Solve a+b/2a-2a-b/3b-3b-a/6a

$$\frac { a + b } { 2 a } - \frac { 2 a - b } { 3 b } - \frac { 3 b - a } { 6 a }$$

Evaluate

$$-\frac{2a}{3b}+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2a and 3b is 6ab. Multiply \frac{a+b}{2a} times \frac{3b}{3b}. Multiply \frac{2a-b}{3b} times \frac{2a}{2a}.

$$\frac{\left(a+b\right)\times 3b}{6ab}-\frac{\left(2a-b\right)\times 2a}{6ab}-\frac{3b-a}{6a}$$

Since \frac{\left(a+b\right)\times 3b}{6ab} and \frac{\left(2a-b\right)\times 2a}{6ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+b\right)\times 3b-\left(2a-b\right)\times 2a}{6ab}-\frac{3b-a}{6a}$$

Do the multiplications in \left(a+b\right)\times 3b-\left(2a-b\right)\times 2a.

$$\frac{3ab+3b^{2}-4a^{2}+2ba}{6ab}-\frac{3b-a}{6a}$$

Combine like terms in 3ab+3b^{2}-4a^{2}+2ba.

$$\frac{3b^{2}+5ab-4a^{2}}{6ab}-\frac{3b-a}{6a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6ab and 6a is 6ab. Multiply \frac{3b-a}{6a} times \frac{b}{b}.

$$\frac{3b^{2}+5ab-4a^{2}}{6ab}-\frac{\left(3b-a\right)b}{6ab}$$

Since \frac{3b^{2}+5ab-4a^{2}}{6ab} and \frac{\left(3b-a\right)b}{6ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3b^{2}+5ab-4a^{2}-\left(3b-a\right)b}{6ab}$$

Do the multiplications in 3b^{2}+5ab-4a^{2}-\left(3b-a\right)b.

$$\frac{3b^{2}+5ab-4a^{2}-3b^{2}+ab}{6ab}$$

Combine like terms in 3b^{2}+5ab-4a^{2}-3b^{2}+ab.

$$\frac{6ab-4a^{2}}{6ab}$$

Factor the expressions that are not already factored in \frac{6ab-4a^{2}}{6ab}.

$$\frac{2a\left(-2a+3b\right)}{6ab}$$

Cancel out 2a in both numerator and denominator.

$$\frac{-2a+3b}{3b}$$

Expand

$$-\frac{2a}{3b}+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2a and 3b is 6ab. Multiply \frac{a+b}{2a} times \frac{3b}{3b}. Multiply \frac{2a-b}{3b} times \frac{2a}{2a}.

$$\frac{\left(a+b\right)\times 3b}{6ab}-\frac{\left(2a-b\right)\times 2a}{6ab}-\frac{3b-a}{6a}$$

Since \frac{\left(a+b\right)\times 3b}{6ab} and \frac{\left(2a-b\right)\times 2a}{6ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+b\right)\times 3b-\left(2a-b\right)\times 2a}{6ab}-\frac{3b-a}{6a}$$

Do the multiplications in \left(a+b\right)\times 3b-\left(2a-b\right)\times 2a.

$$\frac{3ab+3b^{2}-4a^{2}+2ba}{6ab}-\frac{3b-a}{6a}$$

Combine like terms in 3ab+3b^{2}-4a^{2}+2ba.

$$\frac{3b^{2}+5ab-4a^{2}}{6ab}-\frac{3b-a}{6a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6ab and 6a is 6ab. Multiply \frac{3b-a}{6a} times \frac{b}{b}.

$$\frac{3b^{2}+5ab-4a^{2}}{6ab}-\frac{\left(3b-a\right)b}{6ab}$$

Since \frac{3b^{2}+5ab-4a^{2}}{6ab} and \frac{\left(3b-a\right)b}{6ab} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3b^{2}+5ab-4a^{2}-\left(3b-a\right)b}{6ab}$$

Do the multiplications in 3b^{2}+5ab-4a^{2}-\left(3b-a\right)b.

$$\frac{3b^{2}+5ab-4a^{2}-3b^{2}+ab}{6ab}$$

Combine like terms in 3b^{2}+5ab-4a^{2}-3b^{2}+ab.

$$\frac{6ab-4a^{2}}{6ab}$$

Factor the expressions that are not already factored in \frac{6ab-4a^{2}}{6ab}.

$$\frac{2a\left(-2a+3b\right)}{6ab}$$

Cancel out 2a in both numerator and denominator.

$$\frac{-2a+3b}{3b}$$