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PSAT Math : Rectangles

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Example Question #21 : Rectangles

George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 $ft^{2}$ and costs $40. One quart of paint covers 100 $ft^{2}$ and costs $15. How much money will he spend on the blue paint?

Possible Answers:

$\$40$

$\$30$

$\$70$

$\$80$

$\$55$

Correct answer:

$\$70$

Explanation:

The area of the walls is given by $A = 2wh + 2lh = 2(12)(10) + 2(15)(10) = 540 ft^{2}$

One gallon of paint covers 400 $ft^{2}$ and the remaining 140 $ft^{2}$ would be covered by two quarts.

So one gallon and two quarts of paint would cost $\$40 + \$15 + \$15 = \$70$

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

Daisy gets new carpet for her rectangluar room. Her floor is $21\ ft \times 24\ ft$ . The carpet sells for $5 per square yard. How much did she spend on her carpet?

Possible Answers:

$\$225$

$\$350$

$\$120$

$\$310$

$\$280$

Correct answer:

$\$280$

Explanation:

Since $3\ ft=1\ yd$ the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost $5\cdot 56=\$280$ .

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

The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?

Possible Answers:

$150\ ft^{2}$

$125\ ft^{2}$

$75\ ft^{2}$

$100\ ft^{2}$

$50\ ft^{2}$

Correct answer:

$75\ ft^{2}$

Explanation:

For a rectangle, $P=2w+2l$ and $A=lw$ where $w$ is the width and $l$ is the length.

Let $x=width$ and $2x+5=length$ .

So the equation to solve becomes $40=2x+2(2x+5)$ or $40=6x+10$ .

Thus $x=5\ ft$ and $2x+5=15\ ft$ , so the area is $75\ ft^{2}$ .

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Example Question #1 : Rectangles

The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?

Possible Answers:

768 ft²

1536 ft²

192 ft²

1536 ft²

2464 ft²

Correct answer:

1536 ft²

Explanation:

Glass Area per Window = 8 ft x 6 ft = 48 ft²

Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows

Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft²

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Example Question #21 : Rectangles

Rectangle

Note: Figure NOT drawn to scale

AB = 13, BC = 7, AD = 9, DF = 3

What percent of Rectangle ACGF is pink?

Possible Answers:

$48\frac{3}{4} \%$

$50 \%$

$46\frac{2}{3} \%$

$55 \frac{5}{9} \%$

$44 \frac{4}{9} \%$

Correct answer:

$48\frac{3}{4} \%$

Explanation:

The pink region is Rectangle ABED . Its length and width are

L = AB = 13

W = AD = 9

so its area is the product of these, or

$A = 13 \times 9 = 117$ .

The length and width of Rectangle ACGF are

L = AC = AB + BC = 13+7 = 20

W =AF = AD + DF=9+3 = 12

so its area is the product of these, or

$20 \times 12 = 240$ .

We want to know what percent 117 is of 240, which can be answered as follows:

$\frac{117}{240} \times 100 \% = 48\frac{3}{4} \%$

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Example Question #322 : Plane Geometry

Rectangle

Note: Figure NOT drawn to scale

Refer to the above diagram.

40% of Rectangle ACGF is pink. AB = 20, AD = 15, BC = 10 .

Evaluate .

Possible Answers:

DF = 10

DF = 5

DF = 8

DF = 12

DF = 15

Correct answer:

DF = 10

Explanation:

Rectangle ABED has length L = AB = 20 and width W = AD = 15 , so it has area

$LW = 20 \times 15 = 300$ .

300 is 40% of, or 0.40 times, the area of Rectangle ACGF , which we will call . We can determine as follows:

0.40 A = 300

$A = 300 \div 0.4 = 750$ .

The length of Rectangle ACGF is

L = AC = AB + BC = 20 + 10 = 30 ,

so its width is

$W = AF = 750 \div 30 = 25$ .

Since

W = AF= AD + DF ,

25 = 15 + DF

DF = 10

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Example Question #321 : Plane Geometry

Rectangle

Note: Figure NOT drawn to scale

AB = 24, BC = 6, AD = 12, DF = 3

What percent of Rectangle ACGF is white?

