Suresh borrowed X 800 at 6% and Naresh borrowed X 600 at 10%. After how much time, will they both have equal debts?

Options

A. `15(1)/(3)yr`
B. `14(1)/(2)yr`
C. `18(1)/(3)yr`
D. `16(2)/(3)yr`

Answer

To solve the problem of when Suresh and Naresh will have equal debts, we will use the formula for calculating simple interest and the total amount owed after a certain time. Here’s the step-by-step solution:

### Step 1: Define the Variables
- Let Suresh's principal amount (P1) = 800
- Let Naresh's principal amount (P2) = 600
- Suresh's rate of interest (R1) = 6%
- Naresh's rate of interest (R2) = 10%
- Let the time (T) in years be the same for both.

### Step 2: Write the Formula for Total Amount
The total amount (A) after time T can be calculated using the formula:
\[ A = P + \text{Simple Interest} \]
Where Simple Interest (SI) is calculated as:
\[ SI = \frac{P \times R \times T}{100} \]

### Step 3: Calculate Suresh's Total Amount
For Suresh:
\[ A_1 = P_1 + SI_1 \]
\[ A_1 = 800 + \frac{800 \times 6 \times T}{100} \]
\[ A_1 = 800 + \frac{4800T}{100} \]
\[ A_1 = 800 + 48T \]

### Step 4: Calculate Naresh's Total Amount
For Naresh:
\[ A_2 = P_2 + SI_2 \]
\[ A_2 = 600 + \frac{600 \times 10 \times T}{100} \]
\[ A_2 = 600 + \frac{6000T}{100} \]
\[ A_2 = 600 + 60T \]

### Step 5: Set the Amounts Equal
To find out when their debts will be equal, we set \( A_1 \) equal to \( A_2 \):
\[ 800 + 48T = 600 + 60T \]

### Step 6: Rearrange the Equation
Rearranging the equation gives:
\[ 800 - 600 = 60T - 48T \]
\[ 200 = 12T \]

### Step 7: Solve for T
To find T, divide both sides by 12:
\[ T = \frac{200}{12} \]
\[ T = \frac{50}{3} \]

### Step 8: Convert to Mixed Number
Now, converting \( \frac{50}{3} \) into a mixed number:
\[ \frac{50}{3} = 16 \frac{2}{3} \]

### Final Answer
Thus, Suresh and Naresh will have equal debts after **16 years and 8 months** (or **16 2/3 years**).

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