Solve a/a-b(1/b-1/a)+a-1/b

$$\frac { a } { a - b } ( \frac { 1 } { b } - \frac { 1 } { a } ) + \frac { a - 1 } { b }$$

Evaluate

$$\frac{a}{b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{1}{b} times \frac{a}{a}. Multiply \frac{1}{a} times \frac{b}{b}.

$$\frac{a}{a-b}\left(\frac{a}{ab}-\frac{b}{ab}\right)+\frac{a-1}{b}$$

Since \frac{a}{ab} and \frac{b}{ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a}{a-b}\times \frac{a-b}{ab}+\frac{a-1}{b}$$

Multiply \frac{a}{a-b} times \frac{a-b}{ab} by multiplying numerator times numerator and denominator times denominator.

$$\frac{a\left(a-b\right)}{\left(a-b\right)ab}+\frac{a-1}{b}$$

Cancel out a\left(a-b\right) in both numerator and denominator.

$$\frac{1}{b}+\frac{a-1}{b}$$

Since \frac{1}{b} and \frac{a-1}{b} have the same denominator, add them by adding their numerators.

$$\frac{1+a-1}{b}$$

Combine like terms in 1+a-1.

$$\frac{a}{b}$$

Expand

$$\frac{a}{b}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b and a is ab. Multiply \frac{1}{b} times \frac{a}{a}. Multiply \frac{1}{a} times \frac{b}{b}.

$$\frac{a}{a-b}\left(\frac{a}{ab}-\frac{b}{ab}\right)+\frac{a-1}{b}$$

Since \frac{a}{ab} and \frac{b}{ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a}{a-b}\times \frac{a-b}{ab}+\frac{a-1}{b}$$

Multiply \frac{a}{a-b} times \frac{a-b}{ab} by multiplying numerator times numerator and denominator times denominator.

$$\frac{a\left(a-b\right)}{\left(a-b\right)ab}+\frac{a-1}{b}$$

Cancel out a\left(a-b\right) in both numerator and denominator.

$$\frac{1}{b}+\frac{a-1}{b}$$

Since \frac{1}{b} and \frac{a-1}{b} have the same denominator, add them by adding their numerators.

$$\frac{1+a-1}{b}$$

Combine like terms in 1+a-1.

$$\frac{a}{b}$$