When the price of an item was reduced by 20% then its sale increased by x%. If there is an increase of 60% in the receipt of the revenue , then the value of x is :

Options

A. `80 `
B. ` 120 `
C. ` 96`
D. ` 100`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define Variables
Let the original price of the item be \( p \) and the original number of items sold be \( s \).

**Hint:** Start by defining the variables to represent the price and quantity sold.

### Step 2: Calculate Original Revenue
The original revenue can be calculated as:
\[ \text{Original Revenue} = p \times s \]

**Hint:** Revenue is calculated by multiplying the price by the number of items sold.

### Step 3: Calculate Reduced Price
The price is reduced by 20%, so the new price becomes:
\[ \text{Reduced Price} = p - 0.2p = 0.8p = \frac{4p}{5} \]

**Hint:** To find the new price after a percentage reduction, subtract the percentage of the original price from the original price.

### Step 4: Calculate Increased Revenue
It is given that the revenue increases by 60%. Therefore, the new revenue becomes:
\[ \text{Increased Revenue} = \text{Original Revenue} + 60\% \text{ of Original Revenue} = p \times s + 0.6(p \times s) = 1.6ps \]

**Hint:** To find the increased revenue, add the original revenue to the percentage increase of the original revenue.

### Step 5: Relate New Revenue to New Quantity Sold
Let the new number of items sold be \( s' \). The new revenue can also be expressed as:
\[ \text{New Revenue} = \text{Reduced Price} \times \text{New Quantity Sold} = 0.8p \times s' \]
Setting the two expressions for new revenue equal gives:
\[ 1.6ps = 0.8p \times s' \]

**Hint:** Set the expressions for revenue equal to find a relationship between the new quantity sold and the original quantity sold.

### Step 6: Simplify the Equation
We can cancel \( p \) from both sides (assuming \( p \neq 0 \)):
\[ 1.6s = 0.8s' \]
Now, divide both sides by 0.8:
\[ s' = \frac{1.6s}{0.8} = 2s \]

**Hint:** Simplifying the equation helps to find the new quantity sold in terms of the original quantity sold.

### Step 7: Calculate the Increase in Sales
The increase in sales can be expressed as:
\[ s' = s + \frac{x}{100}s \]
Substituting \( s' = 2s \) into the equation gives:
\[ 2s = s + \frac{x}{100}s \]

**Hint:** Express the increase in sales as a percentage of the original sales.

### Step 8: Solve for \( x \)
Now, we can simplify the equation:
\[ 2s - s = \frac{x}{100}s \]
\[ s = \frac{x}{100}s \]
Dividing both sides by \( s \) (assuming \( s \neq 0 \)):
\[ 1 = \frac{x}{100} \]
Multiplying both sides by 100 gives:
\[ x = 100 \]

**Hint:** Isolate \( x \) to find its value.

### Conclusion
The value of \( x \) is \( 100\% \).

**Final Answer:** \( x = 100 \)