# 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

(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

(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

(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

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

(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

(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:

Explanation:

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

Add, then rewrite as a mixed fraction:

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

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.

### 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?

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)

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(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)

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(b) is greater

Explanation:

(a)

(b)

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

Column A                             Column B

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.

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)

(A) and (B) are equal

(A) is greater

(B) is greater

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

(A) is greater

Explanation:

and , so

, the decimal equivalent of (A).

, the value of (B).

(A) is the greater.