Certainly! Let's solve each part of the question step by step.
### Part (i): \( 1 \frac{2}{5} \) of \( 4 \frac{2}{7} \) km
1. Convert the mixed numbers to improper fractions:
- \( 1 \frac{2}{5} = 1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5} \)
- \( 4 \frac{2}{7} = 4 + \frac{2}{7} = \frac{28}{7} + \frac{2}{7} = \frac{30}{7} \)
2. Multiply the improper fractions:
[tex]\[
\left( \frac{7}{5} \right) \times \left( \frac{30}{7} \right) = \frac{7 \times 30}{5 \times 7} = \frac{210}{35} = 6
\][/tex]
So, \( 1 \frac{2}{5} \) of \( 4 \frac{2}{7} \) km is \( 6 \) km.
### Part (ii): \(\frac{1 \frac{1}{2}}{3 \frac{1}{3} + 4 \frac{1}{5} - 6 \frac{1}{2}}\)
1. Convert the mixed numbers to improper fractions:
- \( 1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2} \)
- \( 3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{10}{3} \)
- \( 4 \frac{1}{5} = 4 + \frac{1}{5} = \frac{21}{5} \)
- \( 6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{13}{2} \)
2. Evaluate 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]
To add and subtract these fractions, find a common denominator (30):
[tex]\[
\frac{10}{3} = \frac{100}{30}, \quad \frac{21}{5} = \frac{126}{30}, \quad \frac{13}{2} = \frac{195}{30}
\][/tex]
Combine the fractions:
[tex]\[
\frac{100}{30} + \frac{126}{30} - \frac{195}{30} = \frac{226}{30} - \frac{195}{30} = \frac{31}{30}
\][/tex]
3. Divide \( 1 \frac{1}{2} \) by the result:
[tex]\[
\frac{3}{2} \div \frac{31}{30} = \frac{3}{2} \times \frac{30}{31} = \frac{3 \times 30}{2 \times 31} = \frac{90}{62} = \frac{45}{31}
\][/tex]
In decimal form, this is approximately \( 1.4516129032258067 \).
### Part (iii): \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} \)
1. Simplify the innermost fraction first:
[tex]\[
1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}
\][/tex]
2. Use this result in the next fraction:
[tex]\[
1 + \frac{1}{\frac{3}{2}} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}
\][/tex]
3. Use this result in the outermost fraction:
[tex]\[
1 + \frac{1}{\frac{5}{3}} = 1 + \frac{3}{5} = \frac{5}{5} + \frac{3}{5} = \frac{8}{5}
\][/tex]
In decimal form, this is \( 1.6 \).
### Summary of Results:
1. \( 1 \frac{2}{5} \) of \( 4 \frac{2}{7} \) km is \( 6 \) km.
2. \( \frac{1 \frac{1}{2}}{3 \frac{1}{3} + 4 \frac{1}{5} - 6 \frac{1}{2}} \) is approximately \( 1.4516129032258067 \).
3. \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} \) is \( 1.6 \).
Each part of the question has been meticulously solved step by step.
