If goods are purchased at ₹450 and \(\frac{1}{3}\) are sold at a loss of 10%, at what gain percentage should the remainder be sold so as to gain 20% on the whole transaction?

Given:

Cost Price of total goods = ₹450

Fraction sold at 10% loss = 1/3

Required overall gain = 20%

Formula used:

Total SP = CP × (1 + Gain%)

SP of 1^st part = CP of 1^st part × (1 - Loss%)

SP of 2^nd part = CP of 2^nd part × (1 + x%)

Calculation:

CP of 1^st part = 1/3 × 450 = 150

CP of 2^nd part = 2/3 × 450 = 300

SP of 1^st part = 150 × (1 - 10/100) = 150 × 0.9 = 135

Total SP = 450 × (1 + 20/100) = 450 × 1.2 = 540

Let required gain % on 2^nd part be x%

SP of 2^nd part = 300 × (1 + x/100)

Total SP = SP of 1^st part + SP of 2^nd part

⇒ 540 = 135 + 300 × (1 + x/100)

⇒ 540 - 135 = 300 × (1 + x/100)

⇒ 405 = 300 × (1 + x/100)

⇒ 405 ÷ 300 = 1 + x/100

⇒ x/100 = 0.35

⇒ x = 35%

∴ The remainder should be sold at a gain of 35%.