Hema, Rekha and Jaya enter into partnership. Hema invests 3 times as much as Rekha invests and Rekha invests two-thirds of what Jaya invests. At the end of the year, a total profit of Rs 6600 was recorded. Then :
A. the share of Hema = Rs 4600
B. the share of Rekha = Rs 1200
C. the share of Jaya is = Rs 1800
D. 20% of 30% of $\frac{5}{6}\text{th}$ of total profit = Rs 330
Choose the correct answer from the options given below:
Show Hint
To avoid working with fractions, assume the initial variable is a multiple of the denominator.
In this case, since Rekha's investment is $\frac{2}{3}$ of Jaya's, setting Jaya's investment as $3x$ immediately eliminates fractions and simplifies calculations.
Step 1: Understanding the Question:
This is a problem based on a partnership.
In a simple partnership, where the investment durations are equal, the profit is divided in the direct ratio of the investments made by each partner.
We need to determine the ratio of investment of the three partners, calculate their individual profit shares, and verify the statements. Step 2: Key Formula or Approach:
1. Let Jaya's investment be $J$.
2. Express Rekha's investment ($R$) and Hema's investment ($H$) in terms of $J$:
\[ R = \frac{2}{3}J \]
\[ H = 3 \times R = 3 \times \frac{2}{3}J = 2J \]
3. Find the investment ratio $H : R : J$.
4. Distribute the total profit of Rs 6600 according to this ratio. Step 3: Detailed Explanation:
• Calculate the Investment Ratio:
Let Jaya's investment be $J = 3x$.
Then, Rekha's investment:
\[ R = \frac{2}{3}(3x) = 2x \]
Hema's investment:
\[ H = 3R = 3(2x) = 6x \]
Thus, the ratio of investments of Hema, Rekha, and Jaya is:
\[ H : R : J = 6x : 2x : 3x = 6 : 2 : 3 \]
• Calculate Individual Profit Shares:
The sum of the ratio terms:
\[ 6 + 2 + 3 = 11 \]
Total profit to be distributed is Rs 6600.
One ratio unit corresponds to:
\[ \frac{6600}{11} = \text{Rs } 600 \]
Now, calculate the profit shares:
• Hema's Share:
\[ 6 \times 600 = \text{Rs } 3600 \]
(This means Statement A is False, as it states Hema's share is Rs 4600).
• Rekha's Share:
\[ 2 \times 600 = \text{Rs } 1200 \]
(This means Statement B is True).
• Jaya's Share:
\[ 3 \times 600 = \text{Rs } 1800 \]
(This means Statement C is True).
• Verify Statement D:
We need to calculate:
\[ 20\% \text{ of } 30\% \text{ of } \frac{5}{6}\text{th of } 6600 \]
First, find $\frac{5}{6}\text{th}$ of the total profit:
\[ \frac{5}{6} \times 6600 = 5 \times 1100 = 5500 \]
Next, find 30% of 5500:
\[ \frac{30}{100} \times 5500 = 30 \times 55 = 1650 \]
Finally, find 20% of 1650:
\[ \frac{20}{100} \times 1650 = \frac{1}{5} \times 1650 = 330 \]
Since the calculated value is Rs 330, Statement D is True. Step 4: Final Answer:
Statements B, C, and D are true, while Statement A is false.
Therefore, the correct option is (D).
