Basic Math Examples

$\frac{r}{r + s} + \frac{s}{r - s}$

Step 1

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

$\frac{r}{r + s} \cdot \frac{r - s}{r - s} + \frac{s}{r - s}$

Step 2

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

$\frac{r}{r + s} \cdot \frac{r - s}{r - s} + \frac{s}{r - s} \cdot \frac{r + s}{r + s}$

Step 3

Write each expression with a common denominator of $(r + s) (r - s)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{r}{r + s}$ by $\frac{r - s}{r - s}$ .

$\frac{r (r - s)}{(r + s) (r - s)} + \frac{s}{r - s} \cdot \frac{r + s}{r + s}$

Step 3.2

Multiply $\frac{s}{r - s}$ by $\frac{r + s}{r + s}$ .

$\frac{r (r - s)}{(r + s) (r - s)} + \frac{s (r + s)}{(r - s) (r + s)}$

Step 3.3

Reorder the factors of $(r - s) (r + s)$ .

$\frac{r (r - s)}{(r + s) (r - s)} + \frac{s (r + s)}{(r + s) (r - s)}$

Step 4

Combine the numerators over the common denominator.

$\frac{r (r - s) + s (r + s)}{(r + s) (r - s)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{r \cdot r + r (- s) + s (r + s)}{(r + s) (r - s)}$

Step 5.2

Multiply $r$ by $r$ .

$\frac{r^{2} + r (- s) + s (r + s)}{(r + s) (r - s)}$

Step 5.3

Rewrite using the commutative property of multiplication.

$\frac{r^{2} - r s + s (r + s)}{(r + s) (r - s)}$

Step 5.4

Apply the distributive property.

$\frac{r^{2} - r s + s r + s \cdot s}{(r + s) (r - s)}$

Step 5.5

Multiply $s$ by $s$ .

$\frac{r^{2} - r s + s r + s^{2}}{(r + s) (r - s)}$

Step 5.6

Add $- r s$ and $s r$ .

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Step 5.6.1

Reorder $s$ and $r$ .

$\frac{r^{2} - r s + r s + s^{2}}{(r + s) (r - s)}$

Step 5.6.2

Add $- r s$ and $r s$ .

$\frac{r^{2} + 0 + s^{2}}{(r + s) (r - s)}$

Step 5.7

Add $r^{2}$ and $0$ .

$\frac{r^{2} + s^{2}}{(r + s) (r - s)}$