B is 20% more efficient than A. If B was 60% more efficient than A, then B could complete the work in 18 days lesser than number of days taken by A. What fraction of work would be left after 12 days, if A and B work together?

The correct option is:
$\frac{9}{20}$

Step-by-Step Explanation:

Let the efficiency of A be $x$ units of work per day.
This means A can complete $x$ units of work in one day.

1. Analyze the hypothetical case:
If B was 60% more efficient than A, then B's hypothetical efficiency would be:
$E_{\mathrm{B\text{'}}}=1.6x$ units of work per day.

Let the total work to be completed be $W$ units.
The number of days taken by A to complete the work is:
$D_A=\frac{W}{x}$

The number of days taken by B under the hypothetical efficiency is:
$D_{\mathrm{B\text{'}}}=\frac{W}{1.6x}$

According to the problem, B would take 18 days less than A:
$D_A-D_{\mathrm{B\text{'}}}=18$

Substitute the values of $D_A$ and $D_{\mathrm{B\text{'}}}$ into the equation:
$\frac{W}{x}-\frac{W}{1.6x}=18$

Simplify the equation by factoring out $\frac{W}{x}$:
$\frac{W}{x}(1-\frac{1}{1.6})=18$

Since $1.6=\frac{8}{5}$, we have:
$\frac{W}{x}(1-\frac{5}{8})=18$

$\frac{W}{x}\left(\frac{3}{8}\right)=18$

Solve for $\frac{W}{x}$ (which represents the total days taken by A):
$D_A=\frac{W}{x}=18\times \frac{8}{3}=48\text{days}$

Thus, A alone can complete the work in 48 days.
Let us assume the total work $W=48$ units.
Then A's efficiency $x=1$ unit per day.

2. Determine the actual case:
B is actually 20% more efficient than A.
So, B's actual efficiency is:
$E_B=1.2\times 1=1.2$ units per day.

Combined efficiency of A and B working together:
$E_{A+B}=1+1.2=2.2$ units per day.

3. Calculate the work completed and left after 12 days:
Work completed in 12 days by A and B together:
$\text{Work done}=12\times 2.2=26.4$ units.

Remaining work left:
$\text{Work left}=48-26.4=21.6$ units.

Fraction of work left:
$\text{Fraction left}=\frac{21.6}{48}=\frac{216}{480}$

Dividing the numerator and denominator by 24:
$\text{Fraction left}=\frac{9}{20}$