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Step-by-Step Solution:
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Finding the LCM of Denominators:
- The denominators in the given expression are 11, 3, 5, and 7.
- The LCM of these numbers is 231 (11 x 3 x 5 x 7).
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Converting Fractions to Have a Common Denominator:
- To convert 15/11 to have a denominator of 231, we multiply the numerator and denominator by 21 (since 11 x 21 = 231). This gives us 315/231.
- To convert 10/3 to have a denominator of 231, we multiply the numerator and denominator by 77 (since 3 x 77 = 231). This gives us 770/231.
- To convert 4/5 to have a denominator of 231, we multiply the numerator and denominator by 46 (since 5 x 46 = 231). This gives us 184/231.
- To convert 1/7 to have a denominator of 231, we multiply the numerator and denominator by 33 (since 7 x 33 = 231). This gives us 33/231.
- To convert 1/6 to have a denominator of 231, we multiply the numerator and denominator by 38 (since 6 x 38 = 231). This gives us 38/231.
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Performing the Arithmetic:
- Now, we have: 315/231 - 770/231 + 184/231 - 33/231 + 38/231.
- Combining the numerators, we get: (315 - 770 + 184 - 33 + 38)/231 = (735)/231 = 735/231.
- Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 21, we get: 35/11.
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Therefore, the final result of the expression 15/11 - 10/3 + 4/5 - 1/7 + 1/6 is 35/11.