# ISEE Middle Level Quantitative : Geometry

## Example Questions

### Example Question #1194 : Isee Lower Level (Grades 5 6) Quantitative Reasoning

David wants to put a fence around his backyard. How much fencing will he need if his backyard is  by

Explanation:

The fence is going around the backyard, so this is a perimeter problem.

### Example Question #1195 : Isee Lower Level (Grades 5 6) Quantitative Reasoning

David wants to put a fence around his backyard. How much fencing will he need if his backyard is  by

Explanation:

The fence is going around the backyard, so this is a perimeter problem.

### Example Question #111 : Geometry

What is the length of a rectangular room with a perimeter of  and a width of

Explanation:

We have the perimeter and the width, so we can plug those values into our equation and solve for our unknown.

Subtract  from both sides

Divide  by both sides

### Example Question #501 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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

Explanation:

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 #502 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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)

The quantity in Column B is greater.

The two quantities are equal.

The quantity in Column A is greater.

The relationship cannot be determined from the information given.

The relationship cannot be determined from the information given.

Explanation:

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.

### Example Question #503 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(a) is greater

Explanation:

(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: .

(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 #504 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Column A                                   Column B

The area of a                              The area of a square

rectangle with sides                     with sides 7cm.

11 cm and 5 cm.

The quantities in both columns are equal.

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

The quantity in Column A is greater.

The quantity in Column B is greater.

The quantity in Column A is greater.

Explanation:

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 #505 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

Figure NOT drawn to scale

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

Explanation:

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 #511 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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

(b) is the greater quantity

(a) and (b) are equal

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

(a) is the greater quantity

(a) is the greater quantity

Explanation:

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 #512 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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)

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity