Finite Math Examples

$\frac{m}{6 + m} - \frac{m}{2 + 3} + \frac{m}{3}$

Step 1

Add $2$ and $3$ .

$\frac{m}{6 + m} - \frac{m}{5} + \frac{m}{3}$

Step 2

To write $\frac{m}{6 + m}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{m}{6 + m} \cdot \frac{5}{5} - \frac{m}{5} + \frac{m}{3}$

Step 3

To write $- \frac{m}{5}$ as a fraction with a common denominator, multiply by $\frac{6 + m}{6 + m}$ .

$\frac{m}{6 + m} \cdot \frac{5}{5} - \frac{m}{5} \cdot \frac{6 + m}{6 + m} + \frac{m}{3}$

Step 4

Write each expression with a common denominator of $(6 + m) \cdot 5$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{m}{6 + m}$ by $\frac{5}{5}$ .

$\frac{m \cdot 5}{(6 + m) \cdot 5} - \frac{m}{5} \cdot \frac{6 + m}{6 + m} + \frac{m}{3}$

Step 4.2

Multiply $\frac{m}{5}$ by $\frac{6 + m}{6 + m}$ .

$\frac{m \cdot 5}{(6 + m) \cdot 5} - \frac{m (6 + m)}{5 (6 + m)} + \frac{m}{3}$

Step 4.3

Reorder the factors of $(6 + m) \cdot 5$ .

$\frac{m \cdot 5}{5 (6 + m)} - \frac{m (6 + m)}{5 (6 + m)} + \frac{m}{3}$

Step 5

Combine the numerators over the common denominator.

$\frac{m \cdot 5 - m (6 + m)}{5 (6 + m)} + \frac{m}{3}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $m$ out of $m \cdot 5 - m (6 + m)$ .

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Step 6.1.1

Factor $m$ out of $m \cdot 5$ .

$\frac{m (5) - m (6 + m)}{5 (6 + m)} + \frac{m}{3}$

Step 6.1.2

Factor $m$ out of $- m (6 + m)$ .

$\frac{m (5) + m (- (6 + m))}{5 (6 + m)} + \frac{m}{3}$

Step 6.1.3

Factor $m$ out of $m (5) + m (- (6 + m))$ .

$\frac{m (5 - (6 + m))}{5 (6 + m)} + \frac{m}{3}$

Step 6.2

Apply the distributive property.

$\frac{m (5 - 1 \cdot 6 - m)}{5 (6 + m)} + \frac{m}{3}$

Step 6.3

Multiply $- 1$ by $6$ .

$\frac{m (5 - 6 - m)}{5 (6 + m)} + \frac{m}{3}$

Step 6.4

Subtract $6$ from $5$ .

$\frac{m (- 1 - m)}{5 (6 + m)} + \frac{m}{3}$

Step 7

To write $\frac{m (- 1 - m)}{5 (6 + m)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{m (- 1 - m)}{5 (6 + m)} \cdot \frac{3}{3} + \frac{m}{3}$

Step 8

To write $\frac{m}{3}$ as a fraction with a common denominator, multiply by $\frac{5 (6 + m)}{5 (6 + m)}$ .

$\frac{m (- 1 - m)}{5 (6 + m)} \cdot \frac{3}{3} + \frac{m}{3} \cdot \frac{5 (6 + m)}{5 (6 + m)}$

Step 9

Write each expression with a common denominator of $15 (6 + m)$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{m (- 1 - m)}{5 (6 + m)}$ by $\frac{3}{3}$ .

$\frac{m (- 1 - m) \cdot 3}{5 (6 + m) \cdot 3} + \frac{m}{3} \cdot \frac{5 (6 + m)}{5 (6 + m)}$

Step 9.2

Multiply $3$ by $5$ .

$\frac{m (- 1 - m) \cdot 3}{15 (6 + m)} + \frac{m}{3} \cdot \frac{5 (6 + m)}{5 (6 + m)}$

Step 9.3

Multiply $\frac{m}{3}$ by $\frac{5 (6 + m)}{5 (6 + m)}$ .

$\frac{m (- 1 - m) \cdot 3}{15 (6 + m)} + \frac{m (5 (6 + m))}{3 (5 (6 + m))}$

Step 9.4

Multiply $5$ by $3$ .

$\frac{m (- 1 - m) \cdot 3}{15 (6 + m)} + \frac{m (5 (6 + m))}{15 (6 + m)}$

Step 10

Combine the numerators over the common denominator.

$\frac{m (- 1 - m) \cdot 3 + m (5 (6 + m))}{15 (6 + m)}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $m$ out of $m (- 1 - m) \cdot 3 + m (5 (6 + m))$ .

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Step 11.1.1

Factor $m$ out of $m (- 1 - m) \cdot 3$ .

$\frac{m ((- 1 - m) \cdot 3) + m (5 (6 + m))}{15 (6 + m)}$

Step 11.1.2

Factor $m$ out of $m ((- 1 - m) \cdot 3) + m (5 (6 + m))$ .

$\frac{m ((- 1 - m) \cdot 3 + 5 (6 + m))}{15 (6 + m)}$

Step 11.2

Apply the distributive property.

$\frac{m (- 1 \cdot 3 - m \cdot 3 + 5 (6 + m))}{15 (6 + m)}$

Step 11.3

Multiply $- 1$ by $3$ .

$\frac{m (- 3 - m \cdot 3 + 5 (6 + m))}{15 (6 + m)}$

Step 11.4

Multiply $3$ by $- 1$ .

$\frac{m (- 3 - 3 m + 5 (6 + m))}{15 (6 + m)}$

Step 11.5

Apply the distributive property.

$\frac{m (- 3 - 3 m + 5 \cdot 6 + 5 m)}{15 (6 + m)}$

Step 11.6

Multiply $5$ by $6$ .

$\frac{m (- 3 - 3 m + 30 + 5 m)}{15 (6 + m)}$

Step 11.7

Add $- 3$ and $30$ .

$\frac{m (- 3 m + 27 + 5 m)}{15 (6 + m)}$

Step 11.8

Add $- 3 m$ and $5 m$ .

$\frac{m (2 m + 27)}{15 (6 + m)}$