Answer: Option B
:
B
Method:
Let the distance between home and office =d
Suppose he reaches the office on time, the time taken = x minutes
Case 1: When he reaches office 20 minutes late, time taken = x+20
Case 2: when he reaches office 10 minutes early, time taken = x-10
As the distance traveled is the same, the ratio of speed in case 1 to the speed in case 2 will be the inverse of the time taken in both cases.
To understand this better ,Let us call the horizontal distances as H and the vertical distances as V.
The person needs to first go to B.
Number of horizontal distances = Number of rows =2.
Number of vertical distances =Number of columns =3.
i.e.., HHVVV.
Now the question changes to the number of ways of arranging HHVVV=$\frac{5!}{3!2!}{=}^5{C}_2=10$ ways.
Similarly, from B to C, the number of distances =HHVVVV. Number of arrangements=$\frac{6!}{4!2!}=15$ ways .Total number of ways =$10\times 15=150.$
Ratio of speed in both cases = 4:6 = 2:3
Ratio of time in both cases = 3:2
Therefore,$\frac{x+20}{x-10}=\frac{3}{2}$
2x+40=30x-30$\to$ x=70 minutes.
Taking case 1:$\frac{4}{d}=\frac{60}{90}(90=x+20)d=\frac{360}{60}=6km.$
Shortcut- Constant Product Approach
Assume original speed= 4km/hr.
Percentage increase in speed from 4 kmph to 6 kmph= $50\%or\frac{1}{2}$
$\frac{1}{2}$ increase in speed will result in $\frac{1}{3}$ decrease in original time (from Constant Product Rule)
=30 minutes.(from given data). Thus, Original time= 90 minutes= 1.5 hours
Answer is Distance = 4 $\times$ 1.5=6 km
In case, you want to break down the shortcut in steps, you should proceed like this:
There are 2 cases here,
Case 1 $\Rightarrow$ speed = 4 km/hr , 20 imn late.
Case 2 $\Rightarrow$ speed = 6 km/hr , 10 min early.
Step 1:Calculate the increase or decrease in the parameters speed /time.
To calculate increase / decrease we need to consider a case as original case and then calculate the increase or decrease over it.
We will take the first case as the original case,so we can easily observe that the increase in speed s $50\%$ or $\frac{1}{2}$ from case 1 to case 2 $(\times 100=50\%)$
Step 2:Inverse Proportionality.
Now based on inverse proportionality there will always be a or $33\%$ or decrease in time from case 1 and case 2.
Step 3: Calculating the actual time and distance.
Suppose in the case 1 which we have taken as original, time taken is 'x' mins.
So due,to $\frac{1}{2}$ increase in speed we have decrease in time of 'x' =30 mins.
(form 20 minutes late to 10 mins early there is a decrease of 30 minutes.)
x=90 minutes.
So in the case 1, he is taking 90 mins to travel at a speed of 4 km/hr.So,distance =$1.5\times 4=6$ km.
The explanation may be long, but when you actually apply it; you can solve time consuming questions in very little time,helping you save precious minutes which you can use in your weaker areas !