1. Which of the following inequality signs would make a true statement if it replaced the question mark in the following
equation? 3− 5?− 3
A. < C. ≤
B. > D. =
2. Solve the inequality −4y + 6<−14.
A. y > 2 C. y ≤ 5
B. y < 5 D. y > 5
4. At a baseball game, the ratio of Barons fans to Blue Sox fans is 8 : 3. If you know that at least 22,000 people are attending
the game, at least how many Blue Sox fans are there?
A. 1,000 C. 6,000
B. 2,000 D. 16,000
7. In how many ways can you fill three different positions by choosing from 15 different people?
A. 3⋅ 2⋅1 C. 15⋅14⋅13
B. 15⋅ 3 D. 15⋅14⋅12⋅ 3
8. How many different permutations can you make with the letters in the word s e v e n t e e n ?
A. 3,780 C. 15,120
B. 7,560 D. 17!
9. A teacher has a set of 12 problems to use on a math exam. The teacher makes different versions of the exam by putting
10 questions on each exam. How many different exams can the teacher make?
A.
12
11
!
!
C. 132
B. 66 D. 12!
10. Car thieves steal one automobile out of 400 of a certain type every year in a certain city.What annual net premium
should an owner pay for theft insurance in the amount of $16,000 on the certain type of car?
A. $4 C. $40
B. $16 D. $400
11. Solve −8x ≤ 2.
A. x ≤−.25 C. x ≥−.30
B. x ≥−.25 D. x ≤.25
12. Suppose you draw a card from a well-shuffled pack of playing cards. What is the probability the card you draw will be
an ace?
A.
1
13
C.
4
50
B.
4
13
D.
1
52
13. If 100C2 has a value of 4,950, what is the value of 100C98?
A. 4,950 C. 49,500
B. 9,900 D. 99,000
14. How many arrangements of the letters in the word o l i v e can you make if each arrangement must use three letters?
A. 20 C. 5⋅ 4⋅ 3⋅ 2⋅1
B. 60 D. 8⋅ 7⋅ 6⋅ 5⋅ 4⋅ 3⋅ 2⋅1
1. B. > (Because 3 - 5 is less than 3)
2. A. y > 2 (Solving the inequality gives y > 2)
4. D. 16,000 (The ratio of fans is 8:3, so we can set up the proportion 8/3 = x/22000. Solving for x gives x ≈ 58666, but since we need at least 22000, the answer is 16000)
7. C. 15⋅14⋅13 (We need to choose 3 people out of 15, so we can use the combination formula which is 15C3 = 15! / (3! * (15-3)!) = 15⋅14⋅13)
8. C. 15,120 (Counting the number of letters in the word "seventeen", there are 9 distinct letters, so the number of permutations is 9!)
9. B. 66 (Using the combination formula, we can find the number of ways to choose 10 questions out of 12, which is 12C10 = 66)
10. D. $400 (Since 1 out of 400 cars is stolen, the probability of a car being stolen is 1/400. Multiplied by the value of $16,000, the annual net premium should be $400)
11. A. x ≤−.25 (Solving the inequality gives x ≤ -0.25)
12. B. 4/13 (There are 4 aces in a deck of 52 playing cards, so the probability of drawing an ace is 4/52 = 1/13)
13. C. 49,500 (Using the formula for combinations, 100C98 is equal to 100! / (98! * (100-98)!) = 4950. Multiplying by 10 gives 49500)
14. A. 20 (There are 5 letters in the word "olive" and we need to choose 3 of them, so we can use the combination formula which is 5C3 = 5! / (3! * (5-3)!) = 10. However, each arrangement can be reversed, so we multiply by 2 to get 20)
1. The correct answer is B. >
2. To solve the inequality, we need to isolate the variable y.
-4y + 6 < -14
Subtracting 6 from both sides:
-4y < -20
Dividing both sides by -4 (note that when dividing both sides of an inequality by a negative number, the direction of the inequality sign changes):
y > 5
The correct answer is D. y > 5.
