### All SSAT Middle Level Math Resources

## Example Questions

### Example Question #1 : Sets

Define two sets as follows:

Which of the following is a subset of ?

**Possible Answers:**

Each of the sets listed is a subset of .

**Correct answer:**

is the union of and - that is, it is the set of all elements in one set or the other.

A set is a subset of if and only if every one of its elements is in . Three of the listed sets do not meet this criterion:

, , and , but none of those three elements are in . All of the elements in do appear in , however, so it is the subset.

### Example Question #1 : Sets

Define two sets as follows:

Which of the following numbers is an element of ?

**Possible Answers:**

**Correct answer:**

is the intersection of and - the set of all elements appearing in *both *sets. Thus, an element can be eliminated from by demonstrating either that it is not an element of or that it is not an element of .

is the set of positive integers ending in "5". 513 and 657 are not in , so they are not in .

is the set of muliples of 9. We test the three remaining numbers easily by seeing if 9 divides their digit sum:

425 and 565 are not multiples of 9; neither is in , so neither is in .

and , so . This is the correct choice.

### Example Question #1 : How To Find The Missing Part Of A List

Complete the set by determining the value of .

**Possible Answers:**

**Correct answer:**

The set is composed of consecutive squares.

We can see that will b equal to

Therefore, 36 is the correct answer.

### Example Question #4 : Sets

Define sets and as follows:

How many elements are in the set ?

**Possible Answers:**

Four

Three

One

Two

The correct answer is not given among the other responses.

**Correct answer:**

The correct answer is not given among the other responses.

The elements of the set - that is, the intersection of and - are exactly those in both sets. We can test each of the six elements in for inclusion in set by dividing each by 7 and noting which divisions yield no remainder:

and have no elements in common, so has zero elements. This is not one of the choices.

### Example Question #1 : Sets

Which of the following is a subset of the set

?

**Possible Answers:**

**Correct answer:**

For a set to be a subset of , all of its elements must be elements of - that is, all of its elements must be multiples of 3. A set can therefore be proved to not be a subset of by identifying one element not a multiple of 3.

We can do that with four choices:

:

:

:

:

However, the remaining set, , can be demonstrated to include only multiples of 3:

is the correct choice.

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

Define sets and as follows:

How many elements are in the set ?

**Possible Answers:**

Three

None

Two

One

The correct answer is not given among the other responses.

**Correct answer:**

Three

The elements of the set - that is, the intersection of and - are exactly those in both sets. We can test each of the six elements in for inclusion in set by testing for divisibility by 5 - but this can be accomplished by looking at the last digit. Only 345, 600, and 855 have last digit 5 or 0 so only these three elements are divisible by 5. This makes three the correct answer.

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

Which of the following is a subset of the set

?

**Possible Answers:**

The correct answer is not among the answer choices.

**Correct answer:**

The correct answer is not among the answer choices.

We show that none of the four listed sets can be a subset of the primes by identifying one composite number in each - that is, by proving that there is at least one factor not equal to 1 or itself:

, so 25 has 5 as a factor, and 25 is not prime.

, so 9 has 3 as a factor, and 9 is not prime.

, so 21 has 3 and 7 as factors, and 21 is not prime.

, so 21 has 3 and 9 as factors, and 27 is not prime.

Since each set has at least one element that is not a prime, each has at least one element not in , and none of the sets are subsets of .

### Example Question #1 : How To Find The Missing Part Of A List

How many of the following four numbers are elements of the set

?

(A)

(B)

(C)

(D)

**Possible Answers:**

Three

One

Two

None

Four

**Correct answer:**

Three

By dividing the numerator of each fraction by its denominator, each fraction can be rewritten as its decimal equivalent:

All fractions except can be seen to fall between 0.3 and 0.4, exclusive. Three is the correct answer.

Note that is *equal* to 0.4, so we don't include it. The criterion requires *strict *inequality.

### Example Question #1 : Sets

Define .

How many of the four sets listed are subsets of the set ?

(A)

(B)

(C)

(D)

**Possible Answers:**

Three

Two

None

One

Four

**Correct answer:**

Two

For a set to be a subset of , all of its elements must also be elements of - that is, all of its elements must be multiples of 5. An integer is a multiple of 5 if and only if its last digit is 5 or 0, so all we have to do is examine the last digit of each number in all four sets.

In the sets and , every element ends in a 5 or a 0, so all elements of both sets are in ; both sets are subsets of .

However, includes one element that does not end in either 5 or 0, namely 8934, so 8934 is not an element in ; subsequently, this set is not a subset of . Similarly, is not a subset of , since it includes 7472, which ends in neither 0 nor 5.

The correct answer is therefore two.

### Example Question #1 : How To Find The Missing Part Of A List

What are the next two numbers of this sequence?

**Possible Answers:**

**Correct answer:**

The sequence is formed by alternately adding and adding to each term to get the next term.

and are the next two numbers.

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