A person gave 2/5 part of his income to his elder son and 30% part to his younger son. He saved his remaining money in three trusts A, B and C in the ratio of 3 : 5 : 2. If difference between the amount got by his both sons is X 2000, how much amount he saved in trust C?

Options

A. ? 1000
B. ? 1140
C. X 1200
D. X 1256

Answer

To solve the problem step by step, let's break down the information given and find the amount saved in trust C.

### Step 1: Define the total income
Let the total income of the person be Rs. X.

### Step 2: Calculate the amount given to the elder son
The elder son receives \( \frac{2}{5} \) of the total income.
So, the amount given to the elder son is:
\[ \text{Amount to elder son} = \frac{2}{5} \times X = \frac{2X}{5} \]

### Step 3: Calculate the amount given to the younger son
The younger son receives 30% of the total income.
So, the amount given to the younger son is:
\[ \text{Amount to younger son} = 30\% \times X = \frac{30}{100} \times X = \frac{3X}{10} \]

### Step 4: Calculate the remaining income
To find the remaining income after giving amounts to both sons, we subtract the amounts given to the sons from the total income:
\[ \text{Remaining income} = X - \left(\frac{2X}{5} + \frac{3X}{10}\right) \]
To perform this calculation, we need a common denominator for the fractions. The least common multiple of 5 and 10 is 10.
\[ \frac{2X}{5} = \frac{4X}{10} \]
Now, substituting back:
\[ \text{Remaining income} = X - \left(\frac{4X}{10} + \frac{3X}{10}\right) = X - \frac{7X}{10} = \frac{3X}{10} \]

### Step 5: Set up the equation for the difference between the sons' amounts
The difference between the amounts received by the elder and younger sons is given as Rs. 2000:
\[ \frac{2X}{5} - \frac{3X}{10} = 2000 \]
Again, we convert \( \frac{2X}{5} \) to have a common denominator of 10:
\[ \frac{2X}{5} = \frac{4X}{10} \]
Now, substituting this back into the equation:
\[ \frac{4X}{10} - \frac{3X}{10} = 2000 \]
This simplifies to:
\[ \frac{X}{10} = 2000 \]

### Step 6: Solve for X
Multiplying both sides by 10 gives:
\[ X = 20000 \]

### Step 7: Calculate the remaining income
Now, substituting \( X \) back to find the remaining income:
\[ \text{Remaining income} = \frac{3X}{10} = \frac{3 \times 20000}{10} = 6000 \]

### Step 8: Distribute the remaining income into trusts A, B, and C
The remaining income of Rs. 6000 is distributed in the ratio of 3:5:2 among trusts A, B, and C.
The total parts of the ratio is:
\[ 3 + 5 + 2 = 10 \]
The amount for trust C can be calculated as:
\[ \text{Amount for trust C} = \frac{2}{10} \times 6000 = \frac{12000}{10} = 1200 \]

### Conclusion
The amount saved in trust C is Rs. 1200.

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