4. To find the minimum number of Blue Sox fans, we need to find 3/11th of the total attendance of the game.
Let's set up a proportion:
8/11 = x/22,000
Cross-multiplying:
8 * 22,000 = 11 * x
176,000 = 11 * x
x = 176,000 / 11
x ≈ 16,000
So, at least 16,000 Blue Sox fans are attending the game.
The correct answer is D. 16,000.
7. To fill three different positions by choosing from 15 different people, we need to calculate the number of possible combinations.
The formula for calculating combinations is nCr = n! / (r!(n-r)!)
Plugging in the values:
15C3 = 15! / (3!(15-3)!)
15C3 = 15! / (3!12!)
15C3 = (15 * 14 * 13) / (3!)
15C3 = 15 * 14 * 13 / (3 * 2 * 1)
15C3 = 455
So, there are 455 ways to fill three different positions by choosing from 15 different people.
The correct answer is D. 15 * 14 * 12 * 3.
8. To find the number of different permutations that can be made with the letters in the word "seventeen", we need to calculate the factorial of the number of letters.
The formula for calculating the factorial of a number is n! = n * (n-1)!
Plugging in the values:
7! = 7 * 6!
6! = 6 * 5!
5! = 5 * 4!
4! = 4 * 3!
3! = 3 * 2!
2! = 2 * 1!
1! = 1 * 0!
Using this information, we can calculate:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 * 1
7! = 5,040
So, there are 5,040 different permutations that can be made with the letters in the word "seventeen".
The correct answer is C. 15,120.
9. To find the number of different exams the teacher can make, we need to calculate the number of combinations of 12 problems taken 10 at a time.
The formula for calculating combinations is nCr = n! / (r!(n-r)!)
Plugging in the values:
12C10 = 12! / (10!(12-10)!)
Simplifying:
12C10 = 12! / (10!2!)
12C10 = (12 * 11 * 10!) / (10! * 2 * 1!)
The common factors of 10! cancel out:
12C10 = (12 * 11) / 2
12C10 = 132
So, the teacher can make 132 different exams.
The correct answer is C. 132.
10. The annual net premium for theft insurance can be calculated by dividing the value of the insurance coverage ($16,000) by the probability of the car being stolen (1 out of 400).
Annual net premium = Insurance coverage / Probability of theft
Annual net premium = $16,000 / 1/400
Annual net premium = $16,000 * 400
Annual net premium = $6,400,000
So, the owner should pay an annual net premium of $6,400,000 for theft insurance.
The correct answer is not provided in the options.
11. To solve the inequality, we need to isolate the variable x.
-8x ≤ 2
Dividing both sides by -8 (note that when dividing both sides of an inequality by a negative number, the direction of the inequality sign changes):
x ≥ -0.25
So, the correct answer is A. x ≥ -0.25.
12. There are 52 cards in a playing deck, and 4 of them are aces.
The probability of drawing an ace is the number of favorable outcomes (ace cards) divided by the total number of possible outcomes (cards in the deck):
Probability of drawing an ace = 4/52
Simplifying:
Probability of drawing an ace = 1/13
So, the probability of drawing an ace is 1 out of 13.
The correct answer is B. 4/13.
13. The formula for calculating combinations is nCr = n! / (r!(n-r)!)
Given that 100C2 = 4,950, we can set up the equation:
100! / (2!(100-2)!) = 4,950
Simplifying:
100! / (2!98!) = 4,950
(100 * 99 * 98!)/(2 * 1 * 98!) = 4,950
(100 * 99) / 2 = 4,950
9,900 = 4,950
This is not a true statement, so there must be an error in the question.
The correct answer cannot be determined.
14. To find the number of arrangements of the letters in the word "olive" using three letters, we need to use the formula for permutations:
nPr = n! / (n-r)!
Plugging in the values:
5P3 = 5! / (5-3)!
Simplifying:
5P3 = 5! / 2!
5P3 = (5 * 4 * 3!) / 2!
