### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #11 : Sets

An arithmetic sequence begins as follows:

'

Which is the greater quantity?

(a) The fifth number in the sequence

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

**Correct answer:**

(b) is greater.

The common difference of the sequence is

.

The fifth number in the sequence is

.

This makes (b) greater.

### Example Question #171 : Data Analysis

A geometric sequence begins as follows:

Which is the greater quantity?

(a) The fifth term of the sequence

(b)

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

**Correct answer:**

(a) is greater.

The common ratio of the sequence is

.

The next three terms of the sequence are:

, the fifth term.

, making (a) greater.

### Example Question #20 : How To Find The Missing Part Of A List

A geometric sequence begins as follows:

Which is the greater quantity?

(a) The nine hundred ninety-ninth term of the sequence

(b) The one-thousandth term of the sequence

**Possible Answers:**

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

The common ratio is , which means that the terms will alternate in sign. The first term is negative; therefore, all of the odd-numbered terms will be negative and all of the even-numbered terms will be positive. This makes the one thousandth term greater than the nine hundred ninety-ninth.

### Example Question #21 : Sets

The Fibonacci sequence begins

with each subsequent term being the sum of the previous two.

Which is the greater quantity?

(a) The product of the seventh and eighth terms of the Fibonacci sequence

(b) The product of the sixth and ninth terms of the Fibonacci sequence

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

**Correct answer:**

(a) is greater.

By starting with and adding each pair of consecutive terms to get each subsequent term, we can generate the first nine terms of the Fibonacci sequence:

(a) The seventh and eighth terms are 13 and 21; their product is .

(b) The sixth and ninth terms are 8 and 34; their product is .

(a) is greater.

### Example Question #172 : Data Analysis

The first six terms of a sequence are as follows:

Which is the greater quantity?

(a) The eighth term of the sequence

(b)

**Possible Answers:**

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

**Correct answer:**

(a) is greater.

Each successive term is derived by adding a quantity to the previous term that increments by 1 with each term:

, the eighth term, is greater than .

### Example Question #23 : Sets

Seven freshmen and seven sophomores are running for student council. Each freshman will choose three of the candidates from his or her class; each sophomore will choose four from his or her class.

Which is the greater quantity?

(a) The number of ways a freshman can fill out his or her ballot

(b) The number of ways a sophomore can fill out his or her ballot

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

In both cases, candidates are selected without regard to order, making them combinations (as opposed to permutations).

(a) The freshmen are choosing three elements from a group of seven, or :

(b) The sophomores are choosing four elements from a group of seven, or :

These quantities are equal.

### Example Question #173 : Data Analysis

An arithmetic sequence begins as follows:

The Fibonacci sequence begins

,

with each subsequent term being the sum of the previous two.

Which is the greater quantity?

(a) The tenth term of the arithmetic sequence

(b) The tenth term of the Fibonacci sequence

**Possible Answers:**

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

**Correct answer:**

(a) and (b) are equal.

(a) The arithmetic sequence has common difference ; its tenth term is .

(b) By beginning with and taking the sum of the previous two terms to get each successive term, we can generate the Fibonacci sequence:

Therefore the two are equal.

### Example Question #25 : Sets

Twelve candidates are running for four vacancies on the school board. Six of the candidates are Democrats, and six are Republicans. Samantha wants to vote for two Democrats and two Republicans.

Which is the greater quantity?

(a) The number of ways Samantha can cast her ballot so that she votes for two Democrats and two Republicans

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

The number of ways Samantha can choose two Democrats out of the six is the number of combinations of 2 chosen from a set of 6:

This is also the number of ways Samantha can choose two Republicans out of the six.

These are two independent events, so we multiply them to get the number of total ways Samantha can vote with these specifications:

Therefore (b) is greater.

### Example Question #26 : Sets

The Fibonacci sequence begins

,

with each subsequent term being the sum of the previous two.

Which is the greater quantity?

(a) The sum of the seventh, eighth, ninth, and tenth terms of the Fibonacci sequence

(b) The twelfth term of the Fibonacci sequence

**Possible Answers:**

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

By beginning with and taking the sum of the previous two terms to get each successive term, we can generate the Fibonacci sequence:

The seventh through tenth terms, added together, yield a sum of

,

which is less than the twelfth term, .

### Example Question #27 : Sets

The Student Council comprises thirty students, from which four are chosen to fill the offices of President, Vice-President, Secretary, and Treasurer. Which is the greater quantity?

(a) The number of ways the four offices can be filled

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

(a) There are 30 ways to fill the office of President. Afterward, there will be 29 ways to fill the office of Vice-President, 28 ways to fill the office of Secretary, and 27 ways to fill the office of Treasurer. Multiply:

(b)

This makes (b) greater.

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