Thirty articles are bought at ₹ 450 each. If one-third of these articles are sold at 6% loss; at what price must each of the remaining articles be sold in order to make a profit of 10% on the whole ?

Given:

C.P. of 1 article = ₹ 450

C.P. of 30 articles = ₹ 450 x 30 = ₹ 13,500

Profit desired on the whole = 10%

$\text{Profit\%} = \dfrac{\text{Profit}}{\text{C.P}}\times 100$

Putting the values, we get

$10 = \dfrac{\text{Profit}}{13500} \times 100\\[1em] \Rightarrow \text{Profit} = \dfrac{10 \times 13500}{100}\\[1em] = \dfrac{135000}{100}\\[1em] = 1350$

As we know,

$\text{Profit} = \text{S.P. - C.P.}$

Putting the values, we get

$1350 = \text{S.P.} - 13500\\[1em] \Rightarrow \text{S.P.} = 13500 + 1350\\[1em] = 14850$

C.P. of $\dfrac{1}{3}$articles = ₹ $\Big(\dfrac{1}{3} \times 13500\Big)$

= ₹ $\dfrac{13500}{3}$

= ₹ $4500$

Loss on it = 6%

$\text{Loss\%} = \dfrac{\text{Loss}}{\text{C.P}}\times 100$

Putting the values, we get

$6 = \dfrac{\text{Loss}}{4500} \times 100\\[1em] \Rightarrow \text{Loss} = \dfrac{6 \times 4500}{100}\\[1em] = \dfrac{27000}{100}\\[1em] = 270$

As we know,

$\text{Loss} = \text{C.P. - S.P.}$

Putting the values, we get

$270 = 4500 - \text{S.P}\\[1em] \Rightarrow \text{S.P.} = 4500 - 270\\[1em] = 4230$

The S.P. of the rest of the articles = ₹ 14850 - ₹ 4230 = ₹ 10620

Total articles = 30

Two - third of articles = $\dfrac{2}{3} \times 30$ = 20

The S.P. of each of the 20 articles = ₹ $\dfrac{10620}{20}$ = ₹ 531

The selling price of each of the article = ₹ 531.