Basic Math Examples

$\frac{a}{a + 5} - 1 \div \frac{4 a + 1}{a - 1}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{a}{a + 5} - (1 \frac{a - 1}{4 a + 1})$

Step 1.2

Multiply $\frac{a - 1}{4 a + 1}$ by $1$ .

$\frac{a}{a + 5} - \frac{a - 1}{4 a + 1}$

Step 2

To write $\frac{a}{a + 5}$ as a fraction with a common denominator, multiply by $\frac{4 a + 1}{4 a + 1}$ .

$\frac{a}{a + 5} \cdot \frac{4 a + 1}{4 a + 1} - \frac{a - 1}{4 a + 1}$

Step 3

To write $- \frac{a - 1}{4 a + 1}$ as a fraction with a common denominator, multiply by $\frac{a + 5}{a + 5}$ .

$\frac{a}{a + 5} \cdot \frac{4 a + 1}{4 a + 1} - \frac{a - 1}{4 a + 1} \cdot \frac{a + 5}{a + 5}$

Step 4

Write each expression with a common denominator of $(a + 5) (4 a + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{a + 5}$ by $\frac{4 a + 1}{4 a + 1}$ .

$\frac{a (4 a + 1)}{(a + 5) (4 a + 1)} - \frac{a - 1}{4 a + 1} \cdot \frac{a + 5}{a + 5}$

Step 4.2

Multiply $\frac{a - 1}{4 a + 1}$ by $\frac{a + 5}{a + 5}$ .

$\frac{a (4 a + 1)}{(a + 5) (4 a + 1)} - \frac{(a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 4.3

Reorder the factors of $(a + 5) (4 a + 1)$ .

$\frac{a (4 a + 1)}{(4 a + 1) (a + 5)} - \frac{(a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a (4 a + 1) - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a (4 a) + a \cdot 1 - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.2

Rewrite using the commutative property of multiplication.

$\frac{4 a \cdot a + a \cdot 1 - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.3

Multiply $a$ by $1$ .

$\frac{4 a \cdot a + a - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.4

Multiply $a$ by $a$ by adding the exponents.

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Step 6.4.1

Move $a$ .

$\frac{4 (a \cdot a) + a - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.4.2

Multiply $a$ by $a$ .

$\frac{4 a^{2} + a - (a - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.5

Apply the distributive property.

$\frac{4 a^{2} + a + (- a - - 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.6

Multiply $- 1$ by $- 1$ .

$\frac{4 a^{2} + a + (- a + 1) (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.7

Expand $(- a + 1) (a + 5)$ using the FOIL Method.

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Step 6.7.1

Apply the distributive property.

$\frac{4 a^{2} + a - a (a + 5) + 1 (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.7.2

Apply the distributive property.

$\frac{4 a^{2} + a - a \cdot a - a \cdot 5 + 1 (a + 5)}{(4 a + 1) (a + 5)}$

Step 6.7.3

Apply the distributive property.

$\frac{4 a^{2} + a - a \cdot a - a \cdot 5 + 1 a + 1 \cdot 5}{(4 a + 1) (a + 5)}$

Step 6.8

Simplify and combine like terms.

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Step 6.8.1

Simplify each term.

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Step 6.8.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 6.8.1.1.1

Move $a$ .

$\frac{4 a^{2} + a - (a \cdot a) - a \cdot 5 + 1 a + 1 \cdot 5}{(4 a + 1) (a + 5)}$

Step 6.8.1.1.2

Multiply $a$ by $a$ .

$\frac{4 a^{2} + a - a^{2} - a \cdot 5 + 1 a + 1 \cdot 5}{(4 a + 1) (a + 5)}$

Step 6.8.1.2

Multiply $5$ by $- 1$ .

$\frac{4 a^{2} + a - a^{2} - 5 a + 1 a + 1 \cdot 5}{(4 a + 1) (a + 5)}$

Step 6.8.1.3

Multiply $a$ by $1$ .

$\frac{4 a^{2} + a - a^{2} - 5 a + a + 1 \cdot 5}{(4 a + 1) (a + 5)}$

Step 6.8.1.4

Multiply $5$ by $1$ .

$\frac{4 a^{2} + a - a^{2} - 5 a + a + 5}{(4 a + 1) (a + 5)}$

Step 6.8.2

Add $- 5 a$ and $a$ .

$\frac{4 a^{2} + a - a^{2} - 4 a + 5}{(4 a + 1) (a + 5)}$

Step 6.9

Subtract $a^{2}$ from $4 a^{2}$ .

$\frac{3 a^{2} + a - 4 a + 5}{(4 a + 1) (a + 5)}$

Step 6.10

Subtract $4 a$ from $a$ .

$\frac{3 a^{2} - 3 a + 5}{(4 a + 1) (a + 5)}$