Let's solve these one by one:

(i) $\frac{7}{210} - \frac{2}{5}$

To subtract these fractions, we need to have the same denominator.

1. The LCM (least common multiple) of 210 and 5 is 210.
2. Convert $\frac{2}{5}$ to a fraction with denominator 210: $\frac{2}{5} = \frac{2 \times 42}{5 \times 42} = \frac{84}{210}$
3. Now, subtract the fractions: $\frac{7}{210} - \frac{84}{210} = \frac{7 - 84}{210} = \frac{-77}{210}$
4. Simplify the fraction by dividing both the numerator and denominator by 7: $\frac{-77}{210} = \frac{-11}{30}$ Thus, $\frac{7}{210} - \frac{2}{5} = \frac{-11}{30}$.

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(ii) $5\frac{1}{2} - 1\frac{3}{4}$

1. Convert the mixed numbers to improper fractions: $5\frac{1}{2} = \frac{11}{2}, \quad 1\frac{3}{4} = \frac{7}{4}$
2. To subtract, we need the same denominator. The LCM of 2 and 4 is 4.
3. Convert $\frac{11}{2}$ to a fraction with denominator 4: $\frac{11}{2} = \frac{11 \times 2}{2 \times 2} = \frac{22}{4}$
4. Now subtract the fractions: $\frac{22}{4} - \frac{7}{4} = \frac{22 - 7}{4} = \frac{15}{4}$
5. Convert $\frac{15}{4}$ back to a mixed number: $\frac{15}{4} = 3\frac{3}{4}$ Thus, $5\frac{1}{2} - 1\frac{3}{4} = 3\frac{3}{4}$.

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(iii) $\frac{4}{5} \times \frac{2}{3}$

To multiply fractions, multiply the numerators and denominators:

$\frac{4}{5} \times \frac{2}{3} = \frac{4 \times 2}{5 \times 3} = \frac{8}{15}$

Thus, $\frac{4}{5} \times \frac{2}{3} = \frac{8}{15}$.

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(iv) $\frac{1}{4} \text{ of } \frac{64}{7}$

"Of" in mathematics usually means multiplication. So, we multiply the two fractions:

$\frac{1}{4} \times \frac{64}{7} = \frac{1 \times 64}{4 \times 7} = \frac{64}{28}$

Now, simplify $\frac{64}{28}$ by dividing both numerator and denominator by 4:

$\frac{64}{28} = \frac{16}{7}$

Thus, $\frac{1}{4} \text{ of } \frac{64}{7} = \frac{16}{7}$.

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Summary of Results:

1. (i) $\frac{7}{210} - \frac{2}{5} = \frac{-11}{30}$
2. (ii) $5\frac{1}{2} - 1\frac{3}{4} = 3\frac{3}{4}$
3. (iii) $\frac{4}{5} \times \frac{2}{3} = \frac{8}{15}$
4. (iv) $\frac{1}{4} \text{ of } \frac{64}{7} = \frac{16}{7}$

Would you like more details on any part of the solutions? Here are some related questions:

1. How do you find the least common multiple of two numbers?
2. How can you simplify improper fractions?
3. What are the steps to convert a mixed number into an improper fraction?
4. Why is multiplying fractions easier than adding or subtracting them?
5. How do you check your work when simplifying fractions?

Tip: Always simplify your fractions to their lowest terms to ensure the answer is in its simplest form.