Let's break down the fractions given in the image:

Problem a): $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}$

We need to find the least common denominator (LCD) of these fractions. The denominators are $2, 3, 4, 5, 6, 7, 8$. The least common multiple (LCM) of these numbers is 840.

Now, we rewrite each fraction with 840 as the denominator: $\frac{1}{2} = \frac{420}{840}, \quad \frac{1}{3} = \frac{280}{840}, \quad \frac{1}{4} = \frac{210}{840}, \quad \frac{1}{5} = \frac{168}{840},$ $\frac{1}{6} = \frac{140}{840}, \quad \frac{1}{7} = \frac{120}{840}, \quad \frac{1}{8} = \frac{105}{840}$

Adding all the numerators together: $420 + 280 + 210 + 168 + 140 + 120 + 105 = 1443$

So the result for part (a) is: $\frac{1443}{840} \approx 1.718$

Problem b): $5 + \frac{1}{2} + 2 + \frac{1}{3} + \frac{3}{8} - 4 + \frac{11}{14} + 3 + \frac{3}{7} - \frac{5}{6}$

We start by simplifying the integers: $5 + 2 - 4 + 3 = 6$

Now handle the fractional part: $\frac{1}{2}, \frac{1}{3}, \frac{3}{8}, \frac{11}{14}, \frac{3}{7}, -\frac{5}{6}$

To add and subtract these fractions, we again need to find the least common denominator (LCD). The denominators are $2, 3, 8, 14, 7, 6$. The LCM of these numbers is 168.

Rewriting each fraction with 168 as the denominator: $\frac{1}{2} = \frac{84}{168}, \quad \frac{1}{3} = \frac{56}{168}, \quad \frac{3}{8} = \frac{63}{168}, \quad \frac{11}{14} = \frac{132}{168},$ $\frac{3}{7} = \frac{72}{168}, \quad \frac{5}{6} = \frac{140}{168}$

Now sum and subtract the numerators: $84 + 56 + 63 + 132 + 72 - 140 = 267$

So the fractional part is: $\frac{267}{168} \approx 1.589$

Adding the integer part, 6: $6 + 1.589 = 7.589$

Thus, the answer for part (b) is approximately 7.589.

Would you like more details or have any questions?

Here are five related questions you could explore:

1. How do you find the least common denominator for fractions?
2. What is the importance of simplifying fractions in calculations?
3. How can we convert improper fractions to mixed numbers?
4. Why is it important to perform operations on fractions carefully?
5. What are some common mistakes when adding or subtracting fractions?

Tip: Always simplify your fractions if possible to make your final answer clearer!