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ISEE Middle Level Quantitative : Rectangles

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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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:

$14\: in^{2}$

$18\: in^{2}$

$12\: in^{2}$

$6\: in^{2}$

$10\: in^{2}$

Correct answer:

$18\: in^{2}$

Explanation:

In order to find the area of a rectangle we use the formula $\small A=l\cdot w$ .

In this problem, we know the width is $\small 3\: in$ . We also know that the length is twice as long as the width, which can be written as $\small 2w$ . This means that in order to find the length, we must multiply the width by $\small 2$ .

$\small 2(3\: in)=6\: in$

Now that we know that our length is $\small 6\: in$ , we simply multiply it by our width of $\small 3\: in$ .

$\small \small 6\: in\cdot 3\: in=18\: in^{2}$

The area of the rectangle is $\small 18\: in^{2}$ .

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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 two quantities are equal.

The quantity in Column B is greater.

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.

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:

perimeter=w+w+7w+7w= 16w

$area=w\cdot (7w)= 7w^{2}$

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.

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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:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(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: $60 \times 30 = 1,800 \textrm{ cm}^{2}$ .

Two rectangles have area: $60 \times 15 = 900 \textrm{ cm}^{2}$ .

Two rectangles have area: $15 \times 30 = 450 \textrm{ cm}^{2}$ .

Add the areas: $A = 1,800 + 1,800 + 900 + 900 + 450 + 450 = 6,300 \textrm{ cm}^{2}$

(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:

$A = 6 \times 30 ^{2} = 6 \times 900 = 5,400 \textrm{ cm}^{2}$ .

The first figure has the greater surface area.

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Example Question #4 : Rectangles

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:

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

The quantities in both columns are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

First, you must calculate Column A. The formula for the area of a rectangle is A=lw . Plug in the values given to get $11\cdot 5$ , which gives you 55cm^2 . Then, calculate the area of the square. Since all of the sides of a square are equal, the formula is A=s^2 , or A=lw . Therefore, the area of the square is $7\cdot 7$ , which gives you 49cm^2 . Therefore, the quantity in Column A is greater.

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Example Question #118 : Geometry

Rectangles

Figure NOT drawn to scale

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

Possible Answers:

500

600

1,000

800

Correct answer:

600

Explanation:

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

Since the area of Rectangle ABCD is 1,000,

$20 \cdot (x+10) = 1,000$

$x+10 = 1,000 \div 20 = 50$

x = 50 - 10 = 40

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

$10 (x+20) = 10 (40+20) = 10 \cdot 60 = 600$

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

10 < X < 20

Which is the greater quantity?

(a) The area of a rectangle with length 20 and width X + 10

(b) The area of a rectangle with length 10 and width X+20

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

Correct answer:

(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

$20 \cdot (X+10) = 20 X + 20 \cdot 10 = 20 X +100$

The rectangle described in (b) has area

$10 \cdot (X+20) = 10 X + 10 \cdot 20 = 10 X +100$

20 > 10 , and is positive, so 20 X> 10 X , and 20 X+ 100 > 10 X + 100 . 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.

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

Box

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) 80x

Possible Answers:

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

We can fill in a few edge lengths below:

Box

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 20x . Since the total of these four areas is $4 \cdot 20x = 80 x$ . Since the left and right sides have not been included, the total surface area must be more than 80x .

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

What is the area of the figure below?

Possible Answers:

47in^2

24in^2

35in^2

59in^2

52in^2

Correct answer:

59in^2

Explanation:

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

12.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 5$ $A=6\times 4$

A=35in^2 A=24in^2

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

35in^2+24in^2=59in^2

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

What is the area of the figure below?

Possible Answers:

32in^2

22in^2

10in^2

12in^2

28in^2

Correct answer:

22in^2

Explanation:

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

11.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=6\times 2$ $A=5\times 2$

A=12in^2 A=10in^2

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

12in^2+10in^2=22in^2

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Example Question #5 : How To Find The Area Of A Rectangle

What is the area of the figure below?

Possible Answers:

40in^2

28in^2

18in^2

32in^2

14in^2

Correct answer:

32in^2

Explanation:

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

10.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 2$ $A=6\times 3$

A=14in^2 A=18in^2

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

14in^2+18in^2=32in^2

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : Rectangles

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : Rectangles

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

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

Example Question #4 : Rectangles

Example Question #118 : Geometry

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

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

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

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

Example Question #5 : How To Find The Area Of A Rectangle

All ISEE Middle Level Quantitative Resources

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