Possible Answers:

$33 \frac{1}{3} \%$

$40 \%$

$44 \frac{4}{9} \%$

$36 \%$

$25 \%$

Correct answer:

$36 \%$

Explanation:

The pink region is Rectangle ABED . Its length and width are

L = AB = 24

W = AD = 12

so its area is the product of these, or

$24 \times 12 = 288$ .

The length and width of Rectangle ACGF are

L = AC = AB + BC = 24 + 6 =30

W =AF = AD + DF= 12 + 3 = 15

so its area is the product of these, or

$30 \times 15 = 450$ .

The white region is Rectangle ABED cut from Rectangle ACGF , so its area is the difference of the two:

450 - 288 = 162 .

So we want to know what percent 162 is of 450, which can be answered as follows:

$\frac{162}{450} \times 100 \% = 36 \%$

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Example Question #21 : Rectangles

Rectangle

Note: Figure NOT drawn to scale

AB = 20, BC = 8, AD = 10, DF = 4

Give the ratio of the perimeter of Rectangle ACGF to that of Rectangle ABED .

Possible Answers:

4:3

7:5

5:2

2:1

5:3

Correct answer:

7:5

Explanation:

The perimeter of Rectangle ABED is

P = AB +BE + ED + AD

Opposite sides of a rectangle are congruent, so

AB= ED = 20, AD = BE = 10

and

P = 20 + 10 + 20 + 10 = 60

The perimeter of Rectangle ACGF is

P = AC + CG + GF + AF

Opposite sides of a rectangle are congruent, so

GF = AC = AB + BC = 20 + 8 = 28 ,

CG = AF = AD + DF = 10 + 4 = 14 ,

and

P = 28 + 14+ 28 + 14= 84

The ratio of the perimeters is

$\frac{84}{60} = \frac{84 \div 12}{60 \div 12} = \frac{7}{5}$ - that is, 7 to 5.

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Example Question #21 : Rectangles

Garden

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) eight feet wide throughout. What is the area of that dirt path?

Possible Answers:

The correct area is not given among the other responses.

$1,280\textrm{ ft}^{2}$

$2,304\textrm{ ft}^{2}$

$2,560\textrm{ ft}^{2}$

$1,216\textrm{ ft}^{2}$

Correct answer:

$2,304\textrm{ ft}^{2}$

Explanation:

The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is

$A = 100 \cdot 60 = 6,000$ square feet.

The inner rectangle has length and width $(100 - 2\times 8) = 84$ feet and $(60 - 2\times 8) = 44$ feet, respectively, so its area is

$A = 84 \times 44= 3,696$ square feet.

The area of the path is the difference of the two:

6,000 - 3,696= 2,304 square feet.

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Example Question #21 : Rectangles

Garden

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange). The dirt path is seven feet wide throughout. Which of the following polynomials gives the area of the dirt path in square feet?

Possible Answers:

21x-49

42x - 196

42x - 98

21x

42x

Correct answer:

42x - 196

Explanation:

The area of the dirt path is the difference between the areas of the outer and inner rectangles.

The outer rectangle has area

$A = 2x \cdot x = 2x^{2}$

The area of the inner rectangle can be found as follows:

The length of the garden is $2 \times 7= 14$ feet less than that of the entire lot, or

L = 2x-14 ;

The width of the garden is $2 \times 7= 14$ less than that of the entire lot, or

W = x-14 ;

The area of the garden is their product:

A = LW = (2x-14)(x - 14)

$=2x \cdot x - 2x \cdot 14 - 14 \cdot x + 14 \cdot 14$

$= 2x ^{2} - 28x - 14 x + 196$

$= 2x ^{2} - 42 x + 196$

Now, subtract the areas:

$2x ^{2} - \left (2x ^{2} - 42 x + 196 \right ) = 2x ^{2} - 2x ^{2} + 42 x - 196 = 42x - 196$

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PSAT Math : Rectangles

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #21 : Rectangles

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

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

Example Question #1 : Rectangles

Example Question #21 : Rectangles

Example Question #322 : Plane Geometry

Example Question #321 : Plane Geometry

Example Question #21 : Rectangles

Example Question #21 : Rectangles

Example Question #21 : Rectangles

All PSAT Math Resources

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