)1.8 s
B)1.9 s
C)2.0 s
D)2.1 s
E)2.2 s

Question 37

Use the power principle to solve the following radical equation.

A)16
B)-2
C)100
D)-22
E)10

Question 38

Running at an average rate of 5 meters per second, a sprinter ran to the end of a track. The sprinter then jogged back to the starting point at an average rate of 4 meters per second. The total time for the sprint and the jog back was 2 minutes 15 seconds. Find the length of the track.

A)310 meters
B)315 meters
C)300 meters
D)290 meters
E)320 meters

Question 39

Solve the quadratic equation below by factoring and applying the zero product principle.

A) and
B) and 3
C) and 3
D) and 3
E) and

Question 40

A model rocket is launched upward with an initial velocity of 170 feet per second. The height, in feet, of the rocket t seconds after the launch is given by . How many seconds after the launch will the rocket be 450 feet above the ground? Round to the nearest tenth of a second.

A)5.0 seconds and 5.6 seconds
B)5.0 seconds and 11.3 seconds
C)5.1 seconds and 11.3 seconds
D)5.1 seconds and 5.6 seconds
E)5.0 seconds and 5.1 seconds

Question 41

Write the equation that expresses the relationship between the variables described below, then use the given data to solve for the variation of constant. w varies directly as q, and when

A)
B)
C)
D)
E)

Question 42

Use interval notation to express the solution set of the inequality below.

A)
B)
C)
D)
E)

Question 43

A solid silver sphere has a diameter of 14 millimeters, and a second silver sphere has a diameter of 16 millimeters. The spheres are melted down and recast to form a single cube. What is the length of each edge of the cube? Round your answer to the nearest tenth of a millimeter.

A)59.8 millimeters
B)30.6 millimeters
C)169.3 millimeters
D)15.3 millimeters
E)103.7 millimeters

Question 44

Hooke s Law states that the distance a spring stretches varies directly as the weight on the spring. A weight of 70 pounds stretches a spring 9 inches. How far will a weight of 110 pounds stretch the spring? Round to the nearest tenth of an inch.

A)14.1 inches
B)5.7 inches
C)12.2 inches
D)10.0 inches
E)7.8 inches

Question 45

Use interval notation to express the solution set for the inequality below.

A)
B)
C)
D)
E)

Question 46

A video store offers two rental plans. The yearly membership fee and the daily charge per video for each plan are shown below. How many one-night rentals can be made per year if the No-fee plan is to be the less expensive of the plans? Rental Plan Yearly Fee Daily Charge per Video Low-Rate $30.00 $2.49 No-Fee None $3.99

A)30 rentals or less
B)25 rentals or less
C)20 rentals or more
D)20 rentals or less
E)25 rentals or more

Question 47

On a ship, the distance d that you can see to the horizon is given by , where h is the height of your eye measured in feet above sea level and d is measured in miles. How high is the eye level of a navigator who can see 16 miles to the horizon? Round your answer the nearest foot.

A)114 feet
B)64 feet
C)6 feet
D)5 feet
E)256 feet

Question 48

Find all the real solutions of the following equation by first rewriting the equation as a quadratic equation.

A) ,
B) ,
C) ,
D) ,
E) ,

Question 49

Write the equation that expresses the relationship between the variables, and then use the given data to solve for the variation constant. T varies jointly as r and and the cube of s and when and .

A)k =
B)k =
C)k =
D)k =
E)k =

Question 50

The attendance A , in billions of people, at movie theaters has been declining for a few years. A model of the decline is given by , where corresponds to the year 2000. According to this model, during what year will movie attendance first be less than 1.25 billion people?

A)2014
B)2012
C)2015
D)2016
E)2013

Question 51

Solve the following compound inequality. Write the solution set using set-builder notation.

A)
B)
C)
D)
E)

Question 52

Use the critical value method to solve each rational inequality. Write each solution set in interval notation.

A)
B)
C)
D)
E)

Question 53

Use the critical value method to solve each rational inequality. Write each solution set in interval notation.

A)
B)
C)
D)
E)

Question 54

The pressure a liquid exerts at a given point on a submarine is directly proportional to the depth of the point below the surface of the liquid. If the pressure at a depth of 4 feet is 250 pounds per square foot, find the pressure at a depth of 14 feet.

A)687.5 pounds per square foot
B)1000 pounds per square foot
C)875 pounds per square foot
D)812.5 pounds per square foot
E)1125 pounds per square foot

Question 55

A worker can build a fence in 8 hours. With the help of an assistant, the fence can be built in 5 hours. How long should it take the assistant, working alone, to build the fence?

