Solve +1/3=1/3+1/6+1/12+frac{1}{24}+frac{1}{48}+frac{1}{96}+frac{1}{192}+frac{1}{384}+frac{1}{39}

$$+ \frac { 1 } { 3 } = \frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 24 } + \frac { 1 } { 48 } + \frac { 1 } { 96 } + \frac { 1 } { 192 } + \frac { 1 } { 384 } + \frac { 1 } { 39 }$$

Verify

Solution Steps

Least common multiple of 3 and 6 is 6. Convert \frac{1}{3} and \frac{1}{6} to fractions with denominator 6.

$$\frac{1}{3}=\frac{2}{6}+\frac{1}{6}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{2}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{2+1}{6}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 2 and 1 to get 3.

$$\frac{1}{3}=\frac{3}{6}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}=\frac{1}{2}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 2 and 12 is 12. Convert \frac{1}{2} and \frac{1}{12} to fractions with denominator 12.

$$\frac{1}{3}=\frac{6}{12}+\frac{1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{6}{12} and \frac{1}{12} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{6+1}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 6 and 1 to get 7.

$$\frac{1}{3}=\frac{7}{12}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 12 and 24 is 24. Convert \frac{7}{12} and \frac{1}{24} to fractions with denominator 24.

$$\frac{1}{3}=\frac{14}{24}+\frac{1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{14}{24} and \frac{1}{24} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{14+1}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 14 and 1 to get 15.

$$\frac{1}{3}=\frac{15}{24}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Reduce the fraction \frac{15}{24} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}=\frac{5}{8}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 8 and 48 is 48. Convert \frac{5}{8} and \frac{1}{48} to fractions with denominator 48.

$$\frac{1}{3}=\frac{30}{48}+\frac{1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{30}{48} and \frac{1}{48} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{30+1}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 30 and 1 to get 31.

$$\frac{1}{3}=\frac{31}{48}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 48 and 96 is 96. Convert \frac{31}{48} and \frac{1}{96} to fractions with denominator 96.

$$\frac{1}{3}=\frac{62}{96}+\frac{1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{62}{96} and \frac{1}{96} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{62+1}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 62 and 1 to get 63.

$$\frac{1}{3}=\frac{63}{96}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Reduce the fraction \frac{63}{96} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}=\frac{21}{32}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 32 and 192 is 192. Convert \frac{21}{32} and \frac{1}{192} to fractions with denominator 192.

$$\frac{1}{3}=\frac{126}{192}+\frac{1}{192}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{126}{192} and \frac{1}{192} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{126+1}{192}+\frac{1}{384}+\frac{1}{39}$$

Add 126 and 1 to get 127.

$$\frac{1}{3}=\frac{127}{192}+\frac{1}{384}+\frac{1}{39}$$

Least common multiple of 192 and 384 is 384. Convert \frac{127}{192} and \frac{1}{384} to fractions with denominator 384.

$$\frac{1}{3}=\frac{254}{384}+\frac{1}{384}+\frac{1}{39}$$

Since \frac{254}{384} and \frac{1}{384} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{254+1}{384}+\frac{1}{39}$$

Add 254 and 1 to get 255.

$$\frac{1}{3}=\frac{255}{384}+\frac{1}{39}$$

Reduce the fraction \frac{255}{384} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}=\frac{85}{128}+\frac{1}{39}$$

Least common multiple of 128 and 39 is 4992. Convert \frac{85}{128} and \frac{1}{39} to fractions with denominator 4992.

$$\frac{1}{3}=\frac{3315}{4992}+\frac{128}{4992}$$

Since \frac{3315}{4992} and \frac{128}{4992} have the same denominator, add them by adding their numerators.

$$\frac{1}{3}=\frac{3315+128}{4992}$$

Add 3315 and 128 to get 3443.

$$\frac{1}{3}=\frac{3443}{4992}$$

Least common multiple of 3 and 4992 is 4992. Convert \frac{1}{3} and \frac{3443}{4992} to fractions with denominator 4992.

$$\frac{1664}{4992}=\frac{3443}{4992}$$

Compare \frac{1664}{4992} and \frac{3443}{4992}.

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