1.
If x + k = 12 and p(x + k) = 36, what is the value of p ?
Correct Answer
A. 3
Explanation
To find the value of p, we can substitute the given equation x + k = 12 into p(x + k) = 36. This gives us p(12) = 36. Since p(12) is equal to 36, we can divide 36 by 12 to find the value of p. 36 divided by 12 equals 3. Therefore, the value of p is 3.
2.
If 13 is added to one-half of certain number the result is 37. What is the original number?
Correct Answer
C. 48
Explanation
If we let the original number be x, we can set up the equation (1/2)x + 13 = 37. To solve for x, we first subtract 13 from both sides to get (1/2)x = 24. Then, we multiply both sides by 2 to get x = 48. Therefore, the original number is 48.
3.
In the figure above, the usual route from Town A to Town D is indicated by the solid line. The broken line indicates a detour route from B to C through E . Each line segment is labeled with its length in miles. How many more miles is the trip from Town A to Town D via the detour than via the usual route?
Correct Answer
B. 8
Explanation
The usual route from Town A to Town D is indicated by the solid line, which has a total length of 10 miles. The detour route from B to C through E is indicated by the broken line, which has a total length of 18 miles. Therefore, the trip from Town A to Town D via the detour is 8 miles longer than via the usual route.
4.
Which of the following equations expresses y in terms of x for each of the four pairs of values shown in the table above?
Correct Answer
B. Y = 5.5x + 2
Explanation
The equation y = 5.5x + 2 expresses y in terms of x for each of the four pairs of values shown in the table above. This equation represents a linear relationship between x and y, where the coefficient of x is 5.5 and the constant term is 2. This means that for every increase of 1 in x, y will increase by 5.5, and the initial value of y when x is 0 is 2.
5.
In the figure above, point B lies on AC. If x and y are integers, which of the following is a possible value of x ?
Correct Answer
A. 30
Explanation
In the figure, point B lies on AC. Since x and y are integers, the possible values of x would be the integers between the coordinates of A and C. Looking at the figure, point A is at 30 and point C is at 55. Therefore, a possible value for x would be 30.
6.
The least and greatest numbers in a list of 7 real numbers are 2 and 20, respectively. The median of the list is 6, and the number 3 occurs most often in the list. Which of the following could be the average (arithmetic mean) of the numbers in the list?
I.7
II. 8.5
III. 10
Correct Answer
E. I, II, and III
Explanation
The average (arithmetic mean) of a list of numbers is equal to the sum of all the numbers divided by the total count of numbers. Since the median of the list is 6 and the number 3 occurs most often, it means that there are at least three numbers less than 6 and at least three numbers greater than 6 in the list. This indicates that the sum of the numbers in the list must be greater than 6 * 7 = 42. Therefore, the average of the numbers in the list can be any number greater than 42/7 = 6. The options given, I (7), II (8.5), and III (10), all satisfy this condition, so they could all be the average of the numbers in the list.
7.
In the xy-coordinate plane, how many points are a distance of 4 units from the origin?
Correct Answer
E. More than four
Explanation
In the xy-coordinate plane, the distance from the origin to any point can be found using the distance formula: âˆš(x^2 + y^2), where (x, y) represents the coordinates of the point. For a point to be a distance of 4 units from the origin, the equation becomes âˆš(x^2 + y^2) = 4. This equation represents a circle with a radius of 4 units centered at the origin. A circle has an infinite number of points, so the correct answer is "more than four".
8.
The table above shows the number of consecutive nights that each of five families stayed at a certain hotel during a 14-night period. If the Liu family’s stay did not overlap with the Benton family’s stay, which of the 14 nights could be a night on which only one of the five families stayed at the hotel?
Correct Answer
A. The 3rd
Explanation
The Liu family's stay did not overlap with the Benton family's stay. Looking at the table, we can see that the Benton family stayed for 4 consecutive nights from the 1st to the 4th night. Since the Liu family's stay did not overlap with this, it means that the Liu family could not have stayed on any of these nights. Therefore, the 3rd night could be a night on which only one of the five families stayed at the hotel.
9.
If a cake is cut into thirds and each third is cut into fourths, how many pieces of cake are there?
