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Answer :
Sure! Let's go through each part of the problem step by step.
1. Calculate [tex]\(10 + 12 - (8 - 3)\)[/tex]:
- Start with the expression inside the parentheses: [tex]\(8 - 3 = 5\)[/tex].
- Substitute back: [tex]\(10 + 12 - 5\)[/tex].
- Perform addition: [tex]\(10 + 12 = 22\)[/tex].
- Perform the subtraction: [tex]\(22 - 5 = 17\)[/tex].
So, the result for the first expression is 17.
2. Calculate [tex]\(\frac{10}{\frac{12}{22}}\)[/tex]:
- Divide 12 by 22: [tex]\(\frac{12}{22}\)[/tex].
- The expression [tex]\(10 \div \left(\frac{12}{22}\right)\)[/tex] is equivalent to [tex]\(10 \times \frac{22}{12}\)[/tex].
- Simplify [tex]\(\frac{22}{12}\)[/tex] to get [tex]\(\frac{11}{6}\)[/tex].
- Multiply: [tex]\(10 \times \frac{11}{6} = \frac{110}{6} = 18.333...\)[/tex].
So, the result for the second expression is approximately 18.333.
3. Calculate [tex]\(\frac{-8}{-14}\)[/tex]:
- Division of two negative numbers is positive: [tex]\(\frac{-8}{-14} = \frac{8}{14}\)[/tex].
- Simplify [tex]\(\frac{8}{14}\)[/tex] to [tex]\(\frac{4}{7}\)[/tex].
- As a decimal, [tex]\(\frac{4}{7} \approx 0.571\)[/tex].
So, the result for the third expression is approximately 0.571.
4. Calculate [tex]\(\frac{\frac{10}{12}}{-\frac{21}{-11}}\)[/tex]:
- Simplify the expression: [tex]\(\frac{10}{12} = \frac{5}{6}\)[/tex] and [tex]\(-\frac{21}{-11} = \frac{21}{11}\)[/tex] (since the negative signs cancel).
- The expression is: [tex]\(\frac{5}{6} \div \frac{21}{11} = \frac{5}{6} \times \frac{11}{21}\)[/tex].
- Multiply: [tex]\(\frac{5 \times 11}{6 \times 21} = \frac{55}{126}\)[/tex].
- Simplifying [tex]\(\frac{55}{126}\)[/tex] results in approximately 0.437.
So, the result for the fourth expression is approximately 0.437.
5. Determine the HCF of 210 and 330 using prime factorization:
- Prime factors of 210: [tex]\(2, 3, 5, 7\)[/tex].
- Prime factors of 330: [tex]\(2, 3, 5, 11\)[/tex].
- Common factors: [tex]\(2, 3, 5\)[/tex].
- Multiply these common factors: [tex]\(2 \times 3 \times 5 = 30\)[/tex].
So, the highest common factor (HCF) is 30.
In summary, the results are:
1. [tex]\(10 + 12 - (8 - 3) = 17\)[/tex].
2. [tex]\(\frac{10}{\frac{12}{22}} \approx 18.333\)[/tex].
3. [tex]\(\frac{-8}{-14} \approx 0.571\)[/tex].
4. [tex]\(\frac{\frac{10}{12}}{-\frac{21}{-11}} \approx 0.437\)[/tex].
5. HCF of 210 and 330 is 30.
1. Calculate [tex]\(10 + 12 - (8 - 3)\)[/tex]:
- Start with the expression inside the parentheses: [tex]\(8 - 3 = 5\)[/tex].
- Substitute back: [tex]\(10 + 12 - 5\)[/tex].
- Perform addition: [tex]\(10 + 12 = 22\)[/tex].
- Perform the subtraction: [tex]\(22 - 5 = 17\)[/tex].
So, the result for the first expression is 17.
2. Calculate [tex]\(\frac{10}{\frac{12}{22}}\)[/tex]:
- Divide 12 by 22: [tex]\(\frac{12}{22}\)[/tex].
- The expression [tex]\(10 \div \left(\frac{12}{22}\right)\)[/tex] is equivalent to [tex]\(10 \times \frac{22}{12}\)[/tex].
- Simplify [tex]\(\frac{22}{12}\)[/tex] to get [tex]\(\frac{11}{6}\)[/tex].
- Multiply: [tex]\(10 \times \frac{11}{6} = \frac{110}{6} = 18.333...\)[/tex].
So, the result for the second expression is approximately 18.333.
3. Calculate [tex]\(\frac{-8}{-14}\)[/tex]:
- Division of two negative numbers is positive: [tex]\(\frac{-8}{-14} = \frac{8}{14}\)[/tex].
- Simplify [tex]\(\frac{8}{14}\)[/tex] to [tex]\(\frac{4}{7}\)[/tex].
- As a decimal, [tex]\(\frac{4}{7} \approx 0.571\)[/tex].
So, the result for the third expression is approximately 0.571.
4. Calculate [tex]\(\frac{\frac{10}{12}}{-\frac{21}{-11}}\)[/tex]:
- Simplify the expression: [tex]\(\frac{10}{12} = \frac{5}{6}\)[/tex] and [tex]\(-\frac{21}{-11} = \frac{21}{11}\)[/tex] (since the negative signs cancel).
- The expression is: [tex]\(\frac{5}{6} \div \frac{21}{11} = \frac{5}{6} \times \frac{11}{21}\)[/tex].
- Multiply: [tex]\(\frac{5 \times 11}{6 \times 21} = \frac{55}{126}\)[/tex].
- Simplifying [tex]\(\frac{55}{126}\)[/tex] results in approximately 0.437.
So, the result for the fourth expression is approximately 0.437.
5. Determine the HCF of 210 and 330 using prime factorization:
- Prime factors of 210: [tex]\(2, 3, 5, 7\)[/tex].
- Prime factors of 330: [tex]\(2, 3, 5, 11\)[/tex].
- Common factors: [tex]\(2, 3, 5\)[/tex].
- Multiply these common factors: [tex]\(2 \times 3 \times 5 = 30\)[/tex].
So, the highest common factor (HCF) is 30.
In summary, the results are:
1. [tex]\(10 + 12 - (8 - 3) = 17\)[/tex].
2. [tex]\(\frac{10}{\frac{12}{22}} \approx 18.333\)[/tex].
3. [tex]\(\frac{-8}{-14} \approx 0.571\)[/tex].
4. [tex]\(\frac{\frac{10}{12}}{-\frac{21}{-11}} \approx 0.437\)[/tex].
5. HCF of 210 and 330 is 30.
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