Solve 9991/2+9992/7+9993/7+9994/7+999frac{5}{7}+999frac{6}{7}=?

$$999 \frac { 1 } { 2 } + 999 \frac { 2 } { 7 } + 999 \frac { 3 } { 7 } + 999 \frac { 4 } { 7 } + 999 \frac { 5 } { 7 } + 999 \frac { 6 } { 7 } = ?$$

Evaluate

$$\frac{83963}{14}\approx 5997.357142857$$

Solution Steps

Multiply 999 and 2 to get 1998.

$$\frac{1998+1}{2}+\frac{999\times 7+2}{7}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 1998 and 1 to get 1999.

$$\frac{1999}{2}+\frac{999\times 7+2}{7}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Multiply 999 and 7 to get 6993.

$$\frac{1999}{2}+\frac{6993+2}{7}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 6993 and 2 to get 6995.

$$\frac{1999}{2}+\frac{6995}{7}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Least common multiple of 2 and 7 is 14. Convert \frac{1999}{2} and \frac{6995}{7} to fractions with denominator 14.

$$\frac{13993}{14}+\frac{13990}{14}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Since \frac{13993}{14} and \frac{13990}{14} have the same denominator, add them by adding their numerators.

$$\frac{13993+13990}{14}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 13993 and 13990 to get 27983.

$$\frac{27983}{14}+\frac{999\times 7+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Multiply 999 and 7 to get 6993.

$$\frac{27983}{14}+\frac{6993+3}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 6993 and 3 to get 6996.

$$\frac{27983}{14}+\frac{6996}{7}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Least common multiple of 14 and 7 is 14. Convert \frac{27983}{14} and \frac{6996}{7} to fractions with denominator 14.

$$\frac{27983}{14}+\frac{13992}{14}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Since \frac{27983}{14} and \frac{13992}{14} have the same denominator, add them by adding their numerators.

$$\frac{27983+13992}{14}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 27983 and 13992 to get 41975.

$$\frac{41975}{14}+\frac{999\times 7+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Multiply 999 and 7 to get 6993.

$$\frac{41975}{14}+\frac{6993+4}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 6993 and 4 to get 6997.

$$\frac{41975}{14}+\frac{6997}{7}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Least common multiple of 14 and 7 is 14. Convert \frac{41975}{14} and \frac{6997}{7} to fractions with denominator 14.

$$\frac{41975}{14}+\frac{13994}{14}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Since \frac{41975}{14} and \frac{13994}{14} have the same denominator, add them by adding their numerators.

$$\frac{41975+13994}{14}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Add 41975 and 13994 to get 55969.

$$\frac{55969}{14}+\frac{999\times 7+5}{7}+\frac{999\times 7+6}{7}$$

Multiply 999 and 7 to get 6993.

$$\frac{55969}{14}+\frac{6993+5}{7}+\frac{999\times 7+6}{7}$$

Add 6993 and 5 to get 6998.

$$\frac{55969}{14}+\frac{6998}{7}+\frac{999\times 7+6}{7}$$

Least common multiple of 14 and 7 is 14. Convert \frac{55969}{14} and \frac{6998}{7} to fractions with denominator 14.

$$\frac{55969}{14}+\frac{13996}{14}+\frac{999\times 7+6}{7}$$

Since \frac{55969}{14} and \frac{13996}{14} have the same denominator, add them by adding their numerators.

$$\frac{55969+13996}{14}+\frac{999\times 7+6}{7}$$

Add 55969 and 13996 to get 69965.

$$\frac{69965}{14}+\frac{999\times 7+6}{7}$$

Reduce the fraction \frac{69965}{14} to lowest terms by extracting and canceling out 7.

$$\frac{9995}{2}+\frac{999\times 7+6}{7}$$

Multiply 999 and 7 to get 6993.

$$\frac{9995}{2}+\frac{6993+6}{7}$$

Add 6993 and 6 to get 6999.

$$\frac{9995}{2}+\frac{6999}{7}$$

Least common multiple of 2 and 7 is 14. Convert \frac{9995}{2} and \frac{6999}{7} to fractions with denominator 14.

$$\frac{69965}{14}+\frac{13998}{14}$$

Since \frac{69965}{14} and \frac{13998}{14} have the same denominator, add them by adding their numerators.

$$\frac{69965+13998}{14}$$

Add 69965 and 13998 to get 83963.

$$\frac{83963}{14}$$

Factor

$$\frac{11 \cdot 17 \cdot 449}{2 \cdot 7} = 5997\frac{5}{14} = 5997.357142857143$$