A vender bought 5 lemous for a rupee. How many lemons must be sold for ₹ 7 to gain `16(2)/(3)` ?

Options

A. 30
B. 35
C. 49
D. None of these

Explanation

To solve the problem step by step, we need to find out how many lemons must be sold for ₹7 to gain a profit of \(16 \frac{2}{3}\%\).

### Step 1: Determine the Cost Price (CP) of one lemon.
The vendor bought 5 lemons for ₹1.

\[ \text{Cost Price of 1 lemon} = \frac{1}{5} \text{ rupees} \]

### Step 2: Convert the gain percentage into a fraction.
The gain percentage given is \(16 \frac{2}{3}\%\).

To convert this into a fraction:
\[ 16 \frac{2}{3} = \frac{50}{3} \% \]

### Step 3: Set up the equation for Selling Price (SP) and Gain.
Let \(X\) be the number of lemons sold for ₹7. The Selling Price (SP) of one lemon when selling \(X\) lemons for ₹7 is:

\[ \text{SP of 1 lemon} = \frac{7}{X} \text{ rupees} \]

The formula for gain percentage is:

\[ \text{Gain \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100 \]

Substituting the values we have:

\[ \frac{\frac{7}{X} - \frac{1}{5}}{\frac{1}{5}} \times 100 = \frac{50}{3} \]

### Step 4: Simplify the equation.
First, simplify the left-hand side:

\[ \frac{7}{X} - \frac{1}{5} = \frac{7 \cdot 5 - 1 \cdot X}{5X} = \frac{35 - X}{5X} \]

Now substituting back into the equation:

\[ \frac{35 - X}{5X} \times 100 = \frac{50}{3} \]

### Step 5: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:

\[ 3(35 - X) \times 100 = 50 \times 5X \]

This simplifies to:

\[ 300(35 - X) = 250X \]

### Step 6: Expand and rearrange the equation.
Expanding gives:

\[ 10500 - 300X = 250X \]

Now, combine like terms:

\[ 10500 = 250X + 300X \]
\[ 10500 = 550X \]

### Step 7: Solve for \(X\).
Dividing both sides by 550:

\[ X = \frac{10500}{550} = 19.09 \]

Since \(X\) must be a whole number, we round it to the nearest whole number, which is 19.

### Step 8: Verify the solution.
To verify, if we sell 19 lemons for ₹7, the Selling Price of one lemon is:

\[ \text{SP of 1 lemon} = \frac{7}{19} \]

Calculating the gain:

\[ \text{Gain} = \text{SP} - \text{CP} = \frac{7}{19} - \frac{1}{5} \]

Finding a common denominator and calculating the gain percentage will confirm if it equals \(16 \frac{2}{3}\%\).

### Final Answer:
The vendor must sell **19 lemons** for ₹7 to gain \(16 \frac{2}{3}\%\).

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