### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Venn Diagrams

Let the universal set be the set of all positive integers. Define:

Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?

**Possible Answers:**

None of the other choices is correct.

**Correct answer:**

None of the other choices is correct.

The grayed portion of the Venn diagram corresponds to those integers which are not in any of , , or . Therefore, we eliminate any choices that are in any of the three sets.

is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.

is the set of perfect square integers; we can eliminate 144, since .

is the set of integers which, when divided by 4, yields remainder 2. Since , we can eliminate 154.

All four choices have been eliminated.

### Example Question #2 : Venn Diagrams

In the above Venn diagram, the universal set is defined as . Each of the eight letters is placed in its correct region.

What is ?

**Possible Answers:**

**Correct answer:**

is the union of sets and - that is, the set of all elements of that are elements of either or . We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except . Therefore,

### Example Question #3 : Venn Diagrams

In a school of students, students take Greek, take Old English, and take neither. How many take both?

**Possible Answers:**

No answer is possible.

**Correct answer:**

Based on the information given, you can construct the following Venn Diagram:

In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: . Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that ; therefore, we know that:

Solving for , you get:

### Example Question #4 : How To Use A Venn Diagram

In a group of people, have a laptop and have a tablet. Of those people who have a laptop or a tablet, have both. How many people in the total group have neither a laptop nor a tablet?

**Possible Answers:**

No answer possible

**Correct answer:**

Based on the information given, you can draw the following Venn Diagram:

To solve this, remember that the total number of values in the two circles is:

(We must do this because of the overlap. You need to subtract out one instance of that overlap.)

If we assign the value for the unknown region, we know:

### Example Question #5 : How To Use A Venn Diagram

In a group of plants, are green, have large leaves, and are both green and have large leaves. How many plants are green without large leaves?

**Possible Answers:**

**Correct answer:**

Based on the information, you can draw the following Venn Diagram:

It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely . We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).

### Example Question #6 : How To Use A Venn Diagram

In a group of people, have books, have pens, and have neither books nor pens. How many people in the group have *only* books?

**Possible Answers:**

Cannot be determined

**Correct answer:**

Based on the information given, you can draw the following Venn Diagram:

Now, you must begin by solving for . You know that the two circles together will have in them. This is arrived at by subtracting the people who have neither books nor pens () from the "universe" of people in the sample space (). Now, we know that . This is because of the overlap of in both groups. We have to get rid of one instance of that. Thus we can solve for :

Now, we can find the number of people with *only books* by subtracting from the to get .

### Example Question #7 : How To Use A Venn Diagram

Examine the above Venn diagram. Let be the universal set of the Presidents of the United States. is the set of all Presidents born in Virginia; is the set of all Presidents born after 1850; is the set of all Presidents whose first name was or is James.

James Abram Garfield was born in Ohio in 1831. In which region would he fall?

**Possible Answers:**

I

II

V

IV

III

**Correct answer:**

V

Carter would not fall in set A, since he was not a President born in Virginia.

He would not fall in B, since he was born before 1850.

He would fall in C, since his first name is James.

He would fall in the region included in set C, but *not* A or B - this is Region V.

### Example Question #8 : How To Use A Venn Diagram

Examine the above Venn diagram. Let be the universal set of the Presidents of the United States. is the set of all Presidents born in Virginia; is the set of all Presidents born after 1850; is the set of all Presidents whose first name was or is James.

James Earl Carter was born in Georgia in 1924. In which region would he fall?

**Possible Answers:**

III

IV

I

II

V

**Correct answer:**

III

Carter would not fall in set A, since he was not a President born in Virginia.

He would fall in B, since he was born after 1850.

He would fall in C, since his first name is James.

He would fall in the region included in sets B and C, but *not* A - this is Region III.

### Example Question #9 : How To Use A Venn Diagram

Examine the above Venn diagram. Let universal set represent the set of all words in the English language.

Let be the set of all words whose last letter is a consonant. Let be the set of all words whose first letter is a vowel. Let be the set of all words exactly five letters in length.

Which of the following would be a subset of the set represented by the shaded region in the diagram?

Note: for purposes of this question, "Y" is considered a consonant.

**Possible Answers:**

{usher, aspen, ester, order, earth}

{potato, tomato, breeze, mimosa, magnolia}

{eagle, uvula, apnea, unsee, abide}

{price, value, pinna, trove, three}

{catfish, division, rot, status, giving}

**Correct answer:**

{price, value, pinna, trove, three}

The subset must comprise words that fall inside set , but neither nor .

Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.

The only set among the five choices that matches this description is the set

{price, value, pinna, trove, three}.

### Example Question #10 : How To Use A Venn Diagram

Examine the above Venn diagram. Let universal set represent the set of all words in the English language.

Let be the set of all words whose last letter is a vowel. Let be the set of all words whose first letter is a consonant. Let be the set of all words exactly six letters in length.

Which of the following would be a subset of the set represented by the shaded region in the diagram?

Note: for purposes of this question, "Y" is considered a consonant.

**Possible Answers:**

{plateau, portmanteau, calliope, marionette, taco}

{apnea, esoterica, irradiate, opulence, uvula}

{tomato, potato, ravine, cabana, marine}

{autistic, estrogen, ideology, opal, understand}

{autism, enough, ideals, occult, unduly}

**Correct answer:**

{plateau, portmanteau, calliope, marionette, taco}

The subset must comprise words that fall inside sets and , but *not* . Therefore, all of the words in the subset must begin with a consonant, end with a vowel, and *not* have six letters.

Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.