To solve this problem, let's break down the given information step-by-step and perform the calculations accordingly:
### Step 1: Define Variables and Relations
Let:
- [tex]\(C\)[/tex] be the number of days taken by C to complete the work alone.
- [tex]\(B\)[/tex] be the number of days taken by B to complete the work alone.
- [tex]\(A\)[/tex] be the number of days taken by A to complete the work alone.
From the problem, we know:
1. Efficiency Relation Between A and B:
A is 40% less efficient than B. This means if B can complete the work in [tex]\(B\)[/tex] days, then A, being less efficient, will take more time, specifically:
[tex]\[
\text{Efficiency of A} = \text{Efficiency of B} \times 0.6 \quad \Rightarrow \quad \frac{1}{A} = \frac{1}{B} \cdot 0.6 \quad \Rightarrow \quad \frac{1}{A} = \frac{0.6}{B} \quad \Rightarrow \quad A = \frac{B}{0.6} = \frac{5}{3}B
\][/tex]
2. Efficiency Relation Between B and C:
B can complete the work in 20% less time than C. This means:
[tex]\[
B = 0.8 \cdot C
\][/tex]
### Step 2: Calculate Combined Efficiency of A and B
Together A and B can complete 80% of the work in 12 days. Let's find their combined efficiency:
1. Combined Efficiency:
Since efficiency is a measure of work done per unit time:
[tex]\[
\frac{1}{A} + \frac{1}{B} = \text{Combined Efficiency of A and B}
\][/tex]
We already calculated [tex]\( A = \frac{5}{3}B \)[/tex]:
[tex]\[
\frac{1}{A} + \frac{1}{B} = \frac{3}{5B} + \frac{1}{B} = \frac{3 + 5}{5B} = \frac{8}{5B}
\][/tex]
2. Work Done in 12 Days by A and B:
Since A and B together can complete 80% of the work in 12 days, the combined efficiency in terms of time can be written as:
[tex]\[
\text{Efficiency of A and B} \times 12 = 0.8 \quad \Rightarrow \quad \frac{8}{5B} \times 12 = 0.8
\][/tex]
Solving for [tex]\(B\)[/tex]:
[tex]\[
\frac{96}{5B} = 0.8 \quad \Rightarrow \quad 96 = 4B \quad \Rightarrow \quad B = 24
\][/tex]
Thus, B can complete the work alone in 24 days.
### Step 3: Calculate Time for B and C Together
Now we need to find how many days B and C together will take to complete 60% of the work. First, we find C's time to complete the work alone using [tex]\(B = 0.8 \cdot C\)[/tex]:
[tex]\[
24 = 0.8 \cdot C \quad \Rightarrow \quad C = \frac{24}{0.8} = 30
\][/tex]
1. Efficiency of B and C:
[tex]\[
\frac{1}{B} + \frac{1}{C} = \frac{1}{24} + \frac{1}{30}
\][/tex]
Finding a common denominator:
[tex]\[
\frac{1}{24} + \frac{1}{30} = \frac{5}{120} + \frac{4}{120} = \frac{9}{120} = \frac{3}{40}
\][/tex]
2. Time to Complete 60% of the Work:
[tex]\[
\text{Efficiency of B and C} = \frac{3}{40}
\][/tex]
If 60% of the work is to be completed:
[tex]\[
\left(\frac{3}{40}\right) \times \text{Number of Days} = 0.6 \quad \Rightarrow \quad \text{Number of Days} = \frac{0.6 \times 40}{3} = 8
\][/tex]
Thus, B and C together can complete 60% of the work in [tex]\(\boxed{8 \text{ days}}\)[/tex].
