Solve 2*2/1*3+4*4/3*5+6*6/5*7+8*8/7*9+frac{10*10}{9*11}+frac{12*12}{11*13}

$$\frac { 2 \times 2 } { 1 \times 3 } + \frac { 4 \times 4 } { 3 \times 5 } + \frac { 6 \times 6 } { 5 \times 7 } + \frac { 8 \times 8 } { 7 \times 9 } + \frac { 10 \times 10 } { 9 \times 11 } + \frac { 12 \times 12 } { 11 \times 13 }$$

Evaluate

$$\frac{84}{13}\approx 6.461538462$$

Solution Steps

Multiply 2 and 2 to get 4.

$$\frac{4}{1\times 3}+\frac{4\times 4}{3\times 5}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 1 and 3 to get 3.

$$\frac{4}{3}+\frac{4\times 4}{3\times 5}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 4 and 4 to get 16.

$$\frac{4}{3}+\frac{16}{3\times 5}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 3 and 5 to get 15.

$$\frac{4}{3}+\frac{16}{15}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Least common multiple of 3 and 15 is 15. Convert \frac{4}{3} and \frac{16}{15} to fractions with denominator 15.

$$\frac{20}{15}+\frac{16}{15}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Since \frac{20}{15} and \frac{16}{15} have the same denominator, add them by adding their numerators.

$$\frac{20+16}{15}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Add 20 and 16 to get 36.

$$\frac{36}{15}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Reduce the fraction \frac{36}{15} to lowest terms by extracting and canceling out 3.

$$\frac{12}{5}+\frac{6\times 6}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 6 and 6 to get 36.

$$\frac{12}{5}+\frac{36}{5\times 7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 5 and 7 to get 35.

$$\frac{12}{5}+\frac{36}{35}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Least common multiple of 5 and 35 is 35. Convert \frac{12}{5} and \frac{36}{35} to fractions with denominator 35.

$$\frac{84}{35}+\frac{36}{35}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Since \frac{84}{35} and \frac{36}{35} have the same denominator, add them by adding their numerators.

$$\frac{84+36}{35}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Add 84 and 36 to get 120.

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

Reduce the fraction \frac{120}{35} to lowest terms by extracting and canceling out 5.

$$\frac{24}{7}+\frac{8\times 8}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 8 and 8 to get 64.

$$\frac{24}{7}+\frac{64}{7\times 9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 7 and 9 to get 63.

$$\frac{24}{7}+\frac{64}{63}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Least common multiple of 7 and 63 is 63. Convert \frac{24}{7} and \frac{64}{63} to fractions with denominator 63.

$$\frac{216}{63}+\frac{64}{63}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Since \frac{216}{63} and \frac{64}{63} have the same denominator, add them by adding their numerators.

$$\frac{216+64}{63}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Add 216 and 64 to get 280.

$$\frac{280}{63}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Reduce the fraction \frac{280}{63} to lowest terms by extracting and canceling out 7.

$$\frac{40}{9}+\frac{10\times 10}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 10 and 10 to get 100.

$$\frac{40}{9}+\frac{100}{9\times 11}+\frac{12\times 12}{11\times 13}$$

Multiply 9 and 11 to get 99.

$$\frac{40}{9}+\frac{100}{99}+\frac{12\times 12}{11\times 13}$$

Least common multiple of 9 and 99 is 99. Convert \frac{40}{9} and \frac{100}{99} to fractions with denominator 99.

$$\frac{440}{99}+\frac{100}{99}+\frac{12\times 12}{11\times 13}$$

Since \frac{440}{99} and \frac{100}{99} have the same denominator, add them by adding their numerators.

$$\frac{440+100}{99}+\frac{12\times 12}{11\times 13}$$

Add 440 and 100 to get 540.

$$\frac{540}{99}+\frac{12\times 12}{11\times 13}$$

Reduce the fraction \frac{540}{99} to lowest terms by extracting and canceling out 9.

$$\frac{60}{11}+\frac{12\times 12}{11\times 13}$$

Multiply 12 and 12 to get 144.

$$\frac{60}{11}+\frac{144}{11\times 13}$$

Multiply 11 and 13 to get 143.

$$\frac{60}{11}+\frac{144}{143}$$

Least common multiple of 11 and 143 is 143. Convert \frac{60}{11} and \frac{144}{143} to fractions with denominator 143.

$$\frac{780}{143}+\frac{144}{143}$$

Since \frac{780}{143} and \frac{144}{143} have the same denominator, add them by adding their numerators.

$$\frac{780+144}{143}$$

Add 780 and 144 to get 924.

$$\frac{924}{143}$$

Reduce the fraction \frac{924}{143} to lowest terms by extracting and canceling out 11.

$$\frac{84}{13}$$

Factor

$$\frac{2 ^ {2} \cdot 3 \cdot 7}{13} = 6\frac{6}{13} = 6.461538461538462$$