Solve {l}{(985/466+466/985+899/394)*(769/347+frac{689}{423})}{-(frac{760}{347}+frac{689}{423}+frac{985}{467})*(frac{466}{985}+frac{699}{394})}

$$\left. \begin{array} { l } { ( \frac { 985 } { 466 } + \frac { 466 } { 985 } + \frac { 899 } { 394 } ) \times ( \frac { 769 } { 347 } + \frac { 689 } { 423 } ) } \\ { - ( \frac { 760 } { 347 } + \frac { 689 } { 423 } + \frac { 985 } { 467 } ) \times ( \frac { 466 } { 985 } + \frac { 699 } { 394 } ) } \end{array} \right.$$

Evaluate

$$\frac{28304092692749}{5243938860045}\approx 5.397487165$$

Solution Steps

Least common multiple of 466 and 985 is 459010. Convert \frac{985}{466} and \frac{466}{985} to fractions with denominator 459010.

$$\left(\frac{970225}{459010}+\frac{217156}{459010}+\frac{899}{394}\right)\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Since \frac{970225}{459010} and \frac{217156}{459010} have the same denominator, add them by adding their numerators.

$$\left(\frac{970225+217156}{459010}+\frac{899}{394}\right)\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Add 970225 and 217156 to get 1187381.

$$\left(\frac{1187381}{459010}+\frac{899}{394}\right)\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Least common multiple of 459010 and 394 is 459010. Convert \frac{1187381}{459010} and \frac{899}{394} to fractions with denominator 459010.

$$\left(\frac{1187381}{459010}+\frac{1047335}{459010}\right)\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Since \frac{1187381}{459010} and \frac{1047335}{459010} have the same denominator, add them by adding their numerators.

$$\frac{1187381+1047335}{459010}\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Add 1187381 and 1047335 to get 2234716.

$$\frac{2234716}{459010}\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Reduce the fraction \frac{2234716}{459010} to lowest terms by extracting and canceling out 2.

$$\frac{1117358}{229505}\left(\frac{769}{347}+\frac{689}{423}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Least common multiple of 347 and 423 is 146781. Convert \frac{769}{347} and \frac{689}{423} to fractions with denominator 146781.

$$\frac{1117358}{229505}\left(\frac{325287}{146781}+\frac{239083}{146781}\right)-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Since \frac{325287}{146781} and \frac{239083}{146781} have the same denominator, add them by adding their numerators.

$$\frac{1117358}{229505}\times \frac{325287+239083}{146781}-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Add 325287 and 239083 to get 564370.

$$\frac{1117358}{229505}\times \frac{564370}{146781}-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Multiply \frac{1117358}{229505} times \frac{564370}{146781} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1117358\times 564370}{229505\times 146781}-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Do the multiplications in the fraction \frac{1117358\times 564370}{229505\times 146781}.

$$\frac{630603334460}{33686973405}-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Reduce the fraction \frac{630603334460}{33686973405} to lowest terms by extracting and canceling out 5.

$$\frac{126120666892}{6737394681}-\left(\frac{760}{347}+\frac{689}{423}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Least common multiple of 347 and 423 is 146781. Convert \frac{760}{347} and \frac{689}{423} to fractions with denominator 146781.

$$\frac{126120666892}{6737394681}-\left(\frac{321480}{146781}+\frac{239083}{146781}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Since \frac{321480}{146781} and \frac{239083}{146781} have the same denominator, add them by adding their numerators.

$$\frac{126120666892}{6737394681}-\left(\frac{321480+239083}{146781}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Add 321480 and 239083 to get 560563.

$$\frac{126120666892}{6737394681}-\left(\frac{560563}{146781}+\frac{985}{467}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Least common multiple of 146781 and 467 is 68546727. Convert \frac{560563}{146781} and \frac{985}{467} to fractions with denominator 68546727.

$$\frac{126120666892}{6737394681}-\left(\frac{261782921}{68546727}+\frac{144579285}{68546727}\right)\left(\frac{466}{985}+\frac{699}{394}\right)$$

Since \frac{261782921}{68546727} and \frac{144579285}{68546727} have the same denominator, add them by adding their numerators.

$$\frac{126120666892}{6737394681}-\frac{261782921+144579285}{68546727}\left(\frac{466}{985}+\frac{699}{394}\right)$$

Add 261782921 and 144579285 to get 406362206.

$$\frac{126120666892}{6737394681}-\frac{406362206}{68546727}\left(\frac{466}{985}+\frac{699}{394}\right)$$

Least common multiple of 985 and 394 is 1970. Convert \frac{466}{985} and \frac{699}{394} to fractions with denominator 1970.

$$\frac{126120666892}{6737394681}-\frac{406362206}{68546727}\left(\frac{932}{1970}+\frac{3495}{1970}\right)$$

Since \frac{932}{1970} and \frac{3495}{1970} have the same denominator, add them by adding their numerators.

$$\frac{126120666892}{6737394681}-\frac{406362206}{68546727}\times \frac{932+3495}{1970}$$

Add 932 and 3495 to get 4427.

$$\frac{126120666892}{6737394681}-\frac{406362206}{68546727}\times \frac{4427}{1970}$$

Multiply \frac{406362206}{68546727} times \frac{4427}{1970} by multiplying numerator times numerator and denominator times denominator.

$$\frac{126120666892}{6737394681}-\frac{406362206\times 4427}{68546727\times 1970}$$

Do the multiplications in the fraction \frac{406362206\times 4427}{68546727\times 1970}.

$$\frac{126120666892}{6737394681}-\frac{1798965485962}{135037052190}$$

Reduce the fraction \frac{1798965485962}{135037052190} to lowest terms by extracting and canceling out 2.

$$\frac{126120666892}{6737394681}-\frac{899482742981}{67518526095}$$

Least common multiple of 6737394681 and 67518526095 is 15731816580135. Convert \frac{126120666892}{6737394681} and \frac{899482742981}{67518526095} to fractions with denominator 15731816580135.

$$\frac{294491757192820}{15731816580135}-\frac{209579479114573}{15731816580135}$$

Since \frac{294491757192820}{15731816580135} and \frac{209579479114573}{15731816580135} have the same denominator, subtract them by subtracting their numerators.

$$\frac{294491757192820-209579479114573}{15731816580135}$$

Subtract 209579479114573 from 294491757192820 to get 84912278078247.

$$\frac{84912278078247}{15731816580135}$$

Reduce the fraction \frac{84912278078247}{15731816580135} to lowest terms by extracting and canceling out 3.

$$\frac{28304092692749}{5243938860045}$$

Factor

$$\frac{28304092692749}{3 \cdot 5 \cdot 47 \cdot 197 \cdot 233 \cdot 347 \cdot 467} = 5\frac{2084398392524}{5243938860045} = 5.397487165307284$$