Basic Math Examples

$103.7 = (\frac{1}{1 + r}) \cdot \frac{30.4}{r}$

Step 1

Rewrite the equation as $(\frac{1}{1 + r}) \cdot \frac{30.4}{r} = 103.7$ .

$(\frac{1}{1 + r}) \cdot \frac{30.4}{r} = 103.7$

Step 2

Factor each term.

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Step 2.1

Multiply $\frac{1}{1 + r}$ by $\frac{30.4}{r}$ .

$(\frac{1}{1 + r}) \frac{30.4}{r} = 103.7$

Step 2.2

Multiply $\frac{1}{1 + r}$ by $\frac{30.4}{r}$ .

$\frac{30.4}{(1 + r) r} = 103.7$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(1 + r) r, 1$

Step 3.2

The LCM of one and any expression is the expression.

$(1 + r) r$

Step 4

Multiply each term in $\frac{30.4}{(1 + r) r} = 103.7$ by $(1 + r) r$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{30.4}{(1 + r) r} = 103.7$ by $(1 + r) r$ .

$\frac{30.4}{(1 + r) r} ((1 + r) r) = 103.7 ((1 + r) r)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $(1 + r) r$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{30.4}{(1 + r) r} ((1 + r) r) = 103.7 ((1 + r) r)$

Step 4.2.1.2

Rewrite the expression.

$30.4 = 103.7 ((1 + r) r)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$30.4 = 103.7 (1 r + r \cdot r)$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Multiply $r$ by $1$ .

$30.4 = 103.7 (r + r \cdot r)$

Step 4.3.2.2

Multiply $r$ by $r$ .

$30.4 = 103.7 (r + r^{2})$

Step 4.3.3

Apply the distributive property.

$30.4 = 103.7 r + 103.7 r^{2}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $103.7 r + 103.7 r^{2} = 30.4$ .

$103.7 r + 103.7 r^{2} = 30.4$

Step 5.2

Subtract $30.4$ from both sides of the equation.

$103.7 r + 103.7 r^{2} - 30.4 = 0$

Step 5.3

Factor the left side of the equation.

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Step 5.3.1

Factor $0.1$ out of $103.7 r + 103.7 r^{2} - 30.4$ .

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Step 5.3.1.1

Factor $0.1$ out of $103.7 r$ .

$0.1 (1037 r) + 103.7 r^{2} - 30.4 = 0$

Step 5.3.1.2

Factor $0.1$ out of $103.7 r^{2}$ .

$0.1 (1037 r) + 0.1 (1037 r^{2}) - 30.4 = 0$

Step 5.3.1.3

Factor $0.1$ out of $- 30.4$ .

$0.1 (1037 r) + 0.1 (1037 r^{2}) + 0.1 (- 304) = 0$

Step 5.3.1.4

Factor $0.1$ out of $0.1 (1037 r) + 0.1 (1037 r^{2})$ .

$0.1 (1037 r + 1037 r^{2}) + 0.1 (- 304) = 0$

Step 5.3.1.5

Factor $0.1$ out of $0.1 (1037 r + 1037 r^{2}) + 0.1 (- 304)$ .

$0.1 (1037 r + 1037 r^{2} - 304) = 0$

Step 5.3.2

Reorder terms.

$0.1 (1037 r^{2} + 1037 r - 304) = 0$

Step 5.4

Divide each term in $0.1 (1037 r^{2} + 1037 r - 304) = 0$ by $0.1$ and simplify.

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Step 5.4.1

Divide each term in $0.1 (1037 r^{2} + 1037 r - 304) = 0$ by $0.1$ .

$\frac{0.1 (1037 r^{2} + 1037 r - 304)}{0.1} = \frac{0}{0.1}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $0.1$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{0.1 (1037 r^{2} + 1037 r - 304)}{0.1} = \frac{0}{0.1}$

Step 5.4.2.1.2

Divide $1037 r^{2} + 1037 r - 304$ by $1$ .

$1037 r^{2} + 1037 r - 304 = \frac{0}{0.1}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $0$ by $0.1$ .

$1037 r^{2} + 1037 r - 304 = 0$

Step 5.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.6

Substitute the values $a = 1037$ , $b = 1037$ , and $c = - 304$ into the quadratic formula and solve for $r$ .

$\frac{- 1037 \pm \sqrt{1037^{2} - 4 \cdot (1037 \cdot - 304)}}{2 \cdot 1037}$

Step 5.7

Simplify.

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Step 5.7.1

Simplify the numerator.

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Step 5.7.1.1

Raise $1037$ to the power of $2$ .

$r = \frac{- 1037 \pm \sqrt{1075369 - 4 \cdot 1037 \cdot - 304}}{2 \cdot 1037}$

Step 5.7.1.2

Multiply $- 4 \cdot 1037 \cdot - 304$ .

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Step 5.7.1.2.1

Multiply $- 4$ by $1037$ .

$r = \frac{- 1037 \pm \sqrt{1075369 - 4148 \cdot - 304}}{2 \cdot 1037}$

Step 5.7.1.2.2

Multiply $- 4148$ by $- 304$ .

$r = \frac{- 1037 \pm \sqrt{1075369 + 1260992}}{2 \cdot 1037}$

Step 5.7.1.3

Add $1075369$ and $1260992$ .

$r = \frac{- 1037 \pm \sqrt{2336361}}{2 \cdot 1037}$

Step 5.7.2

Multiply $2$ by $1037$ .

$r = \frac{- 1037 \pm \sqrt{2336361}}{2074}$

Step 5.8

The final answer is the combination of both solutions.

$r = - \frac{1037 - \sqrt{2336361}}{2074}, - \frac{1037 + \sqrt{2336361}}{2074}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{1037 - \sqrt{2336361}}{2074}, - \frac{1037 + \sqrt{2336361}}{2074}$

Decimal Form:

$r = 0.23698936 \dots, - 1.23698936 \dots$