Examples with solutions for The Distributive Property for 7th Grade: Using additional geometric shapes

Exercise #1

Calculate the area of the rectangle below in terms of a and b.

a+3a+3a+3b+8b+8b+8

Video Solution

Step-by-Step Solution

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

S=wh S=w⋅h

When:

S = area

w = width

h = height

We take data from the sides of the rectangle in the figure.w=b+8 w=b+8 h=a+3 h=a+3

We then substitute the above data into the formula in order to calculate the area of the rectangle:

S=wh=(b+8)(a+3) S=w⋅h = (b+8)(a+3)

We use the formula of the extended distributive property:

(a+b)(c+d)=ac+ad+bc+bd (a+b)(c+d)=ac+ad+bc+bd

We substitute once more and solve the problem as follows:

S=(b+8)(a+3)=(b)(a)+(b)(3)+(8)(a)+(8)(3) S=(b+8)(a+3)=(b)(a)+(b)(3)+(8)(a)+(8)(3)

(b)(a)+(b)(3)+(8)(a)+(8)(3)=ab+3b+8a+24 (b)(a)+(b)(3)+(8)(a)+(8)(3)=ab+3b+8a+24

Therefore, the correct answer is option B: ab+8a+3b+24.

Keep in mind that, since there are only addition operations, the order of the terms in the expression can be changed and, therefore,

ab+3b+8a+24=ab+8a+3b+24 ab+3b+8a+24=ab+8a+3b+24

Answer

ab + 8a + 3b + 24

Exercise #2

Calculate the area of the rectangle

y+2y+2y+2x+5x+5x+5

Video Solution

Step-by-Step Solution

Let's begin by reminding ourselves of the formula to calculate the area of a rectangle: width X length

S=wh S=w⋅h

Where:

S = area

w = width

h = height

We extract the data from the sides of the rectangle in the figure.

w=x+5 w=x+5 h=y+2 h=y+2

We then substitute the above data into the formula in order to calculate the area of the rectangle:

S=wh=(x+5)(y+2) S=w⋅h=(x+5)(y+2)

We use the formula of the extended distributive property:

(a+b)(c+d)=ac+ad+bc+bd (a+b)(c+d)=ac+ad+bc+bd

We once again substitute and solve the problem as follows:

S=(x+5)(y+2)=(x)(y)+(x)(2)+(5)(y)+(5)(2) S=(x+5)(y+2)=(x)(y)+(x)(2)+(5)(y)+(5)(2)

(x)(y)+(x)(2)+(5)(y)+(5)(2)=xy+2x+5y+10 (x)(y)+(x)(2)+(5)(y)+(5)(2)=xy+2x+5y+10

Therefore, the correct answer is option C: xy+2x+5y+10.

Answer

xy+2x+5y+10 xy+2x+5y+10

Exercise #3

Express the area of the rectangle below in terms of y and z.

3y3y3yy+3z

Video Solution

Step-by-Step Solution

Let us begin by reminding ourselves of the formula to calculate the area of a rectangle: width X height

S=wh S=w⋅h

Where:

S = area

w = width

h = height

We must first extract the data from the sides of the rectangle shown in the figure.

w=3y w=3y h=y+3z h=y+3z

We then insert the known data into the formula in order to calculate the area of the rectangle:

S=wh=(y+3z)(3y) S=w⋅h=(y+3z)(3y)

We use the distributive property formula:

a(b+c)=ab+ac a\left(b+c\right)=ab+ac

We substitute all known data and solve as follows:

S=(y+3z)(3y)=(3y)(y+3z) S=(y+3z)(3y)=(3y)(y+3z)

(3y)(y+3z)=(3y)(y)+(3y)(3z) (3y)(y+3z)=(3y)(y)+(3y)(3z)

(3y)(y)+(3y)(3z)=3y2+9yz (3y)(y)+(3y)(3z)=3y^2+9yz

Keep in mind that because there is a multiplication operation, the order of the terms in the expression can be changed, hence:

(y+3z)(3y)=(3y)(y+3z) (y+3z)(3y)=(3y)(y+3z)

Therefore, the correct answer is option D: 3y2+9yz 3y^2+9yz

Answer

3y2+9yz 3y^2+9yz