A dishonest dealer professes to sell his goods at cost price but uses a false weight of 950 grams instead of 1 kilogram. His gain percentage is:

The correct answer is $5\frac{5}{19}\%$

The dealer claims to sell goods at cost price, but instead of weighing out 1000 grams (1 kg), he only gives 950 grams while charging for 1000 grams.

Step 1: Understand the Trick
Suppose the cost price of 1000 grams = ₹1000 (for easy calculation).
The dealer charges the customer ₹1000 but actually gives only 950 grams.
So the dealer's actual cost for 950 grams = ₹950.

Step 2: Calculate the Gain
He collects ₹1000 but spends only ₹950.
Gain = ₹1000 - ₹950 = ₹50

Step 3: Apply the Gain Percentage Formula
Gain % is always calculated on the Cost Price (what he actually spends).

$\text{Gain \%}=\frac{\text{Gain}}{\text{Cost Price}}\times 100$

$\text{Gain \%}=\frac{50}{950}\times 100$

$\text{Gain \%}=\frac{5000}{950}$

$\text{Gain \%}=\frac{100}{19}$

Step 4: Convert to Mixed Fraction
Dividing 100 by 19:
19 × 5 = 95, Remainder = 100 - 95 = 5

$\frac{100}{19}=5\frac{5}{19}$

Shortcut Formula:
When a false weight of W grams is used instead of the true weight of T grams, the gain % is:

$\text{Gain \%}=\frac{T-W}{W}\times 100$

$\text{Gain \%}=\frac{1000-950}{950}\times 100=\frac{50}{950}\times 100=5\frac{5}{19}\%$

Therefore, the dealer's gain percentage is $5\frac{5}{19}\%$.