Uday and Amir entered into partnership business Amir invested Rs . 1600 for whole year and uday invested Rs .1200 for first six months and withdraw `1/4` th of his capital and invested remaining amount for next T month . If Uday and Amir got profit in the ratio of `9:16` .Find the value of T ?

Options

A. A)3 months
B. B)4 months
C. C)5 months
D. D)6 months

Answer

To solve the problem step by step, we will break down the investments of Uday and Amir, calculate their effective capital contributions over time, and then use the profit ratio to find the value of T.

### Step 1: Calculate Uday's Investment for the First Six Months
Uday invested Rs. 1200 for the first six months.

- **Investment**: Rs. 1200
- **Time**: 6 months

### Step 2: Calculate Uday's Remaining Investment After Withdrawal
Uday withdrew 1/4 of his capital after six months.

- **Withdrawal Amount**:
\[ \text{Withdrawal} = \frac{1}{4} \times 1200 = 300 \]

- **Remaining Capital**:
\[ \text{Remaining Capital} = 1200 - 300 = 900 \]

### Step 3: Calculate Uday's Investment for the Next T Months
Uday invested the remaining Rs. 900 for T months.

- **Investment**: Rs. 900
- **Time**: T months

### Step 4: Calculate Amir's Investment
Amir invested Rs. 1600 for the whole year (12 months).

- **Investment**: Rs. 1600
- **Time**: 12 months

### Step 5: Calculate Uday's Effective Capital Contribution
Uday's total effective capital contribution can be calculated as:

- For the first 6 months:
\[ \text{Contribution} = 1200 \times 6 = 7200 \]

- For the next T months:
\[ \text{Contribution} = 900 \times T \]

- **Total Contribution by Uday**:
\[ \text{Total Contribution} = 7200 + 900T \]

### Step 6: Calculate Amir's Effective Capital Contribution
Amir's total effective capital contribution is:

- For the whole year:
\[ \text{Contribution} = 1600 \times 12 = 19200 \]

### Step 7: Set Up the Profit Ratio Equation
According to the problem, the profit ratio of Uday and Amir is given as 9:16. Therefore, we can set up the equation:

\[ \frac{7200 + 900T}{19200} = \frac{9}{16} \]

### Step 8: Cross Multiply to Solve for T
Cross multiplying gives us:

\[ 16(7200 + 900T) = 9 \times 19200 \]

Calculating the right side:

\[ 9 \times 19200 = 172800 \]

Now, substituting back into the equation:

\[ 16(7200 + 900T) = 172800 \]

### Step 9: Distribute and Simplify
Distributing the left side:

\[ 115200 + 14400T = 172800 \]

### Step 10: Isolate the T Term
Subtract 115200 from both sides:

\[ 14400T = 172800 - 115200 \]
\[ 14400T = 57600 \]

### Step 11: Solve for T
Now, divide both sides by 14400:

\[ T = \frac{57600}{14400} = 4 \]

### Final Answer
Thus, the value of T is **4 months**.

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