## Thursday, November 16, 2017

### Locations of 4 Friends

Let's represent houses of friends by their initial letter of the name i.e. C is the house of Chris, D is the house of Darren, B is Bill's house & A is name of Alex's house.

Now C & A are located at 60 km away from each other while D & B are 40 km apart i.e. CA = 60 & DB = 40.

As per given data, D is 10 km nearer than B from C. Hence, CB = CD + 10

There are 5 possibilities of the locations of these 4 friends.

Case 1. D & B are in between C & A.

In this case,

CD + DB + BA = 60

but DB = 40,

CD + BA = 20,

CB - 10 + BA = 20

CB + BA = 30 i.e. CA = 30

But CA = 60, hence this combination is not possible. Other way, CB >= 40, since CD = CB-10, CD >= 30. That means CD + DB >= 70 for which B needs to be beyond A as CA = 60.

Case 2 : A is between D & B.

In this case, CD + DA  = 60, DA + AB = 40 . If we subtract second one from first, we get,

CD - AB = 20

CB - 10 - AB = 20

CB - AB = 30

CA = 30 but CA = 60. Hence, this combination too is invalid.

Case 3 : C is between D & B.

I this case, DC + CB = 40 , CB + BA = 60. But DC = CB - 10

CB - 10 + CB = 40

2CB  = 50

CB = 25

Hence DC = 15.

And DA = DC + CA = 60 + 15 = 75. That is Darren's house is 75 km away from Alex's.

Case 4 : C is between B & D.

Now, here, BC + CD = 40 & CD + DA = 60. We know, BC = CD + 10,

CD + CD + 10 = 40

2CD = 30

CD = 15.

Hence, BC = 15 + 10 = 25.

DA = CA - CD = 60 - 15 = 45.

So this another possible location of Darren's house from Alex's house where it is located 45 km away from it.

Case 5 :  B & D are located at left of C.

In this case, BC >= 40, DC >= 30. Now this can be vary from 30 to any value, so we can't be sure with exact value but one thing is sure that DA = DC + CA hence, DA >= 60 + 30. In other words, Darren's house would be at least 90 km away from Alex's house.

Case 6 : D & B are at right side of A.

From figure, CB = CD + 40 but as per the given data is CB = CD + 10. Hence, this possibility too eliminated.

To conclude, Darren's house is at 75 km or 45 km from Alex's house depending on position of Bill's house.