The common factors of 3! cancel out:
5P3 = (5 * 4) / 1!
5P3 = 20
So, there are 20 different arrangements of the letters in the word "olive" using three letters.
The correct answer is A. 20.
1. To determine which inequality sign would make the statement true, let's evaluate the equation first: 3 - 5 ? - 3. Simplifying gives us 0. Now, we need an inequality sign that makes the statement 0 ? 0 true. The correct inequality sign for this would be D. = (equals).
2. Let's solve the inequality -4y + 6 < -14. To isolate y, we'll subtract 6 from both sides of the inequality: -4y < -20. Now, dividing both sides by -4, we get y > 5. Therefore, the correct answer is D. y > 5.
4. The ratio of Barons fans to Blue Sox fans is 8:3. To find the total number of Blue Sox fans, we'll set up the proportion 8/3 = x/22000, where x represents the number of Blue Sox fans. Cross-multiplying gives us 3x = 8 * 22000, and solving for x, we find x = 58666.67. However, since we can't have a fraction of a person, we'll round up to the nearest whole number. Therefore, at least 58667 Blue Sox fans are there. The closest option is C. 6,000.
7. To fill three different positions by choosing from 15 different people, we'll use combinations. The formula for combinations is nCr = n! / r!(n-r)!, where n is the total number of options and r is the number of options chosen. Therefore, the number of ways to fill three different positions is 15C3 = 15! / 3!(15-3)! = (15 * 14 * 13) / (3*2*1) = 455. The correct answer is B. 455.
8. To find the number of different permutations with the letters in the word "seventeen," we'll use the formula for permutations of a word with repeated letters. The formula is n! / (a! * b! * c! * ...), where n is the total number of letters and a, b, c, ... represent the repeated letters. In this case, "seventeen" has two repeated 'e' letters. Therefore, the number of permutations is 9! / (2!). Simplifying, we get 9 * 8 * 7 * 6 * 5 * 4 * 3 = 15,120. The correct answer is C. 15,120.
9. The teacher has a set of 12 problems and needs to make different versions of the exam with 10 questions each. This can be solved using combinations. The formula for combinations is nCr = n! / r!(n-r)!, where n is the total number of options and r is the number of options chosen. In this case, the number of different exams the teacher can make is 12C10 = 12! / (10!(12-10)!) = 12! / (10!2!) = 12 * 11 / (2 * 1) = 66. The correct answer is B. 66.
10. The probability of having an automobile stolen out of a certain type every year is 1 out of 400. To calculate the annual net premium, we divide the value of the insurance ($16,000) by the probability (1/400). Therefore, the annual net premium should be $16,000 / (1/400) = $6,400, which rounded to the nearest dollar is $6,400. The correct answer is D. $400.
11. Let's solve the inequality -8x ≤ 2. To isolate x, we'll divide both sides of the inequality by -8. However, since we're dividing by a negative number, the inequality flips. So we have x ≥ -2/8, which simplifies to x ≥ -0.25. Therefore, the correct answer is A. x ≤ -0.25.
12. A well-shuffled pack of playing cards contains 52 cards, and there are 4 aces in a deck. Therefore, the probability of drawing an ace is 4/52 = 1/13. The correct answer is D. 1/52.
13. The notation 100C2 represents the number of combinations of 100 items taken 2 at a time. The formula for combinations is nCr = n! / r!(n-r)!. Therefore, 100C2 = 100! / (2!(100-2)!) = 100! / (2!98!) = (100 * 99) / (2 * 1) = 4,950. The correct answer is A. 4,950.
14. To find the number of arrangements of the letters in the word "olive" that use three letters, we'll use permutations. The formula for permutations is n! / (n-r)!, where n is the total number of options and r is the number of options chosen. In this case, we have 5 options (the letters "o", "l", "i", "v", "e") and we're choosing 3. Therefore, the number of arrangements is 5! / (5-3)! = 5! / 2! = 5 * 4 * 3 = 60. The correct answer is B. 60.