Basic Math Examples

$\frac{3}{11} = \frac{1}{a + \frac{1}{b + \frac{1}{c}}}$

Step 1

Rewrite the equation as $\frac{1}{a + \frac{1}{b + \frac{1}{c}}} = \frac{3}{11}$ .

$\frac{1}{a + \frac{1}{b + \frac{1}{c}}} = \frac{3}{11}$

Step 2

Simplify both sides.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

To write $b$ as a fraction with a common denominator, multiply by $\frac{c}{c}$ .

$\frac{1}{a + \frac{1}{\frac{b c}{c} + \frac{1}{c}}} = \frac{3}{11}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{1}{a + \frac{1}{\frac{b c + 1}{c}}} = \frac{3}{11}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{a + 1 \frac{c}{b c + 1}} = \frac{3}{11}$

Step 2.2.1.2

Multiply $\frac{c}{b c + 1}$ by $1$ .

$\frac{1}{a + \frac{c}{b c + 1}} = \frac{3}{11}$

Step 2.2.2

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

$\frac{1}{\frac{a (b c + 1)}{b c + 1} + \frac{c}{b c + 1}} = \frac{3}{11}$

Step 2.2.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{a (b c + 1) + c}{b c + 1}} = \frac{3}{11}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Apply the distributive property.

$\frac{1}{\frac{a (b c) + a \cdot 1 + c}{b c + 1}} = \frac{3}{11}$

Step 2.2.4.2

Multiply $a$ by $1$ .

$\frac{1}{\frac{a b c + a + c}{b c + 1}} = \frac{3}{11}$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{b c + 1}{a b c + a + c} = \frac{3}{11}$

Step 2.4

Multiply $\frac{b c + 1}{a b c + a + c}$ by $1$ .

$\frac{b c + 1}{a b c + a + c} = \frac{3}{11}$

Step 3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(b c + 1) \cdot 11 = (a b c + a + c) \cdot 3$

Step 4

Solve the equation for $a$ .

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Step 4.1

Rewrite the equation as $(a b c + a + c) \cdot 3 = (b c + 1) \cdot 11$ .

$(a b c + a + c) \cdot 3 = (b c + 1) \cdot 11$

Step 4.2

Divide each term in $(a b c + a + c) \cdot 3 = (b c + 1) \cdot 11$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $(a b c + a + c) \cdot 3 = (b c + 1) \cdot 11$ by $3$ .

$\frac{(a b c + a + c) \cdot 3}{3} = \frac{(b c + 1) \cdot 11}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{(a b c + a + c) \cdot 3}{3} = \frac{(b c + 1) \cdot 11}{3}$

Step 4.2.2.1.2

Divide $a b c + a + c$ by $1$ .

$a b c + a + c = \frac{(b c + 1) \cdot 11}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move $11$ to the left of $b c + 1$ .

$a b c + a + c = \frac{11 (b c + 1)}{3}$

Step 4.3

Subtract $c$ from both sides of the equation.

$a b c + a = \frac{11 (b c + 1)}{3} - c$

Step 4.4

Factor $a$ out of $a b c + a$ .

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Step 4.4.1

Factor $a$ out of $a b c$ .

$a (b c) + a = \frac{11 (b c + 1)}{3} - c$

Step 4.4.2

Raise $a$ to the power of $1$ .

$a (b c) + a = \frac{11 (b c + 1)}{3} - c$

Step 4.4.3

Factor $a$ out of $a^{1}$ .

$a (b c) + a \cdot 1 = \frac{11 (b c + 1)}{3} - c$

Step 4.4.4

Factor $a$ out of $a (b c) + a \cdot 1$ .

$a (b c + 1) = \frac{11 (b c + 1)}{3} - c$

Step 4.5

Divide each term in $a (b c + 1) = \frac{11 (b c + 1)}{3} - c$ by $b c + 1$ and simplify.

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Step 4.5.1

Divide each term in $a (b c + 1) = \frac{11 (b c + 1)}{3} - c$ by $b c + 1$ .

$\frac{a (b c + 1)}{b c + 1} = \frac{\frac{11 (b c + 1)}{3}}{b c + 1} + \frac{- c}{b c + 1}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $b c + 1$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{a (b c + 1)}{b c + 1} = \frac{\frac{11 (b c + 1)}{3}}{b c + 1} + \frac{- c}{b c + 1}$

Step 4.5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\frac{11 (b c + 1)}{3}}{b c + 1} + \frac{- c}{b c + 1}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Combine the numerators over the common denominator.

$a = \frac{\frac{11 (b c + 1)}{3} - c}{b c + 1}$

Step 4.5.3.2

To write $- c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = \frac{\frac{11 (b c + 1)}{3} - c \cdot \frac{3}{3}}{b c + 1}$

Step 4.5.3.3

Simplify terms.

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Step 4.5.3.3.1

Combine $- c$ and $\frac{3}{3}$ .

$a = \frac{\frac{11 (b c + 1)}{3} + \frac{- c \cdot 3}{3}}{b c + 1}$

Step 4.5.3.3.2

Combine the numerators over the common denominator.

$a = \frac{\frac{11 (b c + 1) - c \cdot 3}{3}}{b c + 1}$

Step 4.5.3.4

Simplify the numerator.

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Step 4.5.3.4.1

Apply the distributive property.

$a = \frac{\frac{11 (b c) + 11 \cdot 1 - c \cdot 3}{3}}{b c + 1}$

Step 4.5.3.4.2

Multiply $11$ by $1$ .

$a = \frac{\frac{11 b c + 11 - c \cdot 3}{3}}{b c + 1}$

Step 4.5.3.4.3

Multiply $3$ by $- 1$ .

$a = \frac{\frac{11 b c + 11 - 3 c}{3}}{b c + 1}$

Step 4.5.3.5

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{11 b c + 11 - 3 c}{3} \cdot \frac{1}{b c + 1}$

Step 4.5.3.6

Multiply $\frac{11 b c + 11 - 3 c}{3}$ by $\frac{1}{b c + 1}$ .

$a = \frac{11 b c + 11 - 3 c}{3 (b c + 1)}$