Rajan borrowed a loan of 10,000 from Ram for 2 years at the rate of 10% simple interest. Immediately, Rajan lent the same sum for the same rate of interest compounded annually to Sham.
i) According to the given context, which interest is more among simple interest and compound interest for two years?
ii) How much profit did Rajan get during the transaction of two years?
iii) How much more interest should Sham need to pay to Rajan if Rajan had lent the amount at semi-annually compounded interest?
Answer (2)
Rajan paid $2000 in simple interest but earned $2100 in compound interest, making the compound interest higher. His profit from the transaction was $100. If Sham paid semi-annually, he would owe Rajan an additional $55.0625 in interest.
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Calculate simple interest: S I = P × R × T = 10000 × 0.10 × 2 = $2000 .
Calculate compound interest: C I = P ( 1 + R ) T − P = 10000 ( 1 + 0.10 ) 2 − 10000 = $2100 . Compound interest is higher.
Calculate Rajan's profit: P ro f i t = C I − S I = 2100 − 2000 = $100 .
Calculate semi-annual compound interest and the difference: D i ff ere n ce = 2155.0625 − 2100 = $55.0625 .
Explanation
Problem Analysis Let's analyze the problem. Rajan borrowed money at simple interest and lent it out at compound interest. We need to determine which interest type yields more, calculate Rajan's profit, and find the difference in interest if compounded semi-annually.
Calculating Simple Interest First, let's calculate the simple interest (SI) that Rajan paid to Ram.
The formula for simple interest is: S I = P × R × T where: P = Principal amount = $10,000 R = Rate of interest = $10% = 0.10 T = Time period = 2 years
So, S I = 10000 × 0.10 × 2 = 2000 Rajan paid Ram $2000 as simple interest.
Calculating Compound Interest Next, let's calculate the compound interest (CI) that Rajan received from Sham.
The formula for compound interest is: C I = P ( 1 + R ) T − P where: P = Principal amount = $10,000 R = Rate of interest = $10% = 0.10 T = Time period = 2 years
So, C I = 10000 ( 1 + 0.10 ) 2 − 10000 C I = 10000 ( 1.1 ) 2 − 10000 C I = 10000 ( 1.21 ) − 10000 C I = 12100 − 10000 = 2100 Rajan received $2100 as compound interest from Sham.
Comparing Simple and Compound Interest Comparing the simple interest ( $2000 ) and compound interest ( $2100 ), we can see that compound interest is higher for two years.
Calculating Rajan's Profit Now, let's calculate Rajan's profit.
Rajan's profit is the difference between the compound interest he received and the simple interest he paid.
Profit = C I − S I Profit = 2100 − 2000 = 100 Rajan's profit is $100 .
Calculating Semi-Annual Compound Interest Next, let's calculate the compound interest if it was compounded semi-annually.
The formula for compound interest compounded semi-annually is: C I = P ( 1 + 2 R ) 2 T − P where: P = Principal amount = $10,000 R = Rate of interest = $10% = 0.10 T = Time period = 2 years
So, C I = 10000 ( 1 + 2 0.10 ) 2 × 2 − 10000 C I = 10000 ( 1 + 0.05 ) 4 − 10000 C I = 10000 ( 1.05 ) 4 − 10000 C I = 10000 ( 1.21550625 ) − 10000 C I = 12155.0625 − 10000 = 2155.0625 The semi-annual compound interest is $2155.0625 .
Calculating the Difference in Interest Finally, let's calculate how much more interest Sham would need to pay if the interest was compounded semi-annually.
Difference = Semi-annual CI − Annual CI Difference = 2155.0625 − 2100 = 55.0625 Sham would need to pay $55.0625 more if the interest was compounded semi-annually.
Final Answers In summary: i) Compound interest is more than simple interest for two years. ii) Rajan's profit is $100 .
iii) Sham would need to pay $55.0625 more if the interest was compounded semi-annually.
Examples
Understanding simple and compound interest is crucial in personal finance. For example, when you deposit money in a savings account, the bank usually offers compound interest, which helps your money grow faster than with simple interest. Conversely, when you borrow money, such as a car loan, understanding the interest rate and compounding period helps you calculate the total cost of the loan and compare different loan offers. Knowing these concepts allows you to make informed decisions about savings, investments, and borrowing, ultimately leading to better financial management.
