A can complete the work in 18 days and A and B together can complete the half of the work in 6 days. C is 20% more efficient than B. In how many days A,B and C together can complete the 84% of the work?

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.

A can complete the work in 18 days and A and B together can complete the half of the work in 6 days. C is 20% more efficient than B. In how many days A, B, and C together can complete the 84% of the work?

Solve how long A, B, and C, with varying efficiencies, take to complete 84% of a task. A works alone in 18 days, A and B complete half the work in 6 days, and C is 20% more efficient than B. Suitable for students in Grades 9-12.

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