A vendor sold 18 pens for ₹1,440 and gained a profit of 20%. How many pens must he sell for ₹1,408 to make a profit of 12%(to the nearest integer)?

The correct answer is Option 19.

Let us solve the problem step-by-step.

Step 1: Find the selling price (SP) and cost price (CP) of 1 pen.

The vendor sold 18 pens for ₹1,440.
Selling price of 1 pen (SP₁) = $\frac{1440}{18}$ = ₹80.

He made a profit of 20% on this sale.
We know that:
$SP=CP\times (1+\frac{\text{Profit \%}}{100})$

$80=\text{CP}\times (1+\frac{20}{100})$

$80=\text{CP}\times \frac{120}{100}$

$\text{CP}=\frac{80\times 100}{120}$

$\text{CP}=\frac{200}{3}$ ₹ per pen.

Step 2: Find the new selling price of 1 pen to make a profit of 12%.

New profit desired = 12%
New selling price per pen (SP₂):
${\text{SP}}_2=\text{CP}\times (1+\frac{12}{100})$

${\text{SP}}_2=\frac{200}{3}\times \frac{112}{100}$

${\text{SP}}_2=\frac{2\times 112}{3}=\frac{224}{3}$ ₹ per pen.

Step 3: Calculate the number of pens to be sold for ₹1,408.

Total Revenue required = ₹1,408
Number of pens = $\frac{\text{Total Revenue}}{{\text{SP}}_2}$

$\text{Number of pens}=\frac{1408}{224/3}$

$\text{Number of pens}=\frac{1408\times 3}{224}$

$\text{Number of pens}=6.2857\times 3=18.857$

Rounding off to the nearest integer, we get 19 pens.