John has chocolates of types A and B in the ratio 3 : 7, while Mike has chocolates of types B and C in the ratio 5 : 4, Ram has chocolates of types C and A in the ratio 3 : 5. If there are more chocolates of type C than of type B, and more of type B than of type A, what is the minimum possible number of chocolates overall?
Again, big thanks to Mukund Sukumar for excellent solution.
Let john have 3x chocolates of type A and 7x of type B
Let Mike have 5y chocolates of type B and 4y of type C
Let john have 3z chocolates of type C and 5z of type A
So in total A=3x+5z ; B=7x+5y ; C=4y+3z
Since C>B we get solving y<3z-7x —>(1)
Since B>A we get solving 5y>5z-4x —> (2)
What gets inferred from above 2 statements is z>=3. so when x=1,z=3, we get only y=1 as choice, for which second condition doesnt satisfy.
So, when x=1,z=4, we get y<5 from first condition and when y > 3.2 from second condition. so which gives choice the only y=4.
Hence x=1,y=4 and z=4 works and is the best possible answer.
For these values, we get A=23,B=27,C=28.
Minimum possible number of chocolates overall is 78.