1. Read the passage below, and then choose the best answer to the question that follows.
Solution
Correct: A
A large body of water, such as a lake or ocean, has a moderating effect on climate because of the way it absorbs, stores, and releases heat from the sun. This process can help regulate the temperature and precipitation patterns in the surrounding area.
2. If a, b, c, and d are positive real numbers such that a + b + c + d = 10, which of the following could be the value of a + c?
Solution
Correct: E
We know that 0 < a < 10, 0 < b < 10, 0 < c < 10, and 0 < d < 10, and a + b + c + d = 10. It follows that a + c < 10. However, since b and d can both be made arbitrarily close to zero, a + c can be made to approach 10. So the correct answer is any of the options less than 10.
3. At a certain company, the average time an employee has been with the company is 10 years. If 10 new employees, each with 0 years of experience, were to join the company, which of the following best describes the effect on the average years of experience of all the employees at the company?
Solution
Correct: C
The addition of 10 new employees with zero experience each would contribute 0 to the total sum of years of experience, but it would increase the number of employees. This means that the average years of experience would decrease, but not significantly, hence, 'less than 1 year'.
4. An empty bucket can be filled from one of three water faucets. Faucet A fills 1/4 of the bucket per minute, faucet B fills 1/6 of the bucket per minute, and faucet C fills 1/8 of the bucket per minute. If two of the faucets are used together, which two will fill the bucket the fastest?
Solution
Correct: A
Faucet A fills 1/4 of the bucket per minute and faucet B fills 1/6 of the bucket per minute, so together, they would fill 1/4 + 1/6 = 5/12 of the bucket per minute. Faucet A and faucet C would fill 1/4 + 1/8 = 3/8 of the bucket per minute. Faucet B and faucet C would fill 1/6 + 1/8 = 7/24 of the bucket per minute. Faucets A and B are the fastest combination, so the correct answer is A.
5. The ratio of males to females at a certain company is 3:5. If there are 80 more females than males at the company, what is the total number of employees at the company?
Solution
Correct: B
Given that the ratio of males to females is 3:5, this means that there are 3x males and 5x females for some value of x. Since there are 80 more females than males, 5x - 3x = 80. This simplifies to 2x = 80. Solving for x, we get x = 40. So the number of males is 3x = 120, and the number of females is 5x = 200. The total number of employees is then 120 + 200 = 320.
6. If it is true that x is an integer and that |x - 1| < |x - 5|, which of the following could be a value of x?
Solution
Correct: C
Given |x - 1| < |x - 5|, x must be closer to 1 than to 5. So the possible integer values for x that satisfy the condition are 0, 1, and 2.
7. A bat and a ball together cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost?
Solution
Correct: A
If the ball costs x dollars, then the bat costs x + $1.00 dollars. We know that x + x + $1.00 = $1.10, or 2x = $0.10. Therefore, the ball costs x = $0.05.
8. If $12 is removed from the sum of the two numbers, the result is 4 times the smaller number, what are the two numbers?
Solution
Correct: D
Let x and y be the two numbers where x < y. Then the given condition becomes x + y - 12 = 4x. This simplifies to y - 12 = 3x, and further to y = 3x + 12. Since y = 3x + 12 and x + y - 12 = 4x, by checking each choice we see that only choice (D) satisfies the equation.
9. In the figure below, the areas of the three squares are 1, 4, and 9, respectively. If the rectangle is formed by combining two of the squares, which two squares would give the largest perimeter for the rectangle?
Solution
Correct: C
To maximize the perimeter of the rectangle formed by combining two of the squares, we want the sides of the rectangle to be as large as possible. Thus we should use the two largest squares (side lengths 3 and 2) to form the rectangle, so the correct answer is the largest and the medium.
10. A company produces a certain product in three different locations. The production cost per unit at location A is $5 and location B is $4. The production cost per unit at location C is 20% higher than at location B.
Solution
Correct: B
The production cost at location C is 20% higher than that at location B, or $4(1 + 0.20) = $4.80.
11. A car rental company has 210 cars to be rented out. They wish to park them in as few rows as possible such that no two adjacent cars are facing the same direction.
Solution
Correct: C
The maximum number of cars that can be in a row is 6 (3 cars facing north and 3 cars facing south in an alternating pattern). Since the company has 210 cars, it must park in at least 210/6 = 35 rows, which is obtained by dividing 210 by 6 and rounding up.
12. What is the sum of all the roots of the given polynomial?
Solution
Correct: A
The given polynomial is x^2 + 3x + 2 = 0. By Vieta's formulas, we have that the sum of all roots of the polynomial ax^2 + bx + c = 0 is -b/a. Therefore, for the given polynomial x^2 + 3x + 2, we have a = 1 and b = 3, so the sum of all roots is -3/1 = -3.
13. If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets?
Solution
Correct: B
If it takes 5 machines 5 minutes to make 5 widgets, then each machine is making 1 widget in 5 minutes. Therefore, if there were 100 machines, it would still take 5 minutes for each machine to make 1 widget, so it would take 5 minutes for 100 machines to make 100 widgets.
14. A survey of a certain population showed that 30% were in favor of a certain issue, 50% were opposed to it, and the remaining percentage was undecided.
Solution
Correct: A
If 30% of the population are in favor of the issue and 50% are opposed, then the percentage of the population that are undecided is 100% - (30% + 50%) = 100% - 80% = 20%.
15. On a map, the distance between City A and City B is 5 inches and the distance between City B and City C is 4 inches. If the distance between City A and City C is 7 inches, what is the measure of angle B?
Solution
Correct: B
The three cities form a triangle with side lengths 5, 4, and 7 inches. By the law of cosines, c^2 = a^2 + b^2 - 2ab * cos(C), where C is the angle opposite side c. We can use this formula to solve for angle B.
16. A bakery sells 250 loaves of bread per day. They make a profit of $0.50 per loaf. What is the total profit the bakery makes in a day?
Solution
Correct: C
To find the total profit the bakery makes in a day, we can multiply the number of loaves of bread sold by the profit per loaf: 250 * $0.50 = $125.
17. Two cars, car A and car B, start at the same time from opposite ends of a 120 mile stretch of a road. Car A travels at 40 mph and car B travels at 60 mph. How long will it take for the two cars to meet?
Solution
Correct: A
Car A travels at a speed of 40 mph, so in t hours, it will travel 40t miles. Car B travels at a speed of 60 mph, so in t hours, it will travel 60t miles. When they meet, the sum of the distances they have traveled must be 120 miles: 40t + 60t = 120. This equation simplifies to 100t = 120, or t = 1.2 hours.
18. A certain bakery sells 240 muffins per day. They are packaged in boxes that hold 12 muffins each. How many boxes are sold per day?
Solution
Correct: B
To find the number of boxes sold per day, we can divide the total number of muffins sold per day by the number of muffins that can fit in one box: 240 muffins / 12 muffins per box = 20 boxes per day.
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