Solve min1/409+1/50+3/10+1/25+frac{9}{16}+frac{3}{40}

$$\min \frac { 1 } { 409 } + \frac { 1 } { 50 } + \frac { 3 } { 10 } + \frac { 1 } { 25 } + \frac { 9 } { 16 } + \frac { 3 } { 40 }$$

Evaluate

$$\frac{163591}{163600}\approx 0.999944988$$

Solution Steps

To find the minimum of \frac{1}{409}, first put the numbers in order from least to greatest. This set is already in order.

$$\frac{1}{409}$$

The minimum is \frac{1}{409}, the leftmost value in the set ordered from least to greatest.

$$\frac{1}{409}+\frac{1}{50}+\frac{3}{10}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Least common multiple of 409 and 50 is 20450. Convert \frac{1}{409} and \frac{1}{50} to fractions with denominator 20450.

$$\frac{50}{20450}+\frac{409}{20450}+\frac{3}{10}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Since \frac{50}{20450} and \frac{409}{20450} have the same denominator, add them by adding their numerators.

$$\frac{50+409}{20450}+\frac{3}{10}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Add 50 and 409 to get 459.

$$\frac{459}{20450}+\frac{3}{10}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Least common multiple of 20450 and 10 is 20450. Convert \frac{459}{20450} and \frac{3}{10} to fractions with denominator 20450.

$$\frac{459}{20450}+\frac{6135}{20450}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Since \frac{459}{20450} and \frac{6135}{20450} have the same denominator, add them by adding their numerators.

$$\frac{459+6135}{20450}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Add 459 and 6135 to get 6594.

$$\frac{6594}{20450}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Reduce the fraction \frac{6594}{20450} to lowest terms by extracting and canceling out 2.

$$\frac{3297}{10225}+\frac{1}{25}+\frac{9}{16}+\frac{3}{40}$$

Least common multiple of 10225 and 25 is 10225. Convert \frac{3297}{10225} and \frac{1}{25} to fractions with denominator 10225.

$$\frac{3297}{10225}+\frac{409}{10225}+\frac{9}{16}+\frac{3}{40}$$

Since \frac{3297}{10225} and \frac{409}{10225} have the same denominator, add them by adding their numerators.

$$\frac{3297+409}{10225}+\frac{9}{16}+\frac{3}{40}$$

Add 3297 and 409 to get 3706.

$$\frac{3706}{10225}+\frac{9}{16}+\frac{3}{40}$$

Least common multiple of 10225 and 16 is 163600. Convert \frac{3706}{10225} and \frac{9}{16} to fractions with denominator 163600.

$$\frac{59296}{163600}+\frac{92025}{163600}+\frac{3}{40}$$

Since \frac{59296}{163600} and \frac{92025}{163600} have the same denominator, add them by adding their numerators.

$$\frac{59296+92025}{163600}+\frac{3}{40}$$

Add 59296 and 92025 to get 151321.

$$\frac{151321}{163600}+\frac{3}{40}$$

Least common multiple of 163600 and 40 is 163600. Convert \frac{151321}{163600} and \frac{3}{40} to fractions with denominator 163600.

$$\frac{151321}{163600}+\frac{12270}{163600}$$

Since \frac{151321}{163600} and \frac{12270}{163600} have the same denominator, add them by adding their numerators.

$$\frac{151321+12270}{163600}$$

Add 151321 and 12270 to get 163591.

$$\frac{163591}{163600}$$

Factor

$$\frac{17 \cdot 9623}{2 ^ {4} \cdot 5 ^ {2} \cdot 409} = 0.9999449877750611$$