Pre-Algebra Examples

$\frac{\frac{9 - s}{3 r s}}{\frac{s^{2} + 2 s - 15}{5 r s}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 - s}{3 r s} \cdot \frac{5 r s}{s^{2} + 2 s - 15}$

Step 2

Cancel the common factor of $r s$ .

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Step 2.1

Factor $r s$ out of $3 r s$ .

$\frac{9 - s}{r s (3)} \cdot \frac{5 r s}{s^{2} + 2 s - 15}$

Step 2.2

Factor $r s$ out of $5 r s$ .

$\frac{9 - s}{r s (3)} \cdot \frac{r s (5)}{s^{2} + 2 s - 15}$

Step 2.3

Cancel the common factor.

$\frac{9 - s}{r s \cdot 3} \cdot \frac{r s \cdot 5}{s^{2} + 2 s - 15}$

Step 2.4

Rewrite the expression.

$\frac{9 - s}{3} \cdot \frac{5}{s^{2} + 2 s - 15}$

Step 3

Multiply $\frac{9 - s}{3}$ by $\frac{5}{s^{2} + 2 s - 15}$ .

$\frac{(9 - s) \cdot 5}{3 (s^{2} + 2 s - 15)}$

Step 4

Factor $s^{2} + 2 s - 15$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(9 - s) \cdot 5}{3 ((s - 3) (s + 5))}$

$\frac{(9 - s) \cdot 5}{3 (s - 3) (s + 5)}$

Step 5

Move $5$ to the left of $9 - s$ .

$\frac{5 (9 - s)}{3 (s - 3) (s + 5)}$

Step 6

Apply the distributive property.

$\frac{5 \cdot 9 + 5 (- s)}{3 (s - 3) (s + 5)}$

Step 7

Multiply $5$ by $9$ .

$\frac{45 + 5 (- s)}{3 (s - 3) (s + 5)}$

Step 8

Multiply $- 1$ by $5$ .

$\frac{45 - 5 s}{3 (s - 3) (s + 5)}$

Step 9

Split the fraction $\frac{45 - 5 s}{3 (s - 3) (s + 5)}$ into two fractions.

$\frac{45}{3 (s - 3) (s + 5)} + \frac{- 5 s}{3 (s - 3) (s + 5)}$

Step 10

Cancel the common factor of $45$ and $3$ .

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Step 10.1

Factor $3$ out of $45$ .

$\frac{3 \cdot 15}{3 (s - 3) (s + 5)} + \frac{- 5 s}{3 (s - 3) (s + 5)}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $3 (s - 3) (s + 5)$ .

$\frac{3 \cdot 15}{3 ((s - 3) (s + 5))} + \frac{- 5 s}{3 (s - 3) (s + 5)}$

Step 10.2.2

Cancel the common factor.

$\frac{3 \cdot 15}{3 ((s - 3) (s + 5))} + \frac{- 5 s}{3 (s - 3) (s + 5)}$

Step 10.2.3

Rewrite the expression.

$\frac{15}{(s - 3) (s + 5)} + \frac{- 5 s}{3 (s - 3) (s + 5)}$

Step 11

Move the negative in front of the fraction.

$\frac{15}{(s - 3) (s + 5)} - \frac{5 s}{3 (s - 3) (s + 5)}$