PSAT Math : Counting / Sets

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

A custom-made ruler is 30\ cm long and for every 2\ cm there's a tick mark. How many tick marks are there on the ruler?

Possible Answers:

16

17

18

15

14

Correct answer:

16

Explanation:

There will be 15 gaps of 2\ cm long but 16 tick marks because there will be a tick mark on each end of the ruler.

Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

How many prime numbers are between 1 to 25?

Possible Answers:

Correct answer:

Explanation:

A prime numbers is a number greater than 1 that can only be divided by 1 and itself. Simply count from 1 to 25 and see how many values fit the criteria.

2, 3, 4, 5, 6, 78, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Prime numbers are bolded. Nine prime numbers are in this set interval.

Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

Four consecutive odd integers sum to 40. How many of these numbers are prime?

Possible Answers:

Correct answer:

Explanation:

Let x equal the smallest of the four numbers. Therefore:

\dpi{100} x + (x+2)+(x+4)+(x+6)=40

\dpi{100} 4x +12 = 40

\dpi{100} 4x + 28

\dpi{100} x=7

Therefore the four odd numbers are 7, 9, 11, and 13. Since all are prime except 9, three of the numbers are prime.

Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

Find the median of this set of numbers

Possible Answers:

Correct answer:

Explanation:

To find the median of a set of numbers, we must first order them from smallest to greatest.

Next, we must find the number directly in the middle, which has an equal number of numbers to the right and to the left.

Therefore, our answer is .

Example Question #1 : How To Find The Missing Number In A Set

Which number completes the following series: 1, 2, 4, 8, 16, 32, 64, _?

Possible Answers:

16

15

Not enough information

128

64

Correct answer:

128

Explanation:

All of the numbers in this series are 2n-1. The number that we are looking for is the eighth number. So 28–1 = 27 = 128.

Example Question #1 : How To Find The Missing Number In A Set

Alhough Danielle’s favorite flowers are tulips, she wants at least one each of three different kinds of flowers in her bouquet. Roses are twice as expensive as lilies and lilies are 25% of the price of tulips. If a rose costs $20 and Danielle only has $130, how many tulips can she buy?

Possible Answers:

3

5

4

1

2

Correct answer:

2

Explanation:

She can only buy 2 tulips at $80, because if she bought 3 she wouldn’t have enough to afford the other 2 kinds of flowers.  She has to spend at least 30 dollars (20 + 10) on 1 rose and 1 lily.

Example Question #1921 : Psat Mathematics

Which of the following is not a rational number?

Possible Answers:

5

0.111...

√2

1.75

.001

Correct answer:

√2

Explanation:

A rational number is a number that can be written in the form of a/b, where a and b are integers, aka a real number that can be written as a simple fraction or ratio. 4 of the 5 answer choices can be written as fractions and are thus rational. 

5 = 5/1, 1.75 = 7/4, .001 = 1/1000, 0.111... = 1/9

√2 cannot be written as a fraction because it is irrational. The two most famous irrational numbers are √2 and pi. 

Example Question #1 : How To Find The Missing Number In A Set

Which set represents all the single-digits integers (0-9) that are either prime, a perfect square, or found in the number 68?

Possible Answers:

\dpi{100} \small \left \{ 0,1,2,3,4,5,6,7,8,9 \right \}

\dpi{100} \small \left \{2,3,4,5,6,7,8,9 \right \}

\dpi{100} \small \left \{0,2,3,5,6,7,8 \right \}

\dpi{100} \small \left \{1,2,3,4,5,6,7,8,9 \right \}

\dpi{100} \small \left \{1,3,4,5,7,9 \right \}

Correct answer:

\dpi{100} \small \left \{ 0,1,2,3,4,5,6,7,8,9 \right \}

Explanation:

The prime digits are 2, 3, 5, and 7.

The perfect square digits are 0, 1, 4, and 9.

The only digits not represent in these two groups are 6 and 8, which are, coincidentally, found in the number 68.

Example Question #1 : Counting / Sets

If four different integers are selected, one from each of the following sets, what is the greater sum that these four integers could have?

W = {4, 6, 9, 10}

X = {4, 5, 8, 10} 

Y = {3, 6, 7, 11}

Z = {5, 8, 10, 11}

Possible Answers:

40

41

38

39

42

Correct answer:

38

Explanation:

By observing each of these sets, we can easily determine the largest number in each; however, the problem is asking for us to find the greatest possible sum if we select four different integers, one from each set. The largest number is 11. We will either select this number from set Y or set Z.

We will select set

Y for the 11

because the next larger number in set

Z is 10, which is greater than the next largest number is set Y, 7.

We can use set Z for 10.

Because the 10 has already been selected, choosing the integer from W and X should be easy because 9 is the next largest integer from set W

and 

8 is the next largest integer from set X.

11 + 10 + 9 + 8 = 38

Example Question #1 : Sets

A team has a win-loss ratio of 4:1. If the team wins six games in a row, the win-loss ratio will now be 6:1. How many losses did the team have initially?

Possible Answers:

4

5

2

1

3

Correct answer:

3

Explanation:

If a team's ratio of wins to losses increases from 4:1 to 6:1 by winning six games, they must have 12 wins initially and 3 losses. 

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