Sets

Arithmetic: Sets

Read the following SAT test question and then select the correct answer.

Approach each math question on the SAT the same way.  Read the question carefully to be sure you take into account all of the information as you solve it, and be sure to identify and note the bottom line.  Assess your options for solving the problem, and then choose the most efficient method to attack the problem.  Never forget to loop back and make sure that your final answer solves for the bottom line, the question that you were asked.

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

Bottom Line: # of intersections = ?

Assess your Options:  When you have a question that asks about number properties, ignore your answer choices!  If you look down and see a 0, you could think to yourself that both 7 and 13 are prime, so they have nothing in common.  Are you looking for factors?  No!  You are looking for multiples.  Think through all of the information that you are given before looking at the answer choices.

Attack the Problem:  A set is just a collection of data.  You are given two different sets and asked to find the intersections, the data that the two have in common.  The only restriction on both sets is that all of the numbers must be positive.

Now think about what multiples are.  Multiples are the product of a number and an integer.  So Set S contains 7, 14, 21, 28… and continues in this manner into infinity.  Set T contains 13, 26, 39, 52… and continues in this manner into infinity.

If you keep listing numbers in each set, it will take you forever to find the answer to this problem.  Instead, think logically about where you know you must have multiples that match.  For example, if you multiply 7 times 13, you will find a number that belongs in both sets.  If you multiply 14 times 13, you will find another intersection.  Notice that you can keep doing this because you will never reach infinity.  The answer to this problem is that there are an infinite number of intersections between S and T.

Loop Back:  You found your bottom line, so look down and see which answer choice it matches.

(A) None
(B) One
(C) Seven
(D) Thirteen
(E) More than thirteen

The correct answer is (E).

On sat.collegeboard.org, 40% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Probability

Data Analysis: Probability

Read the following SAT test question and then select the correct answer.

Read each question carefully to avoid making any mistakes. Identify the bottom line (what the question is asking) and assess your options for reaching it by asking yourself “What could I do?” and “What should I do?” Choose the most efficient method to attack the problem and find an answer. Last, loop back to make sure that your answer addresses the bottom line.

If a number is chosen at random from the set {-10, -5, 0, 5, 10}, what is the probability that it is a member of the solution set of both 3x – 2 <10 and x + 2?

Bottom Line: Prb = ?

Assess Your Options: You cannot solve for a probability until you know whether each number in the set meets the requirements that you are given. You could plug numbers from the set into each inequality and see if they work, but it is much faster to simplify the inequalities before you begin working with them.

Attack the Problem: Simplify the inequalities by solving both for x.

3x – 2 < 10
3x < 12
x < 4

x + 2 > -8
x > -10

You now know that x must be less than 4, but greater than -10. The question asked you to find a number that fits both of these solution sets. Look at the original set that you were given. The only two answers that are between -10 and 4 are -5 and 0 (-10 does not work because it cannot be equal to negative -10; it has to be greater than -10). You found 2 numbers out of 5 that you were given that work. To write this as a probability, you must set the number of relevant outcomes over the number of total possible outcomes. Your answer is .

Loop Back: You found a probability matching the restrictions you were given. Look down at your answer choices.

(A) 0

(B) (C) (D) (E) The correct answer is (C).

On sat.collegeboard.org, 50% of the responses were correct.

For more help with SAT math, visitwww.myknowsys.com!

Sets

Read the following SAT question and then select the correct answer.

As always, remember to follow the Knowsys method for math. Read the problem carefully and identify the bottom line (what you are looking for). Then, consider your options. How could you solve it? How should you solve it? Next, attack the problem using the method that you selected. Finally, loop back and verify that your answer matches the bottom line.

At Central High School, the math club has 15 members and the chess club has 12 members. If a total of 13 students belong to only one of the two clubs, how many students belong to both clubs?

This problem involves sets. The easiest way to solve this problem is to look at the information that you are given and make deductions. By working step by step from the information given, you can find the answer. First of all, you know that there are a total of 15+12=27 memberships. Notice that this isn't the number of students since some students belong to both the chess and math clubs. This is simply the number of memberships. Secondly, you know that 13 students belong to only one club. 27 (the total number of memberships of both clubs) minus 13 (the number of students in only one club) is 14 (this is the number of memberships that come from students who are in both clubs). Since 14 is the number of memberships that come from students who are in both clubs, there must only be 7 students who are members of both clubs (1 student=2 memberships). Now, look at the answers and see which one matches your prediction.

(A) 2
(B) 6
(C) 7
(D) 12
(E) 14

(C) matches your prediction exactly. Interestingly, you can see that (E) is a trap answer. If you had failed to divide 14 by 2 (to account for the fact that the 14 memberships came from students who were in both clubs), you would have chosen (E). Remember to take your time and work the problem carefully. If you work a problem and make a sloppy mistake, you will not only lose a chance to earn a point, but you will also be penalized a quarter of a point (and you will waste valuable time on that problem).

The correct Answer Choice is (C).

On sat.collegeboard.com 37% of the responses were correct.

For more help with math, visit www.myknowsys.com.

Sets

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Mathematics: Standard Multiple Choice

Read the following SAT test question and then select your answer.

Always follow the Knowsys Method to save time and energy on math questions: read carefully, identify the bottom line, assess your options, attack the problem, and loop back to double-check your answer. This will help you find the correct answers more quickly.

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

First, read carefully. Notice that S includes all multiples of 7 and T includes all multiples of 13. Next, identify the bottom line and note it at the top of your scratch work.

Intersection of S and T = ?

You are looking for the intersection, so you need only the numbers that are in both set S and set T. Since neither set has an upper limit, they both have an infinite number of members; therefore, their intersection also has an infinite number of members. Look at the answer choices.

A) None

B) One

C) Seven

D) Thirteen

E) More than thirteen

The answer is E.