Solve (32/5n+23/14)div1/2-(41/5+1frac{3}{10})

$$( 3 \frac { 2 } { 5 n } + 2 \frac { 3 } { 14 } ) \div \frac { 1 } { 2 } - ( 4 \frac { 1 } { 5 } + 1 \frac { 3 } { 10 } )$$

Evaluate

$$-\frac{15}{14}+\frac{12}{5n}$$

Short Solution Steps

Express 3\times \frac{2}{5n} as a single fraction.

$$\frac{\frac{3\times 2}{5n}+\frac{2\times 14+3}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 2 and 14 to get 28.

$$\frac{\frac{3\times 2}{5n}+\frac{28+3}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Add 28 and 3 to get 31.

$$\frac{\frac{3\times 2}{5n}+\frac{31}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5n and 14 is 70n. Multiply \frac{3\times 2}{5n} times \frac{14}{14}. Multiply \frac{31}{14} times \frac{5n}{5n}.

$$\frac{\frac{14\times 3\times 2}{70n}+\frac{31\times 5n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Since \frac{14\times 3\times 2}{70n} and \frac{31\times 5n}{70n} have the same denominator, add them by adding their numerators.

$$\frac{\frac{14\times 3\times 2+31\times 5n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Do the multiplications in 14\times 3\times 2+31\times 5n.

$$\frac{\frac{84+155n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Divide \frac{84+155n}{70n} by \frac{1}{2} by multiplying \frac{84+155n}{70n} by the reciprocal of \frac{1}{2}.

$$\frac{\left(84+155n\right)\times 2}{70n}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{155n+84}{35n}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 4 and 5 to get 20.

$$\frac{155n+84}{35n}-\left(\frac{20+1}{5}+\frac{1\times 10+3}{10}\right)$$

Add 20 and 1 to get 21.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 1 and 10 to get 10.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{10+3}{10}\right)$$

Add 10 and 3 to get 13.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{13}{10}\right)$$

Least common multiple of 5 and 10 is 10. Convert \frac{21}{5} and \frac{13}{10} to fractions with denominator 10.

$$\frac{155n+84}{35n}-\left(\frac{42}{10}+\frac{13}{10}\right)$$

Since \frac{42}{10} and \frac{13}{10} have the same denominator, add them by adding their numerators.

$$\frac{155n+84}{35n}-\frac{42+13}{10}$$

Add 42 and 13 to get 55.

$$\frac{155n+84}{35n}-\frac{55}{10}$$

Reduce the fraction \frac{55}{10} to lowest terms by extracting and canceling out 5.

$$\frac{155n+84}{35n}-\frac{11}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 35n and 2 is 70n. Multiply \frac{155n+84}{35n} times \frac{2}{2}. Multiply \frac{11}{2} times \frac{35n}{35n}.

$$\frac{2\left(155n+84\right)}{70n}-\frac{11\times 35n}{70n}$$

Since \frac{2\left(155n+84\right)}{70n} and \frac{11\times 35n}{70n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(155n+84\right)-11\times 35n}{70n}$$

Do the multiplications in 2\left(155n+84\right)-11\times 35n.

$$\frac{310n+168-385n}{70n}$$

Combine like terms in 310n+168-385n.

$$\frac{-75n+168}{70n}$$

Expand

$$-\frac{15}{14}+\frac{12}{5n}$$

Short Solution Steps

Express 3\times \frac{2}{5n} as a single fraction.

$$\frac{\frac{3\times 2}{5n}+\frac{2\times 14+3}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 2 and 14 to get 28.

$$\frac{\frac{3\times 2}{5n}+\frac{28+3}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Add 28 and 3 to get 31.

$$\frac{\frac{3\times 2}{5n}+\frac{31}{14}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5n and 14 is 70n. Multiply \frac{3\times 2}{5n} times \frac{14}{14}. Multiply \frac{31}{14} times \frac{5n}{5n}.

$$\frac{\frac{14\times 3\times 2}{70n}+\frac{31\times 5n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Since \frac{14\times 3\times 2}{70n} and \frac{31\times 5n}{70n} have the same denominator, add them by adding their numerators.

$$\frac{\frac{14\times 3\times 2+31\times 5n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Do the multiplications in 14\times 3\times 2+31\times 5n.

$$\frac{\frac{84+155n}{70n}}{\frac{1}{2}}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Divide \frac{84+155n}{70n} by \frac{1}{2} by multiplying \frac{84+155n}{70n} by the reciprocal of \frac{1}{2}.

$$\frac{\left(84+155n\right)\times 2}{70n}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{155n+84}{35n}-\left(\frac{4\times 5+1}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 4 and 5 to get 20.

$$\frac{155n+84}{35n}-\left(\frac{20+1}{5}+\frac{1\times 10+3}{10}\right)$$

Add 20 and 1 to get 21.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{1\times 10+3}{10}\right)$$

Multiply 1 and 10 to get 10.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{10+3}{10}\right)$$

Add 10 and 3 to get 13.

$$\frac{155n+84}{35n}-\left(\frac{21}{5}+\frac{13}{10}\right)$$

Least common multiple of 5 and 10 is 10. Convert \frac{21}{5} and \frac{13}{10} to fractions with denominator 10.

$$\frac{155n+84}{35n}-\left(\frac{42}{10}+\frac{13}{10}\right)$$

Since \frac{42}{10} and \frac{13}{10} have the same denominator, add them by adding their numerators.

$$\frac{155n+84}{35n}-\frac{42+13}{10}$$

Add 42 and 13 to get 55.

$$\frac{155n+84}{35n}-\frac{55}{10}$$

Reduce the fraction \frac{55}{10} to lowest terms by extracting and canceling out 5.

$$\frac{155n+84}{35n}-\frac{11}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 35n and 2 is 70n. Multiply \frac{155n+84}{35n} times \frac{2}{2}. Multiply \frac{11}{2} times \frac{35n}{35n}.

$$\frac{2\left(155n+84\right)}{70n}-\frac{11\times 35n}{70n}$$

Since \frac{2\left(155n+84\right)}{70n} and \frac{11\times 35n}{70n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(155n+84\right)-11\times 35n}{70n}$$

Do the multiplications in 2\left(155n+84\right)-11\times 35n.

$$\frac{310n+168-385n}{70n}$$

Combine like terms in 310n+168-385n.

$$\frac{-75n+168}{70n}$$