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Questions and Answers
1.
If x + k = 12 and p(x + k) = 36, what is the value of p ?
A.
3
B.
4
C.
6
D.
9
E.
12
Correct Answer A. 3
Explanation If x + k = 12, we can substitute this value into the second equation. Therefore, p(12) = 36. To find the value of p, we need to divide 36 by 12, which gives us p = 3.
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2.
If 13 is added to one-half of a certain number the result is 37. What is the original number?
A.
24
B.
40
C.
48
D.
61
E.
80
Correct Answer C. 48
Explanation If 13 is added to one-half of a certain number, the result is 37. To find the original number, we can set up the equation: (1/2)x + 13 = 37. Subtracting 13 from both sides gives (1/2)x = 24. Multiplying both sides by 2 gives x = 48. Therefore, the original number is 48.
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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?
A.
4
B.
8
C.
10
D.
12
E.
18
Correct Answer B. 8
Explanation The solid line represents the usual route from Town A to Town D, while the broken line represents the detour route from B to C through E. To find the difference in miles between the two routes, we need to calculate the total distance for each route.
For the usual route, the total distance is the sum of the lengths of the line segments between A, B, C, and D, which is 4 + 6 + 2 + 4 = 16 miles.
For the detour route, the total distance is the sum of the lengths of the line segments between A, B, E, C, and D, which is 4 + 2 + 4 + 2 + 2 = 14 miles.
Therefore, the detour route is 2 miles shorter than the usual route.
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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?
A.
Y = 5x + 7.5
B.
Y = 5.5x + 2
C.
Y = 5.5x + 7.5
D.
Y = 7.5x
E.
Y = 7.5x + 5.5
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 is because the equation has a constant term of 2, which suggests that when x is 0, y will be 2. Additionally, the coefficient of x is 5.5, indicating that for every increase of 1 in x, y will increase by 5.5. Therefore, this equation accurately represents the relationship between x and y in the given table.
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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?
A.
30
B.
35
C.
40
D.
50
E.
55
Correct Answer A. 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
A.
I only
B.
I and II only
C.
I and III only
D.
II and III only
E.
I, II, and III
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 number of numbers in the list. Since the median is 6, it means that there are 3 numbers less than 6 and 3 numbers greater than 6. The least number in the list is 2, and the greatest number is 20. Since the number 3 occurs most often, it means that there are at least 2 occurrences of 3 in the list. Therefore, it is possible to have an average of 7, 8.5, or 10 for the numbers in the list.
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7.
In the XY-coordinate plane, how many points are a distance of 4 units from the origin?
A.
One
B.
Two
C.
Three
D.
Four
E.
More than four
Correct Answer E. More than four
Explanation In the XY-coordinate plane, the distance from the origin to any point (x, y) is given by the formula âˆš(x^2 + y^2). For a point to be a distance of 4 units from the origin, we need to find all the points (x, y) that satisfy the equation âˆš(x^2 + y^2) = 4. This equation represents a circle with radius 4 centered at the origin. A circle has an infinite number of points, so the answer is "more than four".
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8.
A rectangle has a perimeter of 100 units. If the length of the rectangle is twice its width, what is the area of the rectangle?
A.
444.46 square units
B.
555.56 square units
C.
625.25 square units
D.
800.25 square units
Correct Answer B. 555.56 square units
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 If a cake is cut into thirds, it means it is divided into 3 equal parts. Then, each of these thirds is cut into fourths, which means each third is divided into 4 equal parts. Therefore, there will be a total of 3 x 4 = 12 pieces of cake.
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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 relationship where y is inversely proportional to x. In this case, we are given that y = 3 when x = 4. To find the value of y when x = 6, we can use the given relationship and substitute the values. Plugging in x = 6 into the equation y = h/x, we get y = h/6. Since we know that y = 3 when x = 4, we can solve for h by substituting these values into the equation. 3 = h/4, which gives us h = 12. Now, we can substitute h = 12 and x = 6 into the equation y = h/x to find the value of y. y = 12/6 = 2. Therefore, when x = 6, y equals 2.
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11.
In the figure above, point B lies on side AC. If 55 < x < 60, what is one possible value of y ?
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 Since the price of the item has been increasing by $2 per year since 1990, we can calculate the number of years it will take for the price to reach $100 by dividing the difference between $100 and $10 by $2. The difference is $90, and dividing by $2 gives us 45. Therefore, it will take 45 years for the price to reach $100. Adding this to the starting year of 1990, we get the year 2035.
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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 The value of y is greatest when x = 5. This can be observed from the graph, where the vertex of the quadratic function is located at x = 5. At the vertex, the value of y is maximized before it starts decreasing again. Therefore, the value of y is greatest when x = 5.
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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 a number is divided by 10, the remainder is the last digit of the number. In this case, the remainder is 9, which means the last digit of the number is 9. When the number is divided by 9, the remainder is 8, which means the sum of the digits of the number is 8. The only 2-digit number that satisfies both conditions is 89, where the last digit is 9 and the sum of the digits is 8. Therefore, the value of n is 89.
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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 from the set {4, 5, 6} and k is chosen at random from the set {10, 11, 12}, what is the probability that the product of j and k is divisible by 5?
A.
2/3
B.
5/9
C.
7/9
D.
11/9
E.
Option 5
Correct Answer A. 2/3
Explanation To find the probability that the product of j and k is divisible by 5, we need to consider all the possible combinations of j and k and count the favorable outcomes.
The sets for j and k are as follows:
For j: {4, 5, 6} For k: {10, 11, 12}
Now, we can calculate the product of j and k for all possible combinations:
For j = 4:
4 * 10 = 40
4 * 11 = 44
4 * 12 = 48
For j = 5:
5 * 10 = 50
5 * 11 = 55
5 * 12 = 60
For j = 6:
6 * 10 = 60
6 * 11 = 66
6 * 12 = 72
Now, let's count the combinations where the product is divisible by 5:
40, 50, 60, 55, 60, 72
There are a total of 6 combinations where the product is divisible by 5.
Now, let's calculate the total number of possible combinations. Since there are 3 choices for j and 3 choices for k, there are a total of 3 * 3 = 9 possible combinations.
So, the probability that the product of j and k is divisible by 5 is:
(Number of Favorable Outcomes) / (Total Number of Possible Combinations) = 6/9 = 2/3
Therefore, the probability is 2/3.
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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?
Correct Answer 700
Explanation Both Tom and Alison had the same amount of sales and the same compensation for a particular week. Tom's compensation consists of $300 plus 20 percent of his sales, while Alison's compensation consists of $200 plus 25 percent of her sales. Since they both had the same compensation, it means that the additional amount they earned based on their sales was the same. Therefore, the additional amount for both of them must be $400 (25% of sales - 20% of sales = 5% of sales = $400). Adding this additional amount to their base compensation, the total compensation for both Tom and Alison is $700.
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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
Explanation Since the equation represents a line in the XY-plane, we can rewrite it in slope-intercept form (y = mx + b) by solving for y. Rearranging the equation, we have 12y = -tx - 3, which simplifies to y = (-t/12)x - 3/12. Comparing this with the slope-intercept form, we can see that the slope (m) is equal to -t/12. Given that the slope is -10, we can set up the equation -10 = -t/12 and solve for t. Multiplying both sides by 12, we get -120 = -t, which implies that t = 120.
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