Solve {l}{-47/120+1/5*7+1/6*8+1/7*9}{+frac{1}{8*10}+frac{1}{9*11}+frac{1}{10*12}+frac{1}{11*13}}

$$\left. \begin{array} { l } { - \frac { 47 } { 120 } + \frac { 1 } { 5 \times 7 } + \frac { 1 } { 6 \times 8 } + \frac { 1 } { 7 \times 9 } } \\ { + \frac { 1 } { 8 \times 10 } + \frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \frac { 1 } { 11 \times 13 } } \end{array} \right.$$

Evaluate

$$-\frac{15}{52}\approx -0.288461538$$

Solution Steps

Multiply 5 and 7 to get 35.

$$-\frac{47}{120}+\frac{1}{35}+\frac{1}{6\times 8}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Least common multiple of 120 and 35 is 840. Convert -\frac{47}{120} and \frac{1}{35} to fractions with denominator 840.

$$-\frac{329}{840}+\frac{24}{840}+\frac{1}{6\times 8}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Since -\frac{329}{840} and \frac{24}{840} have the same denominator, add them by adding their numerators.

$$\frac{-329+24}{840}+\frac{1}{6\times 8}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Add -329 and 24 to get -305.

$$\frac{-305}{840}+\frac{1}{6\times 8}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Reduce the fraction \frac{-305}{840} to lowest terms by extracting and canceling out 5.

$$-\frac{61}{168}+\frac{1}{6\times 8}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Multiply 6 and 8 to get 48.

$$-\frac{61}{168}+\frac{1}{48}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Least common multiple of 168 and 48 is 336. Convert -\frac{61}{168} and \frac{1}{48} to fractions with denominator 336.

$$-\frac{122}{336}+\frac{7}{336}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Since -\frac{122}{336} and \frac{7}{336} have the same denominator, add them by adding their numerators.

$$\frac{-122+7}{336}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Add -122 and 7 to get -115.

$$-\frac{115}{336}+\frac{1}{7\times 9}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Multiply 7 and 9 to get 63.

$$-\frac{115}{336}+\frac{1}{63}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Least common multiple of 336 and 63 is 1008. Convert -\frac{115}{336} and \frac{1}{63} to fractions with denominator 1008.

$$-\frac{345}{1008}+\frac{16}{1008}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Since -\frac{345}{1008} and \frac{16}{1008} have the same denominator, add them by adding their numerators.

$$\frac{-345+16}{1008}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Add -345 and 16 to get -329.

$$\frac{-329}{1008}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Reduce the fraction \frac{-329}{1008} to lowest terms by extracting and canceling out 7.

$$-\frac{47}{144}+\frac{1}{8\times 10}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Multiply 8 and 10 to get 80.

$$-\frac{47}{144}+\frac{1}{80}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Least common multiple of 144 and 80 is 720. Convert -\frac{47}{144} and \frac{1}{80} to fractions with denominator 720.

$$-\frac{235}{720}+\frac{9}{720}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Since -\frac{235}{720} and \frac{9}{720} have the same denominator, add them by adding their numerators.

$$\frac{-235+9}{720}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Add -235 and 9 to get -226.

$$\frac{-226}{720}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Reduce the fraction \frac{-226}{720} to lowest terms by extracting and canceling out 2.

$$-\frac{113}{360}+\frac{1}{9\times 11}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Multiply 9 and 11 to get 99.

$$-\frac{113}{360}+\frac{1}{99}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Least common multiple of 360 and 99 is 3960. Convert -\frac{113}{360} and \frac{1}{99} to fractions with denominator 3960.

$$-\frac{1243}{3960}+\frac{40}{3960}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Since -\frac{1243}{3960} and \frac{40}{3960} have the same denominator, add them by adding their numerators.

$$\frac{-1243+40}{3960}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Add -1243 and 40 to get -1203.

$$\frac{-1203}{3960}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Reduce the fraction \frac{-1203}{3960} to lowest terms by extracting and canceling out 3.

$$-\frac{401}{1320}+\frac{1}{10\times 12}+\frac{1}{11\times 13}$$

Multiply 10 and 12 to get 120.

$$-\frac{401}{1320}+\frac{1}{120}+\frac{1}{11\times 13}$$

Least common multiple of 1320 and 120 is 1320. Convert -\frac{401}{1320} and \frac{1}{120} to fractions with denominator 1320.

$$-\frac{401}{1320}+\frac{11}{1320}+\frac{1}{11\times 13}$$

Since -\frac{401}{1320} and \frac{11}{1320} have the same denominator, add them by adding their numerators.

$$\frac{-401+11}{1320}+\frac{1}{11\times 13}$$

Add -401 and 11 to get -390.

$$\frac{-390}{1320}+\frac{1}{11\times 13}$$

Reduce the fraction \frac{-390}{1320} to lowest terms by extracting and canceling out 30.

$$-\frac{13}{44}+\frac{1}{11\times 13}$$

Multiply 11 and 13 to get 143.

$$-\frac{13}{44}+\frac{1}{143}$$

Least common multiple of 44 and 143 is 572. Convert -\frac{13}{44} and \frac{1}{143} to fractions with denominator 572.

$$-\frac{169}{572}+\frac{4}{572}$$

Since -\frac{169}{572} and \frac{4}{572} have the same denominator, add them by adding their numerators.

$$\frac{-169+4}{572}$$

Add -169 and 4 to get -165.

$$\frac{-165}{572}$$

Reduce the fraction \frac{-165}{572} to lowest terms by extracting and canceling out 11.

$$-\frac{15}{52}$$

Factor

$$-\frac{15}{52} = -0.28846153846153844$$