Certainly! Let's work through the problems step-by-step.
Part (i)
Evaluate the expression \(1 \frac{1}{2} \div\left(3 \frac{1}{3}+4 \frac{1}{5}-6 \frac{1}{2}\right)\).
1. First, convert the mixed numbers to improper fractions:
- \(1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}\)
- \(3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}\)
- \(4 \frac{1}{5} = 4 + \frac{1}{5} = \frac{20}{5} + \frac{1}{5} = \frac{21}{5}\)
- \(6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2}\)
2. Calculate the expression inside the parentheses:
[tex]\[
3 \frac{1}{3} + 4 \frac{1}{5} - 6 \frac{1}{2} = \frac{10}{3} + \frac{21}{5} - \frac{13}{2}
\][/tex]
3. To add and subtract these fractions, find a common denominator (LCM of 3, 5, and 2 is 30):
- Convert each fraction to have the denominator 30:
- \(\frac{10}{3} = \frac{10 \times 10}{3 \times 10} = \frac{100}{30}\)
- \(\frac{21}{5} = \frac{21 \times 6}{5 \times 6} = \frac{126}{30}\)
- \(\frac{13}{2} = \frac{13 \times 15}{2 \times 15} = \frac{195}{30}\)
4. Add and subtract the numerators:
[tex]\[
\frac{100}{30} + \frac{126}{30} - \frac{195}{30} = \frac{100 + 126 - 195}{30} = \frac{31}{30}
\][/tex]
5. Now, divide \(\frac{3}{2}\) by \(\frac{31}{30}\):
[tex]\[
1 \frac{1}{2} \div\left(3 \frac{1}{3}+4 \frac{1}{5}-6 \frac{1}{2}\right)
= \frac{3}{2} \div \frac{31}{30} = \frac{3}{2} \times \frac{30}{31} = \frac{90}{62} = \frac{45}{31} \approx 1.4516129032258067
\][/tex]
So, the value for part (i) is approximately \(1.4516129032258067\).
Part (ii)
Evaluate the expression \(1 \frac{7}{53}\) of \(\left[1 \frac{1}{5}-\left\{3 \frac{4}{5} \div(\ldots\right.\right. \).
Unfortunately, the given problem for part (ii) is incomplete, so it is impossible to determine a valid solution without additional information. The expression after \(3 \frac{4}{5} \div(\ldots)\) is missing, and therefore part (ii) cannot be solved as presented.
In conclusion:
- The value for part (i) is approximately \(1.4516129032258067\).
- The value for part (ii) cannot be determined as the expression is incomplete or incorrect.
