### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #1 : Rectangles

If the length of a rectangle is twice the width, and the width is three inches, what is the area of the rectangle?

**Possible Answers:**

**Correct answer:**

In order to find the area of a rectangle we use the formula .

In this problem, we know the width is . We also know that the length is twice as long as the width, which can be written as . This means that in order to find the length, we must multiply the width by .

Now that we know that our length is , we simply multiply it by our width of .

The area of the rectangle is .

### Example Question #1 : How To Find The Area Of A Rectangle

Using the information given in each question, compare the quantity in Column A to the quantity in Column B.

A certain rectangle is seven times as long as it is wide.

Column A Column B

the rectangle's the rectangle's

perimeter area

(in units) (in square units)

**Possible Answers:**

The quantity in Column B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

**Correct answer:**

The relationship cannot be determined from the information given.

This type of problem reminds us to be wary of simply plugging in numbers (which works with certain problems). If you were to choose 1 and 7 here, the perimeter would be larger; if you chose 10 and 70, the area would be much larger.

To solve this problem with variables:

From here we can see that smaller values of will lead to a larger perimeter, while larger values of will lead to a larger area.

The answer cannot be determined.

### Example Question #2 : How To Find The Area Of A Rectangle

Which is the greater quantity?

(a) The surface area of a rectangular prism with length 60 centimeters, width 30 centimeters, and height 15 centimeters

(b) The surface area of a cube with sidelength 300 millimeters

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

**Correct answer:**

(a) is greater

(a) The surface of a rectangular prism comprises six rectangles, so we can take the sum of their areas.

Two rectangles have area: .

Two rectangles have area: .

Two rectangles have area: .

Add the areas:

(b) The surface of a cube comprises six squares, so we can square the sidelength - which we rewrite as 30 centimeters - and multiply the result by 6:

.

The first figure has the greater surface area.

### Example Question #3 : How To Find The Area Of A Rectangle

Column A Column B

The area of a The area of a square

rectangle with sides with sides 7cm.

11 cm and 5 cm.

**Possible Answers:**

The quantities in both columns are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

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

**Correct answer:**

The quantity in Column A is greater.

First, you must calculate Column A. The formula for the area of a rectangle is . Plug in the values given to get , which gives you . Then, calculate the area of the square. Since all of the sides of a square are equal, the formula is , or . Therefore, the area of the square is , which gives you . Therefore, the quantity in Column A is greater.

### Example Question #5 : Rectangles

Figure NOT drawn to scale

Refer to the above figure. The area of Rectangle is 1,000. Give the area of Rectangle .

**Possible Answers:**

**Correct answer:**

The area of a rectangle is the product of its length and its width.

Since the area of Rectangle is 1,000,

Substitute 40 for in the height of Rectangle and calculate the area as follows:

### Example Question #4 : How To Find The Area Of A Rectangle

Which is the greater quantity?

(a) The area of a rectangle with length 20 and width

(b) The area of a rectangle with length 10 and width

**Possible Answers:**

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(a) is the greater quantity

The area of a rectangle is the product of its length and its width.

The rectangle described in (a) has area

The rectangle described in (b) has area

, and is positive, so , and . The rectangle from (a) has the greater area.

Note that the value of has no bearing on the answer, except for the fact that it is positive.

### Example Question #7 : Rectangles

Figure NOT drawn to scale

The above diagram shows a rectangular solid. is an integer. Which is the greater quantity?

(a) The surface area of the solid

(b)

**Possible Answers:**

(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

**Correct answer:**

(a) is the greater quantity

We can fill in a few edge lengths below:

All six sides are rectangles, so their areas are equal to the products of their dimensions. We specifically notice that the top, bottom. front, and back each have area . Since the total of these four areas is . Since the left and right sides have not been included, the total surface area must be more than .

### Example Question #8 : Rectangles

What is the area of the figure below?

**Possible Answers:**

**Correct answer:**

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, , we can solve for the area of both of our rectangles

To find our final answer, we need to add the areas together.

### Example Question #9 : Rectangles

What is the area of the figure below?

**Possible Answers:**

**Correct answer:**

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, , we can solve for the area of both of our rectangles

To find our final answer, we need to add the areas together.

### Example Question #10 : Rectangles

What is the area of the figure below?

**Possible Answers:**

**Correct answer:**

To find the area of the figure above, we need to slip the figure into two rectangles.

Using our area formula, , we can solve for the area of both of our rectangles

To find our final answer, we need to add the areas together.