Pre-Algebra Examples

$2420 = 2000 {(1 + r)}^{2}$

Step 1

Rewrite the equation as $2000 {(1 + r)}^{2} = 2420$ .

$2000 {(1 + r)}^{2} = 2420$

Step 2

Divide each term in $2000 {(1 + r)}^{2} = 2420$ by $2000$ and simplify.

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Step 2.1

Divide each term in $2000 {(1 + r)}^{2} = 2420$ by $2000$ .

$\frac{2000 {(1 + r)}^{2}}{2000} = \frac{2420}{2000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2000 {(1 + r)}^{2}}{2000} = \frac{2420}{2000}$

Step 2.2.1.2

Divide ${(1 + r)}^{2}$ by $1$ .

${(1 + r)}^{2} = \frac{2420}{2000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2420$ and $2000$ .

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Step 2.3.1.1

Factor $20$ out of $2420$ .

${(1 + r)}^{2} = \frac{20 (121)}{2000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $20$ out of $2000$ .

${(1 + r)}^{2} = \frac{20 \cdot 121}{20 \cdot 100}$

Step 2.3.1.2.2

Cancel the common factor.

${(1 + r)}^{2} = \frac{20 \cdot 121}{20 \cdot 100}$

Step 2.3.1.2.3

Rewrite the expression.

${(1 + r)}^{2} = \frac{121}{100}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$1 + r = \pm \sqrt{\frac{121}{100}}$

Step 4

Simplify $\pm \sqrt{\frac{121}{100}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{121}{100}}$ as $\frac{\sqrt{121}}{\sqrt{100}}$ .

$1 + r = \pm \frac{\sqrt{121}}{\sqrt{100}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $121$ as $11^{2}$ .

$1 + r = \pm \frac{\sqrt{11^{2}}}{\sqrt{100}}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$1 + r = \pm \frac{11}{\sqrt{100}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $100$ as $10^{2}$ .

$1 + r = \pm \frac{11}{\sqrt{10^{2}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$1 + r = \pm \frac{11}{10}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$1 + r = \frac{11}{10}$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$r = \frac{11}{10} - 1$

Step 5.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$r = \frac{11}{10} - 1 \cdot \frac{10}{10}$

Step 5.2.3

Combine $- 1$ and $\frac{10}{10}$ .

$r = \frac{11}{10} + \frac{- 1 \cdot 10}{10}$

Step 5.2.4

Combine the numerators over the common denominator.

$r = \frac{11 - 1 \cdot 10}{10}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 1$ by $10$ .

$r = \frac{11 - 10}{10}$

Step 5.2.5.2

Subtract $10$ from $11$ .

$r = \frac{1}{10}$

Step 5.3

Next, use the negative value of the $\pm$ to find the second solution.

$1 + r = - \frac{11}{10}$

Step 5.4

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

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Step 5.4.1

Subtract $1$ from both sides of the equation.

$r = - \frac{11}{10} - 1$

Step 5.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$r = - \frac{11}{10} - 1 \cdot \frac{10}{10}$

Step 5.4.3

Combine $- 1$ and $\frac{10}{10}$ .

$r = - \frac{11}{10} + \frac{- 1 \cdot 10}{10}$

Step 5.4.4

Combine the numerators over the common denominator.

$r = \frac{- 11 - 1 \cdot 10}{10}$

Step 5.4.5

Simplify the numerator.

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Step 5.4.5.1

Multiply $- 1$ by $10$ .

$r = \frac{- 11 - 10}{10}$

Step 5.4.5.2

Subtract $10$ from $- 11$ .

$r = \frac{- 21}{10}$

Step 5.4.6

Move the negative in front of the fraction.

$r = - \frac{21}{10}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{1}{10}, - \frac{21}{10}$