Compound Interest: Concepts & Shortcuts
$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$
Scenario-Based Learning for Competitive Exams
By Wisdom Helps Mahesh Sir
Q1. Rahul invested ₹2,000 in a fixed deposit. If the bank offers 10% per annum compounded annually, after how many years will he receive a total maturity amount of ₹2,420?
Solution: $2420 = 2000(1.1)^n \Rightarrow 1.21 = (1.1)^n$. Since $1.1^2 = 1.21$, n = 2 years.
Q2. Anjali's savings account balance grew from ₹1,000 to ₹1,331. If the bank pays 10% p.a. compound interest annually, find the duration of her investment?
Solution: $1331 = 1000(1.1)^n \Rightarrow 1.331 = (1.1)^n$. Since $1.1^3 = 1.331$, n = 3 years.
Q3. A farmer borrowed a certain sum from a cooperative bank at 4% p.a. CI. If he cleared his debt by paying ₹270.40 after 2 years, what was the original sum borrowed?
Solution: $270.40 = P(1.04)^2 \Rightarrow P = 270.40 / 1.0816 = \mathbf{₹250}$.
Q4. A business investment grows to ₹9,680 at the end of 2 years and ₹10,648 at the end of 3 years. What is the annual rate of compound interest being applied?
Solution: Interest for 3rd year = $10648 - 9680 = 968$. Rate = $(968/9680) \times 100 = \mathbf{10\%}$.
Q5. A jewelry shop owner finds that a gold deposit worth ₹2,304 appreciates to ₹2,500 in exactly 2 years under compound growth. Find the rate percent per annum?
Solution:$\sqrt{\frac{2500}{2304}} = 1 + \frac{R}{100} \Rightarrow \frac{50}{48} = 1 + \frac{R}{100} \Rightarrow \frac{R}{100} = \frac{1}{24} \Rightarrow R = \frac{100}{24} = 4 \tfrac{1}{6}\%$.
Q6. Vikram needs ₹1,352 to buy a gadget after 2 years. If his bank offers 4% p.a. compound interest, how much should he deposit today?
Solution: $P = \frac{1352}{(1.04)^2} = \frac{1352}{1.0816} = 1250$ ₹
Q7. The compound interest on a sum for 2 years at 4% is ₹102. What would be the Simple Interest on the same sum at the same rate and for the same period?
Solution: $\text{Effective CI rate} = 4 + 4 + \frac{4 \times 4}{100} = 8.16\% \Rightarrow \text{SI} = \frac{102}{8.16} \times 8 = \mathbf{100}$ .
Q8. A financial consultant tells you that the difference between SI and CI on your principal for 2 years at 4% is ₹4. What is the value of your principal?
Solution:$\text{Diff} = P\left(\frac{R}{100}\right)^2 \Rightarrow 4 = P\left(\frac{4}{100}\right)^2 \Rightarrow 4 = P \times \frac{16}{10000} \Rightarrow P = \frac{4 \times 10000}{16} = 2500$
Q9. In a loan scheme, the difference between CI and SI for 2 years at 8% per annum is ₹768. Find the total sum involved in this scheme.
Solution: $768 = P\left(\frac{8}{100}\right)^2 \Rightarrow 768 = P \times \frac{64}{10000} \Rightarrow P = \frac{768 \times 10000}{64} = 120000$ ₹
Q10. For a long-term bond, the difference between CI and SI for 3 years at 5% p.a. is calculated to be ₹15.25. What is the face value (principal) of the bond?
Solution: $Diff = P(\frac{R}{100})^2 \times \frac{300+R}{100} \Rightarrow 15.25 = P(\frac{1}{400}) \times (\frac{305}{100}) \Rightarrow P = \mathbf{₹2,000}$.
Q11. A real estate property doubles its value every 5 years under compound growth. In how many years will the value of the property become 8 times the original price?
Solution: $2^1$ in 5 years. $8 = 2^3$. So, $5 \times 3 = \mathbf{15 years}$.
Q12. A stock market index doubles itself in 4 years. If it continues to grow at the same compound rate, how many years will it take to quadruple (4 times)?
Solution: $2^1$ in 4 years. $4 = 2^2$. So, $4 \times 2 = \mathbf{8 years}$.
