Finite Math Examples

$(\frac{\frac{m}{n}}{k}) = (\frac{\frac{m}{n}}{k - 1}) \cdot \frac{m - (k - 1) n}{k \cdot n}$

Step 1

Multiply both sides by $k$ .

$\frac{\frac{m}{n}}{k} k = (\frac{\frac{m}{n}}{k - 1}) \cdot \frac{m - (k - 1) n}{k \cdot n} k$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $k$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{\frac{m}{n}}{k} k = (\frac{\frac{m}{n}}{k - 1}) \cdot \frac{m - (k - 1) n}{k \cdot n} k$

Step 2.1.1.2

Rewrite the expression.

$\frac{m}{n} = (\frac{\frac{m}{n}}{k - 1}) \cdot \frac{m - (k - 1) n}{k \cdot n} k$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{\frac{m}{n}}{k - 1}) \cdot \frac{m - (k - 1) n}{k \cdot n} k$ .

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Step 2.2.1.1

Simplify the numerator.

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Step 2.2.1.1.1

Apply the distributive property.

$\frac{m}{n} = \frac{\frac{m}{n}}{k - 1} \cdot \frac{m + (- k - - 1) n}{k \cdot n} k$

Step 2.2.1.1.2

Multiply $- 1$ by $- 1$ .

$\frac{m}{n} = \frac{\frac{m}{n}}{k - 1} \cdot \frac{m + (- k + 1) n}{k \cdot n} k$

Step 2.2.1.1.3

Apply the distributive property.

$\frac{m}{n} = \frac{\frac{m}{n}}{k - 1} \cdot \frac{m - k n + 1 n}{k \cdot n} k$

Step 2.2.1.1.4

Multiply $n$ by $1$ .

$\frac{m}{n} = \frac{\frac{m}{n}}{k - 1} \cdot \frac{m - k n + n}{k \cdot n} k$

Step 2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{m}{n} = \frac{m}{n} \cdot \frac{1}{k - 1} \cdot \frac{m - k n + n}{k \cdot n} k$

Step 2.2.1.3

Multiply $\frac{m}{n}$ by $\frac{1}{k - 1}$ .

$\frac{m}{n} = \frac{m}{n (k - 1)} \cdot \frac{m - k n + n}{k \cdot n} k$

Step 2.2.1.4

Multiply $\frac{m}{n (k - 1)} \cdot \frac{m - k n + n}{k n}$ .

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Step 2.2.1.4.1

Multiply $\frac{m}{n (k - 1)}$ by $\frac{m - k n + n}{k n}$ .

$\frac{m}{n} = \frac{m (m - k n + n)}{n (k - 1) (k n)} k$

Step 2.2.1.4.2

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

$\frac{m}{n} = \frac{m (m - k n + n)}{n^{1} n (k - 1) k} k$

Step 2.2.1.4.3

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

$\frac{m}{n} = \frac{m (m - k n + n)}{n^{1} n^{1} (k - 1) k} k$

Step 2.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{m}{n} = \frac{m (m - k n + n)}{n^{1 + 1} (k - 1) k} k$

Step 2.2.1.4.5

Add $1$ and $1$ .

$\frac{m}{n} = \frac{m (m - k n + n)}{n^{2} (k - 1) k} k$

Step 2.2.1.5

Cancel the common factor of $k$ .

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Step 2.2.1.5.1

Factor $k$ out of $n^{2} (k - 1) k$ .

$\frac{m}{n} = \frac{m (m - k n + n)}{k (n^{2} (k - 1))} k$

Step 2.2.1.5.2

Cancel the common factor.

$\frac{m}{n} = \frac{m (m - k n + n)}{k (n^{2} (k - 1))} k$

Step 2.2.1.5.3

Rewrite the expression.

$\frac{m}{n} = \frac{m (m - k n + n)}{n^{2} (k - 1)}$

Step 3

Solve for $m$ .

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Step 3.1

Multiply both sides by $n$ .

$\frac{m}{n} n = \frac{m (m - k n + n)}{n^{2} (k - 1)} n$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $n$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{m}{n} n = \frac{m (m - k n + n)}{n^{2} (k - 1)} n$

Step 3.2.1.1.2

Rewrite the expression.

$m = \frac{m (m - k n + n)}{n^{2} (k - 1)} n$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $n$ .

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Step 3.2.2.1.1

Factor $n$ out of $n^{2} (k - 1)$ .

$m = \frac{m (m - k n + n)}{n (n (k - 1))} n$

Step 3.2.2.1.2

Cancel the common factor.

$m = \frac{m (m - k n + n)}{n (n (k - 1))} n$

Step 3.2.2.1.3

Rewrite the expression.

$m = \frac{m (m - k n + n)}{n (k - 1)}$

Step 3.3

Solve for $m$ .