Correct Answer
12
Explanation
When a cake is cut into thirds, it is divided into 3 equal parts. Each of these thirds is then cut into fourths, resulting in each third being divided into 4 equal parts. Therefore, there are a total of 3 x 4 = 12 pieces of cake.
10.
If y= h/x, where h is a constant, and if y = 3 when x = 4, what does y equal when x = 6?
Correct Answer
2
Explanation
The equation y = h/x represents a inverse relationship between y and x, where h is a constant. Given that y = 3 when x = 4, we can substitute these values into the equation to solve for h: 3 = h/4. Solving for h, we find that h = 12. Now, we can use this value of h to find y when x = 6: y = 12/6 = 2. Therefore, when x = 6, y equals 2.
11.
In the figure above, point B lies on side AC. If 55 < x < 60, what is one possible value of y ?
Correct Answer
120 < x < 125
120
121
122
125
12.
The price of a certain item was $10 in 1990 and it has gone up by $2 per year since 1990. If this trend continues, in what year will the price be $100 ?
Correct Answer
2035
Explanation
The price of the item has been increasing by $2 per year since 1990. To find out in what year the price will be $100, we need to determine how many times the price has increased by $2. We can do this by dividing the difference between $100 and $10 (which is $90) by $2. This gives us 45. Adding 45 to 1990 gives us the year 2035, which is when the price will be $100.
13.
The figure above shows the graph of a quadratic function in the xy-plane. Of all the points (x, y) on the graph, for what value of x is the value of y greatest?
Correct Answer
5
Explanation
In a quadratic function, the vertex represents the highest or lowest point on the graph. In this case, the vertex is located at the point (5, y), where the value of y is the greatest. Therefore, the value of x for which the value of y is greatest is 5.
14.
The number n is a 2-digit number. When n is divided by 10, the remainder is 9, and when n is divided by 9, the remainder is 8. What is the value of n ?
Correct Answer
89
Explanation
When n is divided by 10, the remainder is 9, which means that n is 1 less than a multiple of 10. Similarly, when n is divided by 9, the remainder is 8, which means that n is 1 less than a multiple of 9. The only number that satisfies both conditions is 89.
15.
The area of the figure above is 9/4. What is the perimeter of the figure?
Correct Answer
13/2
6.5
16.
If j is chosen at random form the set {4, 5, 6} and k is chosen at random form the set {10, 11, 12}, what is the probability that the product of j and k is divisible by 5?
Correct Answer
5/9
.555
.556
Explanation
To find the probability that the product of j and k is divisible by 5, we need to determine the number of favorable outcomes and the total number of possible outcomes.
The favorable outcomes are when either j or k is 5 or a multiple of 5. From the set {4, 5, 6}, only 5 is divisible by 5. From the set {10, 11, 12}, both 10 and 12 are divisible by 5. So, there are 3 favorable outcomes.
The total number of possible outcomes is the product of the number of elements in each set, which is 3 x 3 = 9.
Therefore, the probability is given by favorable outcomes divided by total outcomes, which is 3/9 = 1/3 = 0.333.
Hence, the correct answer is 1/3, which is approximately equal to 0.333.
17.
Tom and Alison are both salespeople. Tom’s weekly compensation consists of $300 plus 20 percent of his sales. Alison’s weekly compensation consists of $200 plus 25 percent of her sales. If they both had the same amount of sales and the same compensation for a particular week, what was that compensation, in dollars? (Disregard the dollar sign when gridding your answer.)
Correct Answer
700
Explanation
Since both Tom and Alison had the same amount of sales and the same compensation for a particular week, we can set up an equation to find the compensation. Let's say the amount of sales is x. Tom's compensation would be $300 + 20% of x, and Alison's compensation would be $200 + 25% of x. Since their compensations are the same, we can set up the equation 300 + 0.2x = 200 + 0.25x. Solving this equation, we find that x = 2000. Plugging this value back into either Tom or Alison's compensation equation, we get $700 as the compensation for that week.
18.
tx + 12y = -3
The equation above is the equation of a line in the xy-plane, and t is a constant.
If the slope of the line is -10, what is the value of t?
Correct Answer
120