Algebra Examples

$\frac{8 b + y}{40 b} - \frac{6 y + b}{30 y}$

Step 1

To write $\frac{8 b + y}{40 b}$ as a fraction with a common denominator, multiply by $\frac{3 y}{3 y}$ .

$\frac{8 b + y}{40 b} \cdot \frac{3 y}{3 y} - \frac{6 y + b}{30 y}$

Step 2

To write $- \frac{6 y + b}{30 y}$ as a fraction with a common denominator, multiply by $\frac{4 b}{4 b}$ .

$\frac{8 b + y}{40 b} \cdot \frac{3 y}{3 y} - \frac{6 y + b}{30 y} \cdot \frac{4 b}{4 b}$

Step 3

Write each expression with a common denominator of $120 b y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8 b + y}{40 b}$ by $\frac{3 y}{3 y}$ .

$\frac{(8 b + y) (3 y)}{40 b (3 y)} - \frac{6 y + b}{30 y} \cdot \frac{4 b}{4 b}$

Step 3.2

Multiply $3$ by $40$ .

$\frac{(8 b + y) (3 y)}{120 b y} - \frac{6 y + b}{30 y} \cdot \frac{4 b}{4 b}$

Step 3.3

Multiply $\frac{6 y + b}{30 y}$ by $\frac{4 b}{4 b}$ .

$\frac{(8 b + y) (3 y)}{120 b y} - \frac{(6 y + b) (4 b)}{30 y (4 b)}$

Step 3.4

Multiply $4$ by $30$ .

$\frac{(8 b + y) (3 y)}{120 b y} - \frac{(6 y + b) (4 b)}{120 y b}$

Step 3.5

Reorder the factors of $120 y b$ .

$\frac{(8 b + y) (3 y)}{120 b y} - \frac{(6 y + b) (4 b)}{120 b y}$

Step 4

Combine the numerators over the common denominator.

$\frac{(8 b + y) (3 y) - (6 y + b) (4 b)}{120 b y}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(8 b \cdot 3 + y \cdot 3) y - (6 y + b) (4 b)}{120 b y}$

Step 5.2

Multiply $3$ by $8$ .

$\frac{(24 b + y \cdot 3) y - (6 y + b) (4 b)}{120 b y}$

Step 5.3

Move $3$ to the left of $y$ .

$\frac{(24 b + 3 \cdot y) y - (6 y + b) (4 b)}{120 b y}$

Step 5.4

Apply the distributive property.

$\frac{24 b y + 3 y \cdot y - (6 y + b) (4 b)}{120 b y}$

Step 5.5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.5.1

Move $y$ .

$\frac{24 b y + 3 (y \cdot y) - (6 y + b) (4 b)}{120 b y}$

Step 5.5.2

Multiply $y$ by $y$ .

$\frac{24 b y + 3 y^{2} - (6 y + b) (4 b)}{120 b y}$

Step 5.6

Apply the distributive property.

$\frac{24 b y + 3 y^{2} + (- (6 y) - b) (4 b)}{120 b y}$

Step 5.7

Multiply $6$ by $- 1$ .

$\frac{24 b y + 3 y^{2} + (- 6 y - b) (4 b)}{120 b y}$

Step 5.8

Apply the distributive property.

$\frac{24 b y + 3 y^{2} - 6 y (4 b) - b (4 b)}{120 b y}$

Step 5.9

Rewrite using the commutative property of multiplication.

$\frac{24 b y + 3 y^{2} - 6 \cdot 4 y b - b (4 b)}{120 b y}$

Step 5.10

Rewrite using the commutative property of multiplication.

$\frac{24 b y + 3 y^{2} - 6 \cdot 4 y b - 1 \cdot 4 b \cdot b}{120 b y}$

Step 5.11

Simplify each term.

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Step 5.11.1

Multiply $- 6$ by $4$ .

$\frac{24 b y + 3 y^{2} - 24 y b - 1 \cdot 4 b \cdot b}{120 b y}$

Step 5.11.2

Multiply $b$ by $b$ by adding the exponents.

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Step 5.11.2.1

Move $b$ .

$\frac{24 b y + 3 y^{2} - 24 y b - 1 \cdot 4 (b \cdot b)}{120 b y}$

Step 5.11.2.2

Multiply $b$ by $b$ .

$\frac{24 b y + 3 y^{2} - 24 y b - 1 \cdot 4 b^{2}}{120 b y}$

Step 5.11.3

Multiply $- 1$ by $4$ .

$\frac{24 b y + 3 y^{2} - 24 y b - 4 b^{2}}{120 b y}$

Step 5.12

Subtract $24 y b$ from $24 b y$ .

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Step 5.12.1

Move $y$ .

$\frac{3 y^{2} + 24 b y - 24 b y - 4 b^{2}}{120 b y}$

Step 5.12.2

Subtract $24 b y$ from $24 b y$ .

$\frac{3 y^{2} + 0 - 4 b^{2}}{120 b y}$

Step 5.13

Add $3 y^{2}$ and $0$ .

$\frac{3 y^{2} - 4 b^{2}}{120 b y}$