# ISEE Middle Level Quantitative : Numbers and Operations

## Example Questions

### Example Question #11 : How To Divide Fractions

is a positive number, and . Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

Possible Answers:

(a) is the greater quantity

It is impossible to determine which quantity is the greater from the information given

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

.

The reciprocal of  is therefore  - that is,  times the reciprocal of . The reciprocal of a positive number must be positive, and , so

The reciprocal of  is the greater number.

### Example Question #671 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is a positive number. Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

Possible Answers:

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

We examine two scenarios to demonstrate that the given information is insufficient.

Case 1:

The reciprocal of this is .

The reciprocal of this is

, so  has the greater reciprocal.

Case 2: .

The reciprocal of this is .

The reciprocal of this is .

, so  has the greater reciprocal.

### Example Question #12 : How To Divide Fractions

is a negative number. Which is the greater quantity?

(a) The reciprocal of

(b) The reciprocal of

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

Correct answer:

(b) is the greater quantity

Explanation:

is negative, so we will let , making  positive. It follows that .

The reciprocal of  is ;

the reciprocal of  is .

and  are both positive; , so , and . Therefore, the reciprocal of  is the greater quantity.

### Example Question #1 : How To Add Fractions

Express the sum as a fraction in lowest terms:

Possible Answers:

Correct answer:

Explanation:

Rewrite the fractions in terms of their least common denominator, 12.

Add, then rewrite as a mixed fraction:

Add the integers:

Now add the sums:

### Example Question #1 : How To Add Fractions

Possible Answers:

Correct answer:

Explanation:

In order to add fractions we must find a common denominator.  Since is a multiple of both and , we must multiply the numerator and denominator of each fraction by a number to get a denomintor of

Since times is , we can multiply the numerator and denominator of the first fraction by .

Since times is , we can multiply the numerator and demonimator of the second fraction by .

Now we add together the numerators.

The answer is .

### Example Question #1 : How To Add Fractions

If a rectangle has a length of and a width of what is the perimeter of the rectangle, in simplest form?

Possible Answers:

Correct answer:

Explanation:

In order to find the perimeter of a rectangle, you add together all the sides.  In this particular case, however, you must first find a common denominator for all of the fractions.  Luckily, is a multiple of , so we can multiply the numerator and denominator of  by to get a denominator of .

Now we simply add all four sides.

Since  can be reduced by dividing the numerator and denominator by , we must simplify.

The perimeter of the rectangle is .

### Example Question #11 : Fractions

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

Add both sides of the two equations:

### Example Question #2 : How To Add Fractions

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

(a)

(b)

### Example Question #1 : How To Add Fractions

Column A                             Column B

Possible Answers:

The quantity in Column A is greater.

There is no way to determine the relationship between the quantities in the columns.

The quantities in each column are equal.

The quantity in Column B is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

First, you must add the fractions in each column. When adding fractions, find the common denominator. The common denominator for Column A is 10. Then, change the numerators to reflect changing the denominators to give you . Combie the numerators to give you Then, add the fractions in Column B. The common denominator for those fractions is 72. Therefore, you get . Combine the numerators to get . Compare those two fractions. Think of them as slices of pizza. There would be way more of Column B. Therefore, it is greater. Also, a little to trick to comparing fractions is cross-multiply. The side that has the biggest product is the greatest.

### Example Question #1 : How To Add Fractions

Which is the greater quantity?

(A)

(B)

Possible Answers:

(A) and (B) are equal

(A) is greater

(B) is greater

It is impossible to determine which is greater from the information given

Correct answer:

(A) is greater

Explanation:

and , so

, the decimal equivalent of (A).

, the value of (B).

(A) is the greater.