Solve 5a+4b/5ac+2a+3c/3ab-3b+2c/3bc

$$\frac { 5 a + 4 b } { 5 a c } + \frac { 2 a + 3 c } { 3 a b } - \frac { 3 b + 2 c } { 3 b c }$$

Evaluate

$$\frac{4b^{2}+5c^{2}}{5abc}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5ac and 3ab is 15abc. Multiply \frac{5a+4b}{5ac} times \frac{3b}{3b}. Multiply \frac{2a+3c}{3ab} times \frac{5c}{5c}.

$$\frac{\left(5a+4b\right)\times 3b}{15abc}+\frac{\left(2a+3c\right)\times 5c}{15abc}-\frac{3b+2c}{3bc}$$

Since \frac{\left(5a+4b\right)\times 3b}{15abc} and \frac{\left(2a+3c\right)\times 5c}{15abc} have the same denominator, add them by adding their numerators.

$$\frac{\left(5a+4b\right)\times 3b+\left(2a+3c\right)\times 5c}{15abc}-\frac{3b+2c}{3bc}$$

Do the multiplications in \left(5a+4b\right)\times 3b+\left(2a+3c\right)\times 5c.

$$\frac{15ab+12b^{2}+10ac+15c^{2}}{15abc}-\frac{3b+2c}{3bc}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15abc and 3bc is 15abc. Multiply \frac{3b+2c}{3bc} times \frac{5a}{5a}.

$$\frac{15ab+12b^{2}+10ac+15c^{2}}{15abc}-\frac{\left(3b+2c\right)\times 5a}{15abc}$$

Since \frac{15ab+12b^{2}+10ac+15c^{2}}{15abc} and \frac{\left(3b+2c\right)\times 5a}{15abc} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15ab+12b^{2}+10ac+15c^{2}-\left(3b+2c\right)\times 5a}{15abc}$$

Do the multiplications in 15ab+12b^{2}+10ac+15c^{2}-\left(3b+2c\right)\times 5a.

$$\frac{15ab+12b^{2}+10ac+15c^{2}-15ba-10ca}{15abc}$$

Combine like terms in 15ab+12b^{2}+10ac+15c^{2}-15ba-10ca.

$$\frac{12b^{2}+15c^{2}}{15abc}$$

Factor the expressions that are not already factored in \frac{12b^{2}+15c^{2}}{15abc}.

$$\frac{3\left(4b^{2}+5c^{2}\right)}{15abc}$$

Cancel out 3 in both numerator and denominator.

$$\frac{4b^{2}+5c^{2}}{5abc}$$

Expand

$$\frac{4b^{2}+5c^{2}}{5abc}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5ac and 3ab is 15abc. Multiply \frac{5a+4b}{5ac} times \frac{3b}{3b}. Multiply \frac{2a+3c}{3ab} times \frac{5c}{5c}.

$$\frac{\left(5a+4b\right)\times 3b}{15abc}+\frac{\left(2a+3c\right)\times 5c}{15abc}-\frac{3b+2c}{3bc}$$

Since \frac{\left(5a+4b\right)\times 3b}{15abc} and \frac{\left(2a+3c\right)\times 5c}{15abc} have the same denominator, add them by adding their numerators.

$$\frac{\left(5a+4b\right)\times 3b+\left(2a+3c\right)\times 5c}{15abc}-\frac{3b+2c}{3bc}$$

Do the multiplications in \left(5a+4b\right)\times 3b+\left(2a+3c\right)\times 5c.

$$\frac{15ab+12b^{2}+10ac+15c^{2}}{15abc}-\frac{3b+2c}{3bc}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15abc and 3bc is 15abc. Multiply \frac{3b+2c}{3bc} times \frac{5a}{5a}.

$$\frac{15ab+12b^{2}+10ac+15c^{2}}{15abc}-\frac{\left(3b+2c\right)\times 5a}{15abc}$$

Since \frac{15ab+12b^{2}+10ac+15c^{2}}{15abc} and \frac{\left(3b+2c\right)\times 5a}{15abc} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15ab+12b^{2}+10ac+15c^{2}-\left(3b+2c\right)\times 5a}{15abc}$$

Do the multiplications in 15ab+12b^{2}+10ac+15c^{2}-\left(3b+2c\right)\times 5a.

$$\frac{15ab+12b^{2}+10ac+15c^{2}-15ba-10ca}{15abc}$$

Combine like terms in 15ab+12b^{2}+10ac+15c^{2}-15ba-10ca.

$$\frac{12b^{2}+15c^{2}}{15abc}$$

Factor the expressions that are not already factored in \frac{12b^{2}+15c^{2}}{15abc}.

$$\frac{3\left(4b^{2}+5c^{2}\right)}{15abc}$$

Cancel out 3 in both numerator and denominator.

$$\frac{4b^{2}+5c^{2}}{5abc}$$