Basic Math Examples

$A = \frac{P {(1 + \frac{r}{n})}^{n t} - 1}{\frac{r}{n}}$

Step 1

Rewrite the equation as $\frac{P {(1 + \frac{r}{n})}^{n t} - 1}{\frac{r}{n}} = A$ .

$\frac{P {(1 + \frac{r}{n})}^{n t} - 1}{\frac{r}{n}} = A$

Step 2

Multiply both sides by $\frac{r}{n}$ .

$\frac{P {(1 + \frac{r}{n})}^{n t} - 1}{\frac{r}{n}} \cdot \frac{r}{n} = A \frac{r}{n}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{P {(1 + \frac{r}{n})}^{n t} - 1}{\frac{r}{n}} \cdot \frac{r}{n}$ .

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Step 3.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$(P {(1 + \frac{r}{n})}^{n t} - 1) \frac{n}{r} \frac{r}{n} = A \frac{r}{n}$

Step 3.1.1.2

Multiply $P {(1 + \frac{r}{n})}^{n t} - 1$ by $\frac{n}{r}$ .

$\frac{(P {(1 + \frac{r}{n})}^{n t} - 1) n}{r} \cdot \frac{r}{n} = A \frac{r}{n}$

Step 3.1.1.3

Combine.

$\frac{(P {(1 + \frac{r}{n})}^{n t} - 1) n r}{r n} = A \frac{r}{n}$

Step 3.1.1.4

Cancel the common factor of $n$ .

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Step 3.1.1.4.1

Cancel the common factor.

$\frac{(P {(1 + \frac{r}{n})}^{n t} - 1) n r}{r n} = A \frac{r}{n}$

Step 3.1.1.4.2

Rewrite the expression.

$\frac{(P {(1 + \frac{r}{n})}^{n t} - 1) r}{r} = A \frac{r}{n}$

Step 3.1.1.5

Cancel the common factor of $r$ .

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Step 3.1.1.5.1

Cancel the common factor.

$\frac{(P {(1 + \frac{r}{n})}^{n t} - 1) r}{r} = A \frac{r}{n}$

Step 3.1.1.5.2

Divide $P {(1 + \frac{r}{n})}^{n t} - 1$ by $1$ .

$P {(1 + \frac{r}{n})}^{n t} - 1 = A \frac{r}{n}$

Step 3.1.1.6

Reorder $1$ and $\frac{r}{n}$ .

$P {(\frac{r}{n} + 1)}^{n t} - 1 = A \frac{r}{n}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $A$ and $\frac{r}{n}$ .

$P {(\frac{r}{n} + 1)}^{n t} - 1 = \frac{A r}{n}$

Step 4

Solve for $P$ .

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Step 4.1

Add $1$ to both sides of the equation.

$P {(\frac{r}{n} + 1)}^{n t} = \frac{A r}{n} + 1$

Step 4.2

Divide each term in $P {(\frac{r}{n} + 1)}^{n t} = \frac{A r}{n} + 1$ by ${(\frac{r}{n} + 1)}^{n t}$ and simplify.

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Step 4.2.1

Divide each term in $P {(\frac{r}{n} + 1)}^{n t} = \frac{A r}{n} + 1$ by ${(\frac{r}{n} + 1)}^{n t}$ .

$\frac{P {(\frac{r}{n} + 1)}^{n t}}{{(\frac{r}{n} + 1)}^{n t}} = \frac{\frac{A r}{n}}{{(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of ${(\frac{r}{n} + 1)}^{n t}$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{P {(\frac{r}{n} + 1)}^{n t}}{{(\frac{r}{n} + 1)}^{n t}} = \frac{\frac{A r}{n}}{{(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.2.1.2

Divide $P$ by $1$ .

$P = \frac{\frac{A r}{n}}{{(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$P = \frac{A r}{n} \cdot \frac{1}{{(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.2

Combine.

$P = \frac{A r \cdot 1}{n {(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.3

Multiply $A$ by $1$ .

$P = \frac{A r}{n {(\frac{r}{n} + 1)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.4

Simplify the denominator.

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Step 4.2.3.1.4.1

Write $1$ as a fraction with a common denominator.

$P = \frac{A r}{n {(\frac{r}{n} + \frac{n}{n})}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.4.2

Combine the numerators over the common denominator.

$P = \frac{A r}{n {(\frac{r + n}{n})}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.4.3

Apply the product rule to $\frac{r + n}{n}$ .

$P = \frac{A r}{n \frac{{(r + n)}^{n t}}{n^{n t}}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.5

Combine $n$ and $\frac{{(r + n)}^{n t}}{n^{n t}}$ .

$P = \frac{A r}{\frac{n {(r + n)}^{n t}}{n^{n t}}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$P = A r \frac{n^{n t}}{n {(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.7

Cancel the common factor of $n^{n t}$ and $n$ .

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Step 4.2.3.1.7.1

Factor $n$ out of $n^{n t}$ .

$P = A r \frac{n \cdot n^{n t - 1}}{n {(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.7.2

Cancel the common factors.

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Step 4.2.3.1.7.2.1

Factor $n$ out of $n {(r + n)}^{n t}$ .

$P = A r \frac{n \cdot n^{n t - 1}}{n ({(r + n)}^{n t})} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.7.2.2

Cancel the common factor.

$P = A r \frac{n \cdot n^{n t - 1}}{n {(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.7.2.3

Rewrite the expression.

$P = A r \frac{n^{n t - 1}}{{(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.8

Multiply $A r \frac{n^{n t - 1}}{{(r + n)}^{n t}}$ .

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Step 4.2.3.1.8.1

Combine $A$ and $\frac{n^{n t - 1}}{{(r + n)}^{n t}}$ .

$P = r \frac{A n^{n t - 1}}{{(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.8.2

Combine $r$ and $\frac{A n^{n t - 1}}{{(r + n)}^{n t}}$ .

$P = \frac{r (A n^{n t - 1})}{{(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + 1)}^{n t}}$

Step 4.2.3.1.9

Simplify the denominator.

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Step 4.2.3.1.9.1

Write $1$ as a fraction with a common denominator.

$P = \frac{r A n^{n t - 1}}{{(r + n)}^{n t}} + \frac{1}{{(\frac{r}{n} + \frac{n}{n})}^{n t}}$

Step 4.2.3.1.9.2

Combine the numerators over the common denominator.

$P = \frac{r A n^{n t - 1}}{{(r + n)}^{n t}} + \frac{1}{{(\frac{r + n}{n})}^{n t}}$

Step 4.2.3.1.9.3

Apply the product rule to $\frac{r + n}{n}$ .

$P = \frac{r A n^{n t - 1}}{{(r + n)}^{n t}} + \frac{1}{\frac{{(r + n)}^{n t}}{n^{n t}}}$

Step 4.2.3.1.10

Multiply the numerator by the reciprocal of the denominator.

$P = \frac{r A n^{n t - 1}}{{(r + n)}^{n t}} + 1 \frac{n^{n t}}{{(r + n)}^{n t}}$

Step 4.2.3.1.11

Multiply $\frac{n^{n t}}{{(r + n)}^{n t}}$ by $1$ .

$P = \frac{r A n^{n t - 1}}{{(r + n)}^{n t}} + \frac{n^{n t}}{{(r + n)}^{n t}}$

Step 4.2.3.2

Combine the numerators over the common denominator.

$P = \frac{r A n^{n t - 1} + n^{n t}}{{(r + n)}^{n t}}$