# ISEE Upper Level Quantitative : Rectangles

## Example Questions

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

Which is the greater quantity?

(a) The area of the rectangle on the coordinate plane with vertices

(b) The area of the rectangle on the coordinate plane with vertices

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

Explanation:

(a) The first rectangle has width  and height ; its area is .

(b) The second rectangle has width  and height ; its area is .

The areas of the rectangle are the same.

### Example Question #2 : Rectangles

A rectangle on the coordinate plane has its vertices at the points .

Which is the greater quantity?

(a) The area of the portion of the rectangle in Quadrant I

(b) The area of the portion of the rectangle in Quadrant III

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Explanation:

(a) The portion of the rectangle in Quadrant I is a rectangle with vertices , so its area is .

(a) The portion of the rectangle in Quadrant III is a rectangle with vertices , so its area is .

### Example Question #1 : Rectangles

. Which is the greater quantity?

(a) The area of the square on the coordinate plane with vertices

(b) The area of the rectangle on the coordinate plane with vertices

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Explanation:

(a) The square has sidelength , and therefore has area .

(b) The rectangle has width  and height , and therefore has area .

Since , the square in (a) has the greater area.

### Example Question #4 : Rectangles

The perimeter of a rectangle is one yard. The rectangle is three times as long as it is wide. Which is the greater quantity?

(a) The area of the rectangle

(b) 60 square inches

(b) is greater

(a) and (b) are equal

It is impossible to tell form the information given

(a) is greater

(a) is greater

Explanation:

Let  be the width of the rectangle. Then its length is , and its perimeter is

Since the perimeter is one yard, or 36 inches,

inches is the wiidth, and  inches is the length, so the area is

square inches. (a) is the greater quantity.

### Example Question #5 : Rectangles

If one rectangular park measures  and another rectangular park measures , how many times greater is the area of the second park than the area of the first park?

Explanation:

First, you must compute the area of both parks. The area of a rectangle is length times width. Therefore, the area of park one is , which is . The area of park two is , which is Then, divide the area of the second park by the area of the first park (). This yields 3 as the answer.

### Example Question #6 : Rectangles

The sum of the lengths of three sides of a rectangle is 572 inches; the width of the rectangle is 60% of its length. Give its area in square inches.

It is impossible to determine the area from the given information.

It is impossible to determine the area from the given information.

Explanation:

Since the width of the rectangle is 60% of its length, we can write .

However, it is not clear from the problem which three sides - two lengths and a width or two widths and a length - we are choosing to have sum 572 inches. Depending on the three sides chosen, we can either set up

or

Since the length cannot be determined with certainty, neither can the width, and, subsequently, neither can the area.

### Example Question #7 : Rectangles

Five rectangles each have the same length, which we will call . The widths of the five rectangles are 7, 5, 8, 10, and 12. Which of the following expressions is equal to the mean of their areas?

Explanation:

The area of a rectangle is the product of its width and its length. The areas of the five rectangles, therefore, are . The mean of these five areas is their sum divided by 5, or

### Example Question #8 : Rectangles

Two rectangles, A and B, each have perimeter 32 feet. Rectangle A has length 12 feet; Rectangle B has length 10 feet. The area of Rectangle A is what percent of the area of Rectangle B?

Explanation:

The perimeter of a rectangle can be given by the formula

Since for both rectangles, 30 is the perimeter, this becomes

, and subsequently

.

Rectangle A has length 12 feet and, subsequently. width 4 feet, making its area

square feet

Rectangle B has length 10 feet and, subsequently. width 6 feet, making its area

square feet

The area of Rectangle A is

of the area of Rectangle B.

### Example Question #9 : Rectangles

Give the area of the above rectangle in terms of

Explanation:

The area of a rectangle is equal to the product of its length and height, which here are 5 and . This product is .

### Example Question #1 : Rectangles

A rectangle is two feet shorter than twice its width; its perimeter is six yards. Give its area in square inches.