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Locations of 4 Friends

**What was the given data & question?**
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.**
** **
In 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. **
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