Pre-Algebra Examples

$\frac{80}{1 + r} = 70 + \frac{90}{(1 + r) \cdot 2}$

Step 1

Simplify $70 + \frac{90}{(1 + r) \cdot 2}$ .

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Step 1.1

Cancel the common factor of $90$ and $2$ .

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Step 1.1.1

Factor $2$ out of $90$ .

$\frac{80}{1 + r} = 70 + \frac{2 \cdot 45}{(1 + r) \cdot 2}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $(1 + r) \cdot 2$ .

$\frac{80}{1 + r} = 70 + \frac{2 \cdot 45}{2 \cdot (1 + r)}$

Step 1.1.2.2

Cancel the common factor.

$\frac{80}{1 + r} = 70 + \frac{2 \cdot 45}{2 \cdot (1 + r)}$

Step 1.1.2.3

Rewrite the expression.

$\frac{80}{1 + r} = 70 + \frac{45}{1 + r}$

Step 1.2

To write $70$ as a fraction with a common denominator, multiply by $\frac{1 + r}{1 + r}$ .

$\frac{80}{1 + r} = \frac{70 (1 + r)}{1 + r} + \frac{45}{1 + r}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{80}{1 + r} = \frac{70 (1 + r) + 45}{1 + r}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $5$ out of $70 (1 + r) + 45$ .

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Step 1.4.1.1

Factor $5$ out of $70 (1 + r)$ .

$\frac{80}{1 + r} = \frac{5 (14 (1 + r)) + 45}{1 + r}$

Step 1.4.1.2

Factor $5$ out of $45$ .

$\frac{80}{1 + r} = \frac{5 (14 (1 + r)) + 5 (9)}{1 + r}$

Step 1.4.1.3

Factor $5$ out of $5 (14 (1 + r)) + 5 (9)$ .

$\frac{80}{1 + r} = \frac{5 (14 (1 + r) + 9)}{1 + r}$

Step 1.4.2

Apply the distributive property.

$\frac{80}{1 + r} = \frac{5 (14 \cdot 1 + 14 r + 9)}{1 + r}$

Step 1.4.3

Multiply $14$ by $1$ .

$\frac{80}{1 + r} = \frac{5 (14 + 14 r + 9)}{1 + r}$

Step 1.4.4

Add $14$ and $9$ .

$\frac{80}{1 + r} = \frac{5 (14 r + 23)}{1 + r}$

Step 2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$80 = 5 (14 r + 23)$

Step 3

Rewrite the equation as $5 (14 r + 23) = 80$ .

$5 (14 r + 23) = 80$

Step 4

Divide each term in $5 (14 r + 23) = 80$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 (14 r + 23) = 80$ by $5$ .

$\frac{5 (14 r + 23)}{5} = \frac{80}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 (14 r + 23)}{5} = \frac{80}{5}$

Step 4.2.1.2

Divide $14 r + 23$ by $1$ .

$14 r + 23 = \frac{80}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $80$ by $5$ .

$14 r + 23 = 16$

Step 5

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

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Step 5.1

Subtract $23$ from both sides of the equation.

$14 r = 16 - 23$

Step 5.2

Subtract $23$ from $16$ .

$14 r = - 7$

Step 6

Divide each term in $14 r = - 7$ by $14$ and simplify.

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Step 6.1

Divide each term in $14 r = - 7$ by $14$ .

$\frac{14 r}{14} = \frac{- 7}{14}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $14$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{14 r}{14} = \frac{- 7}{14}$

Step 6.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 7}{14}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $- 7$ and $14$ .

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Step 6.3.1.1

Factor $7$ out of $- 7$ .

$r = \frac{7 (- 1)}{14}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $7$ out of $14$ .

$r = \frac{7 \cdot - 1}{7 \cdot 2}$

Step 6.3.1.2.2

Cancel the common factor.

$r = \frac{7 \cdot - 1}{7 \cdot 2}$

Step 6.3.1.2.3

Rewrite the expression.

$r = \frac{- 1}{2}$

Step 6.3.2

Move the negative in front of the fraction.

$r = - \frac{1}{2}$