Solve 76*(1/23-1/53)+23*(1/53-1/76)-53*(frac{1}{23}-frac{1}{76})

$$76 \times ( \frac{ 1 }{ 23 } - \frac{ 1 }{ 53 } )+23 \times ( \frac{ 1 }{ 53 } - \frac{ 1 }{ 76 } )-53 \times ( \frac{ 1 }{ 23 } - \frac{ 1 }{ 76 } )$$

Evaluate

$$\frac{15}{38}\approx 0.394736842$$

Solution Steps

Least common multiple of 23 and 53 is 1219. Convert \frac{1}{23} and \frac{1}{53} to fractions with denominator 1219.

$$76\left(\frac{53}{1219}-\frac{23}{1219}\right)+23\left(\frac{1}{53}-\frac{1}{76}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Since \frac{53}{1219} and \frac{23}{1219} have the same denominator, subtract them by subtracting their numerators.

$$76\times \frac{53-23}{1219}+23\left(\frac{1}{53}-\frac{1}{76}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Subtract 23 from 53 to get 30.

$$76\times \frac{30}{1219}+23\left(\frac{1}{53}-\frac{1}{76}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Express 76\times \frac{30}{1219} as a single fraction.

$$\frac{76\times 30}{1219}+23\left(\frac{1}{53}-\frac{1}{76}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Multiply 76 and 30 to get 2280.

$$\frac{2280}{1219}+23\left(\frac{1}{53}-\frac{1}{76}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Least common multiple of 53 and 76 is 4028. Convert \frac{1}{53} and \frac{1}{76} to fractions with denominator 4028.

$$\frac{2280}{1219}+23\left(\frac{76}{4028}-\frac{53}{4028}\right)-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Since \frac{76}{4028} and \frac{53}{4028} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2280}{1219}+23\times \frac{76-53}{4028}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Subtract 53 from 76 to get 23.

$$\frac{2280}{1219}+23\times \frac{23}{4028}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Express 23\times \frac{23}{4028} as a single fraction.

$$\frac{2280}{1219}+\frac{23\times 23}{4028}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Multiply 23 and 23 to get 529.

$$\frac{2280}{1219}+\frac{529}{4028}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Least common multiple of 1219 and 4028 is 92644. Convert \frac{2280}{1219} and \frac{529}{4028} to fractions with denominator 92644.

$$\frac{173280}{92644}+\frac{12167}{92644}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Since \frac{173280}{92644} and \frac{12167}{92644} have the same denominator, add them by adding their numerators.

$$\frac{173280+12167}{92644}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Add 173280 and 12167 to get 185447.

$$\frac{185447}{92644}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Reduce the fraction \frac{185447}{92644} to lowest terms by extracting and canceling out 53.

$$\frac{3499}{1748}-53\left(\frac{1}{23}-\frac{1}{76}\right)$$

Least common multiple of 23 and 76 is 1748. Convert \frac{1}{23} and \frac{1}{76} to fractions with denominator 1748.

$$\frac{3499}{1748}-53\left(\frac{76}{1748}-\frac{23}{1748}\right)$$

Since \frac{76}{1748} and \frac{23}{1748} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3499}{1748}-53\times \frac{76-23}{1748}$$

Subtract 23 from 76 to get 53.

$$\frac{3499}{1748}-53\times \frac{53}{1748}$$

Express 53\times \frac{53}{1748} as a single fraction.

$$\frac{3499}{1748}-\frac{53\times 53}{1748}$$

Multiply 53 and 53 to get 2809.

$$\frac{3499}{1748}-\frac{2809}{1748}$$

Since \frac{3499}{1748} and \frac{2809}{1748} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3499-2809}{1748}$$

Subtract 2809 from 3499 to get 690.

$$\frac{690}{1748}$$

Reduce the fraction \frac{690}{1748} to lowest terms by extracting and canceling out 46.

$$\frac{15}{38}$$

Factor

$$\frac{3 \cdot 5}{2 \cdot 19} = 0.39473684210526316$$