Q13. Mrs. Sharma borrowed ₹2,550 at 4% p.a. CI to be paid back in 2 equal yearly instalments. What is the value of each instalment?
Solution: $2550 = \frac{x}{1.04} + \frac{x}{(1.04)^2} \Rightarrow 2550 = \frac{25x}{26} + \frac{625x}{676} \Rightarrow 2550 = \frac{1275x}{676} \Rightarrow x = \frac{2550 \times 676}{1275} = 1352$
Q14. In a private lending agreement, the difference between CI and SI at 10% for 2 years is ₹464. Find the total sum lent?
Solution: $464 = P(\frac{10}{100})^2 \Rightarrow P = 464 \times 100 = \mathbf{₹46,400}$.
Q15. Find the compound interest on a sum of ₹16,000 for 9 months at 20% p.a., if the interest is payable quarterly?
Solution:$R = 5\% \text{ (quarterly)},\; n = 3 \text{ periods} \Rightarrow A = 16000(1.05)^3 = 18522 \Rightarrow \text{CI} = 18522 - 16000 = 2522$ ₹
Q16. What sum of money, if invested today at 6% p.a. compound interest, will grow to exactly ₹5,618 in 2 years?
Solution:$A = P \left(1 + \frac{R}{100}\right)^T,\; 5618 = P \left(1 + \frac{6}{100}\right)^2,\; 5618 = P (1.06)^2,\; 5618 = P \times 1.1236,\; P = \frac{5618}{1.1236} = 5000$
Q17. Ashok has ₹1612 with him. He divided it amongst his sons Raj and Varun and asked them to invest it at 8% rate of interest compounded annually. It was seen that Raj and Varun got same amount after 14 and 15 years respectively. How much (in ₹) did Ashok give to Raj? (Most Important Problem) (SSC, Railway, Bank)
⚡ Shortcut Trick (Very Important)
When the rate is the same, amounts become equal, and time differs by 1 year:
Step 1: Use the direct ratio
Ratio of investments (Lower Time : Higher Time) = $(1 + R) : 1$
$= 1.08 : 1$
Step 2: Calculate the Shares
Since Raj invested for less time (14 years) but his final amount is equal to Varun's (15 years), Raj must have invested a higher principal amount.
Varun's Share (for 15 years) = $\frac{1}{1 + 1.08} \times 1612$
$= \frac{1}{2.08} \times 1612 = 775$
Final Answer:
Since Raj invested less time but is equal to amount invested by Varun because of high amount so take the difference:
Raj's Share = $1612 - 775 = 837$
Correct Option: (B)
When the rate is the same, amounts become equal, and time differs by 1 year:
Step 1: Use the direct ratio
Ratio of investments (Lower Time : Higher Time) = $(1 + R) : 1$
$= 1.08 : 1$
Step 2: Calculate the Shares
Since Raj invested for less time (14 years) but his final amount is equal to Varun's (15 years), Raj must have invested a higher principal amount.
Varun's Share (for 15 years) = $\frac{1}{1 + 1.08} \times 1612$
$= \frac{1}{2.08} \times 1612 = 775$
Final Answer:
Since Raj invested less time but is equal to amount invested by Varun because of high amount so take the difference:
Raj's Share = $1612 - 775 = 837$
Correct Option: (B)
Q17. Ashok has ₹1612 with him. He divided it amongst his sons Raj and Varun and asked them to invest it at 8% rate of interest compounded annually. It was seen that Raj and Varun got same amount after 14 and 15 years respectively. How much (in ₹) did Ashok give to Raj? (Most Important Problem)
Q18. Gopal invests a sum of ₹5400 and Akshay invests a sum of ₹10200 at the same rate of simple interest per annum. If, at the end of 4 years, Akshay gets ₹720 more interest than Gopal, then find the rate of interest per annum.
Q19. Mahesh has two grandsons Kiran and Tushar. 11 year old Kiran gets some money from Mandar’s wealth and 12 year old Tushar gets rest of the money. Both will get money only when they turn 22 years old. Till then the money is in a bank getting interest at 8% compounded annually. When both turn 22, they receive the same amount. How much had Mandar given Tushar (in ₹) initially, if total money was ₹24700? (Very Important)
by Wisdom helps mahesh sir , by wisdom helps maths
By Wisdom Helps Mahesh Sir
