To find the sum of money lent at 4%, let's follow the given steps.
Step 1: Understand the Given Information
- Difference in interest earned: ₹72
- Interest rates: [tex]\( 5.5\% \)[/tex] per annum and [tex]\( 4\% \)[/tex] per annum
- Time period: 3 years
Step 2: Let the sum of money lent be [tex]\( x \)[/tex] rupees.
Let:
[tex]\( I_1 \)[/tex] be the interest earned from lending money at [tex]\( 5.5\% \)[/tex].
[tex]\( I_2 \)[/tex] be the interest earned from lending money at [tex]\( 4\% \)[/tex].
Step 3: Calculate the Interest Earned Using Simple Interest Formula
The simple interest formula is given by:
[tex]\[ I = P \times R \times T \][/tex]
where [tex]\( P \)[/tex] is the principal amount, [tex]\( R \)[/tex] is the interest rate, and [tex]\( T \)[/tex] is the time period in years.
Interest earned at [tex]\( 5.5\% \)[/tex] p.a. for 3 years:
[tex]\[ I_1 = x \times 5.5\% \times 3 \][/tex]
[tex]\[ I_1 = x \times \frac{5.5}{100} \times 3 \][/tex]
[tex]\[ I_1 = x \times \frac{5.5}{100} \times 3 \][/tex]
[tex]\[ I_1 = x \times 0.055 \times 3 \][/tex]
[tex]\[ I_1 = 0.165x \][/tex]
Interest earned at [tex]\( 4\% \)[/tex] p.a. for 3 years:
[tex]\[ I_2 = x \times 4\% \times 3 \][/tex]
[tex]\[ I_2 = x \times \frac{4}{100} \times 3 \][/tex]
[tex]\[ I_2 = x \times \frac{4}{100} \times 3 \][/tex]
[tex]\[ I_2 = x \times 0.04 \times 3 \][/tex]
[tex]\[ I_2 = 0.12x \][/tex]
Step 4: Use the Given Difference in Interest to Form an Equation
We know that the difference in interest earned is ₹72:
[tex]\[ I_1 - I_2 = 72 \][/tex]
[tex]\[ 0.165x - 0.12x = 72 \][/tex]
Step 5: Solve for [tex]\( x \)[/tex]
[tex]\[ 0.045x = 72 \][/tex]
[tex]\[ x = \frac{72}{0.045} \][/tex]
[tex]\[ x = 1600 \][/tex]
So, the sum of money lent at [tex]\( 4\% \)[/tex] is ₹1600.
