Let's break the problem into manageable parts and solve it step-by-step.

Step 1: A's work rate

A can complete the work in 18 days. Therefore, A's work rate is:

$\text{A's rate} = \frac{1}{18} \text{ work per day}.$

Step 2: A and B's combined rate

A and B together can complete half of the work in 6 days. So, A and B together can complete $\frac{1}{2}$ of the work in 6 days. Their combined work rate is:

$\text{(A + B)'s rate} = \frac{1/2}{6} = \frac{1}{12} \text{ work per day}.$

We already know A's rate is $\frac{1}{18}$. To find B's rate, we subtract A's rate from the combined rate:

$\text{B's rate} = \frac{1}{12} - \frac{1}{18}.$

To subtract, find a common denominator:

$\text{B's rate} = \frac{3}{36} - \frac{2}{36} = \frac{1}{36} \text{ work per day}.$

Step 3: C's rate

C is 20% more efficient than B, meaning C's work rate is 120% of B's rate:

$\text{C's rate} = 1.2 \times \text{B's rate} = 1.2 \times \frac{1}{36} = \frac{1.2}{36} = \frac{1}{30} \text{ work per day}.$

Step 4: Combined rate of A, B, and C

Now, let's find the combined rate of A, B, and C:

$\text{(A + B + C)'s rate} = \text{A's rate} + \text{B's rate} + \text{C's rate}.$ Substitute the values:

$\text{(A + B + C)'s rate} = \frac{1}{18} + \frac{1}{36} + \frac{1}{30}.$

To add these, find the common denominator, which is 180:

$\frac{1}{18} = \frac{10}{180}, \quad \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{30} = \frac{6}{180}.$

Now, add the fractions:

$\text{(A + B + C)'s rate} = \frac{10}{180} + \frac{5}{180} + \frac{6}{180} = \frac{21}{180} = \frac{7}{60} \text{ work per day}.$

Step 5: Time to complete 84% of the work

A, B, and C together complete $\frac{7}{60}$ of the work in 1 day. To find how many days it takes them to complete 84% (or $0.84$) of the work, we use the formula:

$\text{Time} = \frac{0.84}{\text{(A + B + C)'s rate}} = \frac{0.84}{\frac{7}{60}} = 0.84 \times \frac{60}{7} = \frac{50.4}{7} = 7.2 \text{ days}.$

Final Answer:

It will take A, B, and C together 7.2 days to complete 84% of the work.

Would you like more details or have any questions? Here are 5 related questions:

1. How would the total work time change if C were 30% more efficient than B?
2. How long would it take for only B and C to complete the entire work together?
3. If A worked alone, how long would it take to complete 75% of the work?
4. What if A, B, and C worked together for 5 days—how much work would be completed?
5. How much time would B need alone to finish 50% of the work?

Tip: When working with rates, always remember that time and efficiency are inversely related. More efficiency means less time needed to complete the same work.