Ranbeer sold a music system to Rajbir at 15% gain and Rajbir sold it to Dharambir at 30% gain. If Dharambir paid Rs 8970 for the music system, what amount (in Rs) did Ranbeer pay for the same?
Show Hint
Using fractional representations can make calculation easier:
15% gain = factor of $\frac{115}{100} = \frac{23}{20}$.
30% gain = factor of $\frac{13}{10}$.
The equation is:
\[ x \times \frac{23}{20} \times \frac{13}{10} = 8970 \]
\[ x \times \frac{299}{200} = 8970 \]
Since $299 \times 3 = 897$, we can see that $299 \times 30 = 8970$.
So:
\[ x = 30 \times 200 = 6000 \]
Working with fractions avoids tedious decimal division.
Step 1: Understanding the Question:
This is a successive transaction problem in profit and loss.
Ranbeer sells an item to Rajbir at a profit, and Rajbir subsequently sells it to Dharambir at another profit.
The final price paid by Dharambir is given, and we need to work backward to find the original cost price paid by Ranbeer. Step 2: Key Formula or Approach:
Let the price Ranbeer paid be $x$ (Ranbeer's Cost Price).
1. Ranbeer sells to Rajbir at a 15% gain:
\[ \text{Rajbir's Cost Price} = x \times \left(1 + \frac{15}{100}\right) = 1.15x \]
2. Rajbir sells to Dharambir at a 30% gain:
\[ \text{Dharambir's Cost Price} = \text{Rajbir's Cost Price} \times \left(1 + \frac{30}{100}\right) = 1.15x \times 1.30 \]
3. Equate this to the given price paid by Dharambir (Rs 8970) and solve for $x$. Step 3: Detailed Explanation:
• Formulate the Equation:
\[ x \times 1.15 \times 1.30 = 8970 \]
Multiply the decimals:
\[ 1.15 \times 1.30 = 1.495 \]
So the equation becomes:
\[ 1.495x = 8970 \]
• Solve for $x$:
To eliminate decimals, multiply both the numerator and denominator by 1000:
\[ x = \frac{8970}{1.495} = \frac{8970 \times 1000}{1495} \]
Let us check if 1495 divides 8970:
\[ 1495 \times 6 = 8970 \]
Yes, it is exactly divisible:
\[ x = 6 \times 1000 = 6000 \]
Therefore, Ranbeer paid Rs 6000 for the music system. Step 4: Final Answer:
The cost price for Ranbeer was Rs 6000.
Therefore, the correct option is (A).