A) hours
B) hours
C) hour
D) hour
E) hours

Question 56

Use the critical value method to solve the polynomial inequality below. Use interval notation to write the solution set.

A)
B)
C)
D)
E)

Question 57

If a researcher wanted to know the mean weight (the mean is the sum of all measurements divided by the number of measurements)of women in the U.S., the weight of every woman would have to be measured, and then the mean weight calculated, an impossible task. Instead, researchers find a representative sample of women and find the mean weight of the sample. Because the entire population of women is not used, there is a possibility that the calculated mean weight is not the true mean weight. For one study, researchers used the formula where m is the true mean weight, in pounds, of all women, to be 98% sure of the range of values for the true mean weight. Using this inequality, what is the range of mean weights of women in the U.S.? Round to the nearest tenth of a pound.

A)
B)
C)
D)
E)

Question 58

You can rent a car for the day from Company A for $28 plus $0.11 a mile. Company B charges $22.00 plus $0.22 a mile. Find the number of miles m (to the nearest mile)per day for which it is cheaper to rent from Company A.

A)less than 152 miles
B)less than 55 miles
C)more than 55 miles
D)less than 104 miles
E)more than 152 miles

Question 59

Solve the compound inequality below. Write the solution set using set-builder notation. or

A)
B)
C)
D)
E)

Question 60

Write an equation that represents the relationship between the given variables below. Use k as the variation constant. d varies jointly as r as well as t

A)
B)
C)
D)
E)

Question 61

The load L that a horizontal beam can safely support varies jointly as the width w and the square of the depth d . If a beam with width 2 inches and depth 12 inches safely supports up to 798 pounds, how many pounds can a beam of the same length that has a width of 4 inches and depth 10 inches be expected to support? Round to the nearest pound. Assume the two beams are made of the same material.

A)884 lbs
B)776 lbs
C)1209 lbs
D)1108 lbs
E)443 lbs

Question 62

The volume V of a right circular cone varies jointly as the square of the radius r and the height h . Tell what happens to V when r is doubled.

A)V is 4 times as large.
B)V is 3 times as large.
C)V is 2 times as large.
D)V is 9 times as large.
E)V is 6 times as large.

Question 63

The range of a projectile is directly proportional to the square of its velocity. If a motorcyclist can make a jump of 188.5 feet by coming off a ramp at 67 mph, find the distance the motorcyclist could expect to jump if the speed coming off the ramp were increased to 72 mph. Round to the nearest tenth of a foot.

A)220.4 feet
B)214.8 feet
C)213.4 feet
D)214.2 feet
E)217.7 feet

Question 64

The amount of juice in a grapefruit is directly proportional to the cube of its diameter. A grapefruit with a 4-inch diameter contains 5 fluid ounces of juice. How much juice is contained in a grapefruit with a 6-inch diameter? Round to the nearest tenth of a fluid ounce.

A)19.4 fluid ounces
B)18.4 fluid ounces
C)16.9 fluid ounces
D)14.4 fluid ounces
E)17.9 fluid ounces

Question 65

The speed of a bicycle gear, in revolutions per minute, is inversely proportional to the number of teeth on the gear. If a gear with 50 teeth has a speed of 24 revolutions per minute, what will be the speed of a gear with 66 teeth? Round your answer to the nearest revolution per minute.

A)19 revolutions per minute
B)18 revolutions per minute
C)23 revolutions per minute
D)22 revolutions per minute
E)17 revolutions per minute

Question 66

The frequency of vibration of a guitar string under constant tension varies inversely as the length of the string. A guitar string with a length of 20 inches has a frequency of 270 vibrations per second. Find the frequency of a guitar string with a length of 16 inches. Assume the tension is the same for both strings. Round answer to nearest whole vibration.

A)360 vibrations/s
B)277 vibrations/s
C)338 vibrations/s
D)216 vibrations/s
E)349 vibrations/s

Question 67

The load L a horizontal beam can safely support varies jointly as the width w and the square of the depth d and inversely as the length l . If a 16-foot beam with width 7 inches and depth 15 inches safely supports 272 pounds, how many pounds can a 20-foot beam that has width 8 inches and depth 9 inches be expected to support? Round to the nearest pound. Assume the two beams are made of the same material.

A)184 lbs
B)559 lbs
C)89 lbs
D)79 lbs
E)73 lbs