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ISEE Upper Level Quantitative Reasoning › Sets

Questions 1 - 10

Which of the following can be the next number after 47 in the sequence below?

$\{2, 5, 11, 23, 47...\}\textup{\ ?}$

Explanation

Each number in the set is the value of the preceding number multiplied by 2, plus 1:

Therefore, the next number in the set will be 47 times 2, plus 1. This gives us:

Examine the sequence:

$5, 6, 8, 11, 15, 20, 26, \square, \bigcirc,...$

What number goes into the circle?

Explanation

Each element is obtained by adding a number to the previous one; the number added increases by 1 each time:

- this number replaces the square

- this number replaces the circle

Which of the following can be the next number after 47 in the sequence below?

$\{2, 5, 11, 23, 47...\}\textup{\ ?}$

Explanation

Each number in the set is the value of the preceding number multiplied by 2, plus 1:

Therefore, the next number in the set will be 47 times 2, plus 1. This gives us:

Examine the sequence:

$5, 6, 8, 11, 15, 20, 26, \square, \bigcirc,...$

What number goes into the circle?

Explanation

Each element is obtained by adding a number to the previous one; the number added increases by 1 each time:

- this number replaces the square

- this number replaces the circle

Find the missing part of the list:

Explanation

The sequence pattern follows the rule:
$x_n=3x_{n-1}-1$

Therefore,

Find the missing part of the list:

Explanation

The sequence pattern follows the rule:
$x_n=3x_{n-1}-1$

Therefore,

Given a set $A = \left \{ 1,3, 6, 7, 8\right \}$ , which of the following sets could we assign to such that $A \cup B = \left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ ?

$\left \{ 2, 9,5, 4, 10, 8\right \}$

$\left \{ 6,5,4, 9, 10\right \}$

$\left \{ 2, 4, 9, 10 \right \}$

$\left \{1, 2, 4, 5, 6, 9, 10, 12\right \}$

None of the choices answer the question correctly.

Explanation

$A \cup B$ is the union of the sets, i.e. the set of all elements in , , or both.

For $A \cup B = \left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ , must include all elements in $\left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ that are not in , or $\left \{ 2, 4, 5, 9, 10\right \}$ . This allows us to eliminate $\left \{ 2, 4, 9, 10 \right \}$ , which excludes 5, and $\left \{ 6,5,4, 9, 10\right \}$ , which excludes 2.

Also, cannot include any element not in $\left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ . This allows us to eliminate $\left \{1, 2, 4, 5, 6, 9, 10, 12\right \}$ , which includes 12.

This leaves $\left \{ 2, 9,5, 4, 10, 8\right \}$ .

If $B = \left \{ 2, 9,5, 4, 10, 8\right \}$ , then

$A \cup B =\left \{ 1,3, 6, 7, 8\right \} \cup \left \{ 2, 9,5, 4, 10, 8\right \}$

$= \left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ .

This is the correct choice.

Given a set $A = \left \{ 1,3, 6, 7, 8\right \}$ , which of the following sets could we assign to such that $A \cup B = \left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ ?

$\left \{ 2, 9,5, 4, 10, 8\right \}$

$\left \{ 6,5,4, 9, 10\right \}$

$\left \{ 2, 4, 9, 10 \right \}$

$\left \{1, 2, 4, 5, 6, 9, 10, 12\right \}$

None of the choices answer the question correctly.

Explanation

$A \cup B$ is the union of the sets, i.e. the set of all elements in , , or both.

Also, cannot include any element not in $\left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ . This allows us to eliminate $\left \{1, 2, 4, 5, 6, 9, 10, 12\right \}$ , which includes 12.

This leaves $\left \{ 2, 9,5, 4, 10, 8\right \}$ .

If $B = \left \{ 2, 9,5, 4, 10, 8\right \}$ , then

$A \cup B =\left \{ 1,3, 6, 7, 8\right \} \cup \left \{ 2, 9,5, 4, 10, 8\right \}$

$= \left \{ 1,2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$ .

This is the correct choice.

Nine students are running for student council; each member of the student body will choose four.

The candidates include two of Eileen's sisters, Maureen and Colleen. Eileen wants to vote for one or both of them. How many ways can Eileen fill out her ballot so that one or both of her sisters is among her choices?

Explanation

If Eileen wants to choose both sisters, then she will choose two out of the other seven candidates. This is the number of combinations of two out of seven:

$C (7,2) = \frac{7!}{(7-2)!; 2!} = \frac{7!}{5!; 2!} = \frac{5,040}{120 \times 2} = 21$

If Eileen wants to choose one sister, then she will choose three out of the other eight candidates. This is the number of combinations of three out of eight:

$C (8,3) = \frac{8!}{(8-3)!; 3!} = \frac{8!}{5!; 3!} = \frac{40,320}{120 \times 6} = 56$

Since there are also two ways Eileen can choose one sister, by the multiplication principle, she can fill out the ballot $56\times 2 = 112$ ways that include one sister.

Add these two:

Nine students are running for student council; each member of the student body will choose four.

Explanation

If Eileen wants to choose both sisters, then she will choose two out of the other seven candidates. This is the number of combinations of two out of seven:

$C (7,2) = \frac{7!}{(7-2)!; 2!} = \frac{7!}{5!; 2!} = \frac{5,040}{120 \times 2} = 21$

If Eileen wants to choose one sister, then she will choose three out of the other eight candidates. This is the number of combinations of three out of eight:

$C (8,3) = \frac{8!}{(8-3)!; 3!} = \frac{8!}{5!; 3!} = \frac{40,320}{120 \times 6} = 56$

Since there are also two ways Eileen can choose one sister, by the multiplication principle, she can fill out the ballot $56\times 2 = 112$ ways that include one sister.

Add these two:

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