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I thought of solving this ques by forming an equation as =>10x+20y=k(x+y) =>x(10-k)=y(k-20) =>x/y=k-20/10-k now since x<y =>k-20<10-k =>2k<30 =>k<15

I know i am unable to reach a unique answer with this...but would like to know is there something else going wrong with this approach?

the step highlighted above is incorrect. This does not follow from x/y=(k-20)/(10-k) Eg. If x/y=2/3=(k-20)/(10-k) Now if k-20=2 then 20-k=3 so indeed your inequality holds But if k-20=-2 then 20-k=-3 so the inequality you have concluded is wrong

In general when manipulating around with inequalities you have to be very careful and be-ware that you are not implicitly assuming the positive or negative nature of any of the terms o/w you can get such wrong results
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I got that the expression reduces to: k =10(1+ y/x+y) Can somebody please elaborate after that?How are we getting values?

After this you need to think what values the expression y/(x+y) can take. Since x<y so x "limits to " y and expression can reach 0.5. So expression is greater than 0.5

x limits to 0 and expression can reach 1. So expression is smaller than 1. Hope this answers how we got the range.

Now 10 + 10 multiplied by range 0.5 to 1 will give you the limit. Pick answer choice fitting the range
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The way I thought about it was that k is made up of (x/x+y) percent of 10 and (y/x+y) percent of 20. That is how interpolation is done. Then since y is greater than x, you will have the number close to 20 than to 10. It also has to be between 20 and 10, since it is made up of certain percentages of these two number.