Question

If 8 skiers raced down a course multiple times in one day, and each skier used a different pair of skis on each run, what is the total number of different skis used by the 8 skiers during the day?

• A. No skier shared any skis with any other skier.
• B. Each skier made exactly 12 runs during the day.
• C.
• D.
• E.
• A. \( 4 \oplus 7<1 \)
• B. \( 4 \oplus 3>1 \)
• C.
• D.
• J.
• A. One salesperson exceeded the quota by 8 cars and received a total monthly commission of $7,500.
• B. One salesperson achieved only half of the quota; he received a commission of $1,750 and a warning that he will be fired unless he meets the next month's quota.
• C.
• D.
• O.
• A. \( b>a^2 \)
• B. \( a - b>1 \)
• C.
• D.
• T.
• A. There are twice as many halibut as cod in the hold, and twice as many haddock as halibut.
• B. Cod account for 1/7 of the fish by number in the hold.
• C.
• D.
• Y.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that no skier shared any skis with any other skier. This means that each skier used a unique pair of skis for each run, but it doesn't tell us how many runs each skier made, so we can't calculate the total number of skis used. Therefore, statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that each skier made exactly 12 runs. This means that each skier used 12 different pairs of skis, but it doesn't give us any information about whether any skiers shared skis. Hence, statement (2) alone is not sufficient.
Step 3: Combine both statements.
From statement (1), we know that no skier shared skis, and from statement (2), we know that each skier made 12 runs. Thus, each skier used 12 different pairs of skis, and since there are 8 skiers, the total number of skis used is: \[ 8 \times 12 = 96 \, \text{different skis}. \] Conclusion:
Both statements together are sufficient to answer the question. The total number of skis used by the 8 skiers is 96. Therefore, the correct answer is (C).

Answer 2

Step 1: Analyze statement (1).
Statement (1) tells us that \( 4 \oplus 7<1 \), which provides us with some constraints on what \( \oplus \) could represent. We know that the result of \( 4 \oplus 7 \) must be less than 1, so we can infer that \( \oplus \) could be either subtraction or division. This helps us to narrow down the possible operations for \( \oplus \). Step 2: Analyze statement (2).
Statement (2) tells us that \( 4 \oplus 3>1 \), but this alone doesn't provide enough information about the specific operation \( \oplus \). It doesn't tell us whether we are dealing with addition, multiplication, subtraction, or division. Hence, statement (2) alone is insufficient to determine the value of \( 7 \oplus 4 \).
Conclusion:
Statement (1) alone is sufficient to determine the value of \( 7 \oplus 4 \), but statement (2) alone is not. The correct answer is (A).

Answer 3

Step 1: Analyze statement (1).
Let \( q \) represent the monthly quota. From statement (1), we know that one salesperson exceeded the quota by 8 cars, and their total commission was $7,500. The commission breakdown is: \[ \text{Commission for quota} = 250q, \quad \text{Commission for exceeding quota} = 500 \times 8 = 4,000. \] Thus, the total commission is: \[ 250q + 4,000 = 7,500 \quad \Rightarrow \quad 250q = 7,500 - 4,000 = 3,500 \quad \Rightarrow \quad q = \frac{3,500}{250} = 14. \] So the monthly quota is 14 cars. Therefore, statement (1) alone is sufficient to determine the quota. Step 2: Analyze statement (2).
Let \( q \) represent the monthly quota. From statement (2), we know that the salesperson achieved half the quota and received $1,750 in commission. The commission for half the quota is: \[ \text{Commission for quota} = 250 \times \frac{q}{2} = 125q. \] Since the total commission is $1,750, we have: \[ 125q = 1,750 \quad \Rightarrow \quad q = \frac{1,750}{125} = 14. \] Thus, the monthly quota is 14 cars. Statement (2) alone is also sufficient to determine the quota. Conclusion:
Both statements together provide sufficient information to determine the quota, and either statement alone is sufficient. Therefore, the correct answer is (C).

Answer 4

Step 1: Analyze the given equation.
We are given that \( a - b = 1 \). The expression we need to analyze is: \[ \frac{a - b}{b + a} = \frac{1}{b + a}. \] We need to determine if this expression is less than 1, i.e., if: \[ \frac{1}{b + a}<1 \quad \text{or equivalently} \quad b + a>1. \] Step 2: Analyze statement (1).
Statement (1) tells us that \( b>a^2 \), but this doesn't provide a specific relationship between \( a \) and \( b + a \), so statement (1) alone is not sufficient to determine whether \( b + a>1 \).
Step 3: Analyze statement (2).
Statement (2) tells us that \( a - b>1 \), but this contradicts the given condition \( a - b = 1 \). Therefore, statement (2) alone is inconsistent with the problem and cannot be used to determine the value of the expression.
Conclusion:
Both statements together are not sufficient to answer the question. Thus, the correct answer is (E).

Answer 5

Step 1: Analyze statement (1).
Statement (1) tells us the following relationships between the numbers of fish in the hold: - There are twice as many halibut as cod: \[ \text{Halibut} = 2 \times \text{Cod}. \] - There are twice as many haddock as halibut: \[ \text{Haddock} = 2 \times \text{Halibut}. \] This implies that the total number of fish in the hold is: \[ \text{Total} = \text{Cod} + \text{Halibut} + \text{Haddock} = x + 2x + 4x = 7x. \] Thus, the probability of selecting a halibut or a haddock is: \[ P(\text{Halibut or Haddock}) = \frac{2x + 4x}{7x} = \frac{6x}{7x} = \frac{6}{7}. \] Thus, statement (1) is sufficient to determine the probability. Step 2: Analyze statement (2).
Statement (2) tells us that cod account for \( \frac{1}{7} \) of the fish in the hold. This means the total number of fish in the hold is \( 7 \times \) the number of cod, but we still need information about the number of haddock and halibut to calculate the probability. Therefore, statement (2) alone is insufficient. Step 3: Combine both statements.
Combining both statements, we can calculate the total number of halibut and haddock, and the total number of fish in the hold. Thus, both statements together provide sufficient information to determine the probability. Conclusion:
Thus, both statements together are sufficient to answer the question, and the correct answer is (C).