The questions that follow will give you an idea of the type of mathematical thinking required to solve problems on the SAT. First, try to answer each question yourself, and then read the solutions that follow. These solutions may give you new insights into solving the problems or point out techniques you'll be able to use again. Most problems can be solved in a variety of ways, so don't be concerned if your method is different from the one given. Note that the directions indicate that you are to select the best of the choices given.
On the following pages are seven examples of standard multiple-choice questions.
Directions
For this section, solve each problem and decide which is the best of the choices given. Fill in the corresponding circle on the answer sheet. You may use any available space for scratchwork.
Notes
- The use of a calculator is permitted.
- All numbers used are real numbers.
- Figures that accompany problems in this test are intended to provide information useful in solving problems.
- They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that the figure is not drawn to scale. All figures lie in a plane unless otherwise indicated.
- Unless otherwise specified, the domain of any function ƒ is assumed to be the set of all real numbers x for which ƒ(x) is a real number.
Reference Information
The number of degrees of arc in a circle is 360.
The sum of the measures in degrees of the angles of a triangle is 180.
1. A special lottery is to be held to select the student who will live in the only deluxe room in a dormitory. There are 100 seniors, 150 juniors, and 200 sophomores who applied. Each senior's name is placed in the lottery 3 times; each junior's name, 2 times; and each sophomore's name, 1 time. What is the probability that a senior's name will be chosen?
2.
3. The projected sales volume of a video game cartridge is given by the function 
where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 100,000 cartridges are sold at $10 per cartridge, how many cartridges will be sold at $20 per cartridge?
4.
In the xy-coordinate plane above, line 
contains the points (0,0) and (1,2). If line m (not shown) contains the point (0,0) and is perpendicular to , what is an equation of m?
5.
Note: Figure not drawn to scale.
If two sides of the triangle above have lengths 5 and 6, the perimeter of the triangle could be which of the following?
- 11
- 15
- 24
6.
If 
and 
what is the value of m?
7.
If k is divisible by 2, 3, and 15, which of the following is also divisible by these numbers?
Answers
1.
Correct Answer: D
Explanation:
To determine the probability that a senior's name will be chosen, you must determine the total number of seniors' names that are in the lottery and divide this number by the total number of names in the lottery. Since each senior's name is placed in the lottery 3 times, there are 3 × 100 = 300 seniors' names. Likewise, there are 2 × 150 = 300 juniors' names and 1 × 200 = 200 sophomores' names in the lottery. The probability that a senior's name will be chosen is 
.
2.
Correct Answer: A
Explanation:
To determine the correct order of m, ƒ, and a, it is helpful to first place the seven temperatures in ascending order as shown below.
66 69 70 75 77 78 78
The median temperature is the middle temperature in the ordered list, which is 75, so m = 75. The temperature that occurs most often, or the mode, is 78, so f = 78. To determine the average, you can add the seven numbers together and divide by 7. However, you can determine the relationship between the average and the median by inspection. The three numbers greater than 75 are closer to 75 than are the three numbers smaller than 75. Therefore, the average of the seven numbers will be less than 75. The correct order of m, ƒ, and a is a < m < f.
3.
Correct Answer: C
Explanation:
For 100,000 cartridges sold at $10 per cartridge, s = 100 (since s is the number of cartridges sold, in thousands) and p = 10. Substituting into the equation yields
Solving this equation for a yields
Since a is a constant, the function can be written as 
To determine how many cartridges will be sold at $20 per cartridge, you need to evaluate 
Since s is given in thousands, there will be 60,000 cartridges sold at $20 per cartridge.
4.
Correct Answer: A
Explanation:
Since the coordinates of two points on line are given, the slope of is 
Line m, which is perpendicular to will have a slope of 
since slopes of perpendicular lines are negative reciprocals of each other. The equation of m can be written as 
Since line m also contains point (0,0), it follows that 
Therefore, an equation of line m is 
5.
Correct Answer: B
Explanation:
In questions of this type, statements I, II, and III should each be considered independently of the others. You must determine which of those statements could be true.
- Statement I cannot be true. The perimeter of the triangle cannot be 11, since the sum of the two given sides is 11 without even considering the third side of the triangle.
- Continuing to work the problem, you see that in II, if the perimeter were 15, then the third side of the triangle would be 15 – (6 + 5), or 4. A triangle can have side lengths of 4, 5, and 6. So the perimeter of the triangle could be 15.
- Finally, consider whether it is possible for the triangle to have a perimeter of 24. In this case, the third side of the triangle would be 24 – (6 + 5)= 13. The third side of this triangle cannot be 13, since the sum of the other two sides is not greater than 13. By the Triangle Inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So the correct answer is II only.
6.
Correct Answer: C
Explanation:
Since 
can be written as 
and 
can be written as 
the left side of the equation is 
Since 
the value of m is 
7.
Correct Answer: D
Explanation:
Since k is divisible by 2, 3, and 15, k must be a multiple of 30, as 30 is the least common multiple of 2, 3, and 15. Some multiples of 30 are 0, 30, 60, 90, and 120.
- If you add two multiples of 30, the sum will also be a multiple of 30. For example, 60 and 90 are multiples of 30 and their sum, 150, is also a multiple of 30.
- If you add a multiple of 30 to a number that is not a multiple of 30, the sum will not be a multiple of 30. For example, 60 is a multiple of 30 and 45 is not. Their sum, 105, is not a multiple of 30.
- The question asks which answer choice is divisible by 2, 3, and 15; that is, which answer choice is a multiple of 30. All the answer choices are in the form of "k plus a number." Only choice (D), k + 30, has k added to a multiple of 30. The sum of k and 30 is also a multiple of 30, so the correct answer is choice (D).
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