Solve a/a+b+b/a+b*b/a+2b

$$\frac { a } { a + b } + \frac { b } { a + b } \cdot \frac { b } { a + 2 b }$$

Evaluate

$$\frac{a+b}{a+2b}$$

Short Solution Steps

Multiply \frac{b}{a+b} times \frac{b}{a+2b} by multiplying numerator times numerator and denominator times denominator.

$$\frac{a}{a+b}+\frac{bb}{\left(a+b\right)\left(a+2b\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and \left(a+b\right)\left(a+2b\right) is \left(a+b\right)\left(a+2b\right). Multiply \frac{a}{a+b} times \frac{a+2b}{a+2b}.

$$\frac{a\left(a+2b\right)}{\left(a+b\right)\left(a+2b\right)}+\frac{bb}{\left(a+b\right)\left(a+2b\right)}$$

Since \frac{a\left(a+2b\right)}{\left(a+b\right)\left(a+2b\right)} and \frac{bb}{\left(a+b\right)\left(a+2b\right)} have the same denominator, add them by adding their numerators.

$$\frac{a\left(a+2b\right)+bb}{\left(a+b\right)\left(a+2b\right)}$$

Do the multiplications in a\left(a+2b\right)+bb.

$$\frac{a^{2}+2ab+b^{2}}{\left(a+b\right)\left(a+2b\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}+2ab+b^{2}}{\left(a+b\right)\left(a+2b\right)}.

$$\frac{\left(a+b\right)^{2}}{\left(a+b\right)\left(a+2b\right)}$$

Cancel out a+b in both numerator and denominator.

$$\frac{a+b}{a+2b}$$

Factor

$$\frac{a+b}{a+2b}$$