A man sells 12 articles for ₹ 80 gaining $33\dfrac{1}{3}\%$. Find the number of articles bought by the man for ₹ 90.

Given:

S.P. of 12 articles = ₹ 80

S.P. of 1 article = ₹ $\dfrac{80}{12}$ = ₹ $\dfrac{20}{3}$

Gain = $33\dfrac{1}{3}\%$ = $\dfrac{100}{3}\%$

Let C.P. of 1 article = $₹x$.

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

Putting the values, we get

$\Rightarrow \dfrac{100}{3} = \dfrac{\text{Profit}}{x} \times 100\\[1em] \Rightarrow \text{Profit} = \dfrac{100 \times x}{3 \times 100}\\[1em] = \dfrac{\cancel{100} \times x}{3 \times \cancel{100}}\\[1em] = \dfrac{x}{3}$

As we know:

$\text{Profit} = \text{S.P. - C.P.}\\[1em] \Rightarrow \dfrac{x}{3} = \dfrac{20}{3} - x\\[1em] \Rightarrow \dfrac{20}{3} = \dfrac{x}{3} + x\\[1em] \Rightarrow \dfrac{20}{3} = \dfrac{x}{3} + \dfrac{3x}{3}\\[1em] \Rightarrow \dfrac{20}{3} = \dfrac{(x + 3x)}{3} \\[1em] \Rightarrow \dfrac{20}{3} = \dfrac{4x}{3}\\[1em] \Rightarrow x = \dfrac{20 \times 3}{4 \times 3}\\[1em] \Rightarrow x = \dfrac{60}{12}\\[1em] \Rightarrow x = 5$

C.P. of 1 article = ₹ 5

Let number of articles bought by the man for ₹ 90 be $y$

C.P. of $y$ articles = ₹ 5 x $y$ = ₹ 90

⇒ $y = \dfrac{90}{5}$

⇒ $y = 18$

Hence, 18 articles are bought by the man for ₹ 90.