A sum of money is lend at 15% per annum for `3 (1)/(2)` years and same sum lend for 5 years at 15% per annum. If the difference of S.I. from both cases is Rs 144. Find the sum lend.

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI), which is given by:

\[ SI = \frac{P \times R \times T}{100} \]

where:
- \( P \) = Principal amount (the sum of money lent)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Define the variables
Let the principal amount be \( P \) (the sum lent).

### Step 2: Calculate Simple Interest for the first case
The first case is for \( 3 \frac{1}{2} \) years at 15% per annum. Convert \( 3 \frac{1}{2} \) years into an improper fraction:
\[ 3 \frac{1}{2} = \frac{7}{2} \text{ years} \]

Now, using the SI formula:
\[ SI_1 = \frac{P \times 15 \times \frac{7}{2}}{100} \]
\[ SI_1 = \frac{P \times 15 \times 7}{200} = \frac{105P}{200} \]

### Step 3: Calculate Simple Interest for the second case
The second case is for 5 years at 15% per annum:
\[ SI_2 = \frac{P \times 15 \times 5}{100} \]
\[ SI_2 = \frac{75P}{100} = \frac{75P}{100} \]

### Step 4: Set up the equation based on the difference in Simple Interest
According to the problem, the difference between the two Simple Interests is Rs 144:
\[ SI_2 - SI_1 = 144 \]
Substituting the values we calculated:
\[ \frac{75P}{100} - \frac{105P}{200} = 144 \]

### Step 5: Simplify the equation
To simplify, we need a common denominator. The common denominator between 100 and 200 is 200:
\[ \frac{150P}{200} - \frac{105P}{200} = 144 \]
\[ \frac{150P - 105P}{200} = 144 \]
\[ \frac{45P}{200} = 144 \]

### Step 6: Solve for \( P \)
Now, multiply both sides by 200:
\[ 45P = 144 \times 200 \]
\[ 45P = 28800 \]
Now, divide both sides by 45:
\[ P = \frac{28800}{45} \]
\[ P = 640 \]

### Final Answer
The sum lent is Rs 640.

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