### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Sets

A class has 25 students. If 60% of them are boys, how many students are girls?

**Possible Answers:**

10

15

12

18

**Correct answer:**

10

If 60% of the students are boys, 40% are girls (100 – 60 = 40). Multiply 25 by 40% (25 * 0.4 = 10); therefore, 10 students are girls.

### Example Question #2 : How To Find The Missing Part Of A List

We can divide the natural numbers into four sets:

In which of these sets would 197 be a member?

**Possible Answers:**

It cannot be determined from the information given.

**Correct answer:**

The sets are divided according to the remainder obtained when each element is divided by 4. 197 divided by 4 yields a remainder of 1; all of the elements of match this description, so .

### Example Question #3 : How To Find The Missing Part Of A List

Define the universal set, , as follows:

Also, let

and

Which of these sets represents the *complement* of ?

**Possible Answers:**

**Correct answer:**

, the union of and is the set of all elements in , , or both. Merge the two sets to get:

The complement of a set is the set of all elements in the universal set that are *not* in the set. The only elements in not in are 4 and 8, so

### Example Question #4 : How To Find The Missing Part Of A List

Which of the following is an example of two sets and such that ?

**Possible Answers:**

**Correct answer:**

refers to the empty set, the set with no elements; if and only if the two sets have no elements in common. In four of these cases, and share an element, which in each of these four choices is underlined:

and do not have an element in common, so this is the right choice.

### Example Question #1 : Sets

Define set . How could we define set so that ?

**Possible Answers:**

**Correct answer:**

is the set of all elements in both and .

We can test each set and determine which elements are shared by that set and :

If :

then

If :

then

If :

then

If :

then

If :

then

This is the correct choice.

### Example Question #2 : Sets

A pair of fair dice are rolled. Which is the greater quantity?

(a) The probability that at least one die comes up 5 or 6

(b)

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given.

(a) and (b) are equal

(b) is greater

**Correct answer:**

(a) is greater

For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are ways out of 36 that this can happen, so there are ways for either or both of the two dice to be 5 or 6. Since half of 36 is 18, the probability of this event is greater than .

### Example Question #3 : Sets

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the product of the two dice will be an even number

(b)

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

**Correct answer:**

(a) is greater

The product of the two dice will be an odd number only if both dice are odd; the rolls favorable to that event are:

,

or nine out of the thirty-six possible rolls. This makes twenty-seven of the thirty-six equally probable rolls favorable to getting an even product. Since half of 36 is 18, the probability of getting an even product is greater than .

This makes (a) the greater quantity

### Example Question #8 : How To Find The Missing Part Of A List

Which quantity is greater?

(a) The number of even integers that satisfy the inequality

(b) The number of multiples of 4 that satisfy the inequality

**Possible Answers:**

(b) is greater

(a) is greater

It is impossible to tell from the information given.

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

The easiest way to answer this is to try to match each element in the first set to one the second set as follows:

...

In other words, each element in the set in (a) is paired with the element in the set in (b) that is its double. Since there is a one-to-one correspondence, the two sets are of equal aize, and (a) and (b) are equal quantities.

### Example Question #9 : How To Find The Missing Part Of A List

Which quantity is greater?

(a) The number of even integers that satisfy the inequality

(b) The number of odd integers that satisfy the inequality

**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.

The easiest way to answer this is to try to match each element in the first set to one in the second set as follows:

...

Since there is a one-to-one correspondence between the elements of the two sets, (a) and (b) are equal.

### Example Question #3 : How To Find The Missing Part Of A List

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the product of the numbers will be at least .

(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.

Out of a possible thirty-six rolls, the following result in a product of or greater:

This is ten equally probable rolls out of thirty-six, resulting in a probability of

.

Since , (b) is the greater quantity.

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