If goods be purchased for 450 and one third sold at a loss of 10%. At what gain percent should the remainder be sold so as to gain 20% on the whole transaction ?

Let no. of goods purchased be 3 for 450

=> Price of 1 good = 150

One third i.e. $$\frac{3}{3} = 1$$ is sold at 10% discount.

=> 10% of 150 = $$\frac{10}{100}$$ * 150 = 15

=> S.P. of 1 good = 150-15 = 135

Overall gain % = 20%

=> $$\frac{20}{100}$$ * 450 = 90

=> Total S.P. for 3 goods = 450+90 = 540

Since 1 good is sold at 135 => S.P. of remaining 2 = 540-135 = 405

C.P. of remaining 2 goods = 150*2 = 300

=> Required profit % = $$\frac{405-300}{300}$$ * 100

= 35%