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Step 3.3.1

Multiply both sides by $n (k - 1)$ .

$m (n (k - 1)) = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Simplify $m (n (k - 1))$ .

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Step 3.3.2.1.1.1

Simplify by multiplying through.

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Step 3.3.2.1.1.1.1

Apply the distributive property.

$m (n k + n \cdot - 1) = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.1.2

Move $- 1$ to the left of $n$ .

$m (n k - 1 \cdot n) = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.2

Rewrite $- 1 n$ as $- n$ .

$m (n k - n) = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.3

Simplify by multiplying through.

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Step 3.3.2.1.1.3.1

Apply the distributive property.

$m (n k) + m (- n) = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.3.2

Simplify the expression.

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Step 3.3.2.1.1.3.2.1

Rewrite using the commutative property of multiplication.

$m n k - m n = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.3.2.2

Move $n$ .

$m k n - m n = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.1.1.3.2.3

Reorder $m$ and $k$ .

$k m n - m n = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $\frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$ .

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Step 3.3.2.2.1.1

Cancel the common factor of $n (k - 1)$ .

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Step 3.3.2.2.1.1.1

Cancel the common factor.

$k m n - m n = \frac{m (m - k n + n)}{n (k - 1)} (n (k - 1))$

Step 3.3.2.2.1.1.2

Rewrite the expression.

$k m n - m n = m (m - k n + n)$

Step 3.3.2.2.1.2

Apply the distributive property.

$k m n - m n = m \cdot m + m (- k n) + m n$

Step 3.3.2.2.1.3

Simplify.

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Step 3.3.2.2.1.3.1

Multiply $m$ by $m$ .

$k m n - m n = m^{2} + m (- k n) + m n$

Step 3.3.2.2.1.3.2

Rewrite using the commutative property of multiplication.

$k m n - m n = m^{2} - m k n + m n$

Step 3.3.2.2.1.4

Move $m$ .

$k m n - m n = m^{2} - k m n + m n$

Step 3.3.2.2.1.5

Move $m^{2}$ .

$k m n - m n = - k m n + m n + m^{2}$

Step 3.3.3

Solve for $m$ .

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Step 3.3.3.1

Since $m$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- k m n + m n + m^{2} = k m n - m n$

Step 3.3.3.2

Move all terms containing $m$ to the left side of the equation.

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Step 3.3.3.2.1

Subtract $k m n$ from both sides of the equation.

$- k m n + m n + m^{2} - k m n = - m n$

Step 3.3.3.2.2

Add $m n$ to both sides of the equation.

$- k m n + m n + m^{2} - k m n + m n = 0$

Step 3.3.3.2.3

Subtract $k m n$ from $- k m n$ .

$m n + m^{2} - 2 k m n + m n = 0$

Step 3.3.3.2.4

Add $m n$ and $m n$ .

$m^{2} - 2 k m n + 2 m n = 0$

Step 3.3.3.3

Factor $m$ out of $m^{2} - 2 k m n + 2 m n$ .

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Step 3.3.3.3.1

Factor $m$ out of $m^{2}$ .

$m \cdot m - 2 k m n + 2 m n = 0$

Step 3.3.3.3.2

Factor $m$ out of $- 2 k m n$ .

$m \cdot m + m (- 2 k n) + 2 m n = 0$

Step 3.3.3.3.3

Factor $m$ out of $2 m n$ .

$m \cdot m + m (- 2 k n) + m (2 n) = 0$

Step 3.3.3.3.4

Factor $m$ out of $m \cdot m + m (- 2 k n)$ .

$m (m - 2 k n) + m (2 n) = 0$

Step 3.3.3.3.5

Factor $m$ out of $m (m - 2 k n) + m (2 n)$ .

$m (m - 2 k n + 2 n) = 0$

Step 3.3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$m = 0$

$m - 2 k n + 2 n = 0$

Step 3.3.3.5

Set $m$ equal to $0$ .

$m = 0$

Step 3.3.3.6

Set $m - 2 k n + 2 n$ equal to $0$ and solve for $m$ .

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Step 3.3.3.6.1

Set $m - 2 k n + 2 n$ equal to $0$ .

$m - 2 k n + 2 n = 0$

Step 3.3.3.6.2

Move all terms not containing $m$ to the right side of the equation.

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Step 3.3.3.6.2.1

Add $2 k n$ to both sides of the equation.

$m + 2 n = 2 k n$

Step 3.3.3.6.2.2

Subtract $2 n$ from both sides of the equation.

$m = 2 k n - 2 n$

Step 3.3.3.7

The final solution is all the values that make $m (m - 2 k n + 2 n) = 0$ true.

$m = 0$

$m = 2 k n - 2 n$

$m = 0$

$m = 2 k n - 2 n$

$m = 0$

$m = 2 k n - 2 n$

$m = 0$

$m = 2 k n - 2 n$