Solve 1/2+3/4-4/3=5/4-frac{7}{5}+frac{5}{6}-frac{6}{5}=1+frac{1}{2}+frac{1}{4}+frac{1}{8}+frac{1}{16}

$$\frac { 1 } { 2 } + \frac { 3 } { 4 } - \frac { 4 } { 3 } = \frac { 5 } { 4 } - \frac { 7 } { 5 } + \frac { 5 } { 6 } - \frac { 6 } { 5 } = 1 + \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 }$$

Verify

Solution Steps

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

$$\frac{2}{4}+\frac{3}{4}-\frac{4}{3}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{2}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\frac{2+3}{4}-\frac{4}{3}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Add 2 and 3 to get 5.

$$\frac{5}{4}-\frac{4}{3}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 4 and 3 is 12. Convert \frac{5}{4} and \frac{4}{3} to fractions with denominator 12.

$$\frac{15}{12}-\frac{16}{12}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{15}{12} and \frac{16}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15-16}{12}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Subtract 16 from 15 to get -1.

$$-\frac{1}{12}=\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 4 and 5 is 20. Convert \frac{5}{4} and \frac{7}{5} to fractions with denominator 20.

$$-\frac{1}{12}=\frac{25}{20}-\frac{28}{20}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{25}{20} and \frac{28}{20} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{12}=\frac{25-28}{20}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Subtract 28 from 25 to get -3.

$$-\frac{1}{12}=-\frac{3}{20}+\frac{5}{6}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 20 and 6 is 60. Convert -\frac{3}{20} and \frac{5}{6} to fractions with denominator 60.

$$-\frac{1}{12}=-\frac{9}{60}+\frac{50}{60}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since -\frac{9}{60} and \frac{50}{60} have the same denominator, add them by adding their numerators.

$$-\frac{1}{12}=\frac{-9+50}{60}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Add -9 and 50 to get 41.

$$-\frac{1}{12}=\frac{41}{60}-\frac{6}{5}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 60 and 5 is 60. Convert \frac{41}{60} and \frac{6}{5} to fractions with denominator 60.

$$-\frac{1}{12}=\frac{41}{60}-\frac{72}{60}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{41}{60} and \frac{72}{60} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{1}{12}=\frac{41-72}{60}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Subtract 72 from 41 to get -31.

$$-\frac{1}{12}=-\frac{31}{60}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 12 and 60 is 60. Convert -\frac{1}{12} and -\frac{31}{60} to fractions with denominator 60.

$$-\frac{5}{60}=-\frac{31}{60}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Compare -\frac{5}{60} and -\frac{31}{60}.

$$\text{false}\text{ and }\frac{5}{4}-\frac{7}{5}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 4 and 5 is 20. Convert \frac{5}{4} and \frac{7}{5} to fractions with denominator 20.

$$\text{false}\text{ and }\frac{25}{20}-\frac{28}{20}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{25}{20} and \frac{28}{20} have the same denominator, subtract them by subtracting their numerators.

$$\text{false}\text{ and }\frac{25-28}{20}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Subtract 28 from 25 to get -3.

$$\text{false}\text{ and }-\frac{3}{20}+\frac{5}{6}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 20 and 6 is 60. Convert -\frac{3}{20} and \frac{5}{6} to fractions with denominator 60.

$$\text{false}\text{ and }-\frac{9}{60}+\frac{50}{60}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since -\frac{9}{60} and \frac{50}{60} have the same denominator, add them by adding their numerators.

$$\text{false}\text{ and }\frac{-9+50}{60}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Add -9 and 50 to get 41.

$$\text{false}\text{ and }\frac{41}{60}-\frac{6}{5}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 60 and 5 is 60. Convert \frac{41}{60} and \frac{6}{5} to fractions with denominator 60.

$$\text{false}\text{ and }\frac{41}{60}-\frac{72}{60}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{41}{60} and \frac{72}{60} have the same denominator, subtract them by subtracting their numerators.

$$\text{false}\text{ and }\frac{41-72}{60}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Subtract 72 from 41 to get -31.

$$\text{false}\text{ and }-\frac{31}{60}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Convert 1 to fraction \frac{2}{2}.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{2}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{2+1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Add 2 and 1 to get 3.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{3}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{1}{4} to fractions with denominator 4.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{6}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{6}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{6+1}{4}+\frac{1}{8}+\frac{1}{16}$$

Add 6 and 1 to get 7.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{7}{4}+\frac{1}{8}+\frac{1}{16}$$

Least common multiple of 4 and 8 is 8. Convert \frac{7}{4} and \frac{1}{8} to fractions with denominator 8.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{14}{8}+\frac{1}{8}+\frac{1}{16}$$

Since \frac{14}{8} and \frac{1}{8} have the same denominator, add them by adding their numerators.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{14+1}{8}+\frac{1}{16}$$

Add 14 and 1 to get 15.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{15}{8}+\frac{1}{16}$$

Least common multiple of 8 and 16 is 16. Convert \frac{15}{8} and \frac{1}{16} to fractions with denominator 16.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{30}{16}+\frac{1}{16}$$

Since \frac{30}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{30+1}{16}$$

Add 30 and 1 to get 31.

$$\text{false}\text{ and }-\frac{31}{60}=\frac{31}{16}$$

Least common multiple of 60 and 16 is 240. Convert -\frac{31}{60} and \frac{31}{16} to fractions with denominator 240.

$$\text{false}\text{ and }-\frac{124}{240}=\frac{465}{240}$$

Compare -\frac{124}{240} and \frac{465}{240}.

$$\text{false}\text{ and }\text{false}$$

The conjunction of \text{false} and \text{false} is \text{false}.

$$\text{false}$$