Compound Interest Class 8th CBSE By Abhinav Sir

Class 8 CBSE Notes: Compound Interest (CI)

Concept| Formulas| C.I. vs S.I.| Examples| 40+ Solved Questions| Tips| Extra Practice

Concept & Key Points

• Compound Interest is the interest calculated on the initial sum plus all previously accumulated interest.
• Unlike Simple Interest (SI), where interest is only on principal, every period in CI, interest is added to principal and new interest is calculated. This is called "interest on interest".
• Used in bank savings, fixed deposits, loans, investments, etc.

Compound Interest Formulas

\[ \text{Amount} (A) = P \left(1 + \frac{r}{n \times 100} \right)^{nt} \]

• P = Principal (initial sum)
• r = Rate of interest (% per annum)
• t = Time (in years)
• n = Number of times interest is compounded per year
  (1 for annually, 2 for half-yearly, 4 for quarterly)

\[ \text{Compound Interest (CI)} = \text{Amount} - \text{Principal} = A - P \]

For fractional years (like 1.5 years, etc.) – calculate in terms of half/quarters if asked, else split in years & fractions as shown in later answers.

• If compounded annually, \( n=1 \): \( A = P (1+\frac{r}{100})^t \)
• Half-yearly: rate = r/2, n = 2, t in years \( \rightarrow \) total periods = 2t
• Quarterly: rate = r/4, n = 4, t in years \( \rightarrow \) total periods = 4t

Compound Interest vs Simple Interest

Feature	Simple Interest (SI)	Compound Interest (CI)
Interest is calculated on	Original Principal	Principal + Accumulated Interest
Formula	\( SI = \frac{P \times r \times t}{100} \)	\( CI = P \left(1+\frac{r}{n \times 100}\right)^{nt} - P \)
Growth	Linear	Exponential
Amount Earned Over Years	Lower	Higher for longer period

Examples & Illustrations

Example: Find the CI on Rs. 5000 for 2 years at 10% p.a. compounded annually.

P = 5000, r = 10%, t = 2
\( A = 5000 \times (1+\frac{10}{100})^2 = 5000 \times (1.1)^2 = 5000 \times 1.21 = \text{Rs. } 6050 \)
\( \boxed{CI = A - P = 6050 - 5000 = \text{Rs. } 1050} \)

40+ Most Important Solved Questions (RD Sharma, RS Aggarwal, NCERT)

1. Find the compound interest on Rs. 4000 for 2 years at 5% per annum compounded annually.
   \( A = 4000 \times (1+0.05)^2 = 4000 \times 1.1025 = \text{Rs. } 4410 \)
   \( \text{CI} = 4410 - 4000 = \boxed{Rs. 410} \)
2. Find the compound interest on Rs. 6000 for 2 years at 8% per annum compounded annually.
   \( A = 6000 \times (1 + 0.08)^2 = 6000 \times 1.1664 = \text{Rs. } 6998.4 \)
   \( \text{CI} = 6998.4 - 6000 = \boxed{Rs. 998.4} \)
3. Calculate CI on Rs. 12,000 for 3 years at 10% per annum compounded annually.
   \( (1.1)^3 = 1.331 \)
   \( A = 12000 \times 1.331 = \text{Rs. } 15972 \)
   \( \text{CI} = 15972 - 12000 = \boxed{Rs. 3972} \)
4. Find the CI and Total Amount for Rs. 5,000 at 12% p.a. for 2 years compounded annually.
   \( A = 5000 \times (1.12)^2 = 5000 \times 1.2544 = \text{Rs. } 6272 \)
   \( \text{CI} = 1272 \)
5. Calculate CI for Rs. 15,625 for 9 months at 4% p.a., compounded quarterly.
   Quarterly rate = \( \frac{4}{4} = 1\% \); total quarters = 9/3 = 3
   \( A = 15625 \times (1+0.01)^3 = 15625 \times 1.030301 = \text{Rs. } 16098.14 \)
   \( \text{CI} = 16098.14 - 15625 = \boxed{Rs. 473.14} \)
6. Find the CI and amount on Rs. 6,400 for 1 year at 10% per annum compounded quarterly.
   Quarterly rate = 2.5%, periods = 4
   \( A = 6400 \times (1.025)^4 = 6400 \times 1.1038 = \text{Rs. } 7064.32 \)
   \( \text{CI} = 7064.32 - 6400 = \boxed{Rs. 664.32} \)
7. Find CI on Rs. 1,000 at 8% per annum for 1.5 years, compounded half-yearly.
   Half-yearly rate = 4%, periods = 3
   \( A = 1000 \times (1.04)^3 = 1000 \times 1.124864 = Rs. 1124.86 \)
   CI = Rs. 124.86
8. Compare CI for 2 years on Rs. 10,000 at 20% per annum compounded annually and half-yearly.
   Annually: \(A = 10000 \times 1.44 = Rs. 14400\), CI = Rs. 4400
   Half-yearly: Rate = 10%, periods = 4: \(A = 10000 \times (1.1)^4 = 10000 \times 1.4641 = Rs. 14641\), CI = Rs. 4641
9. Find the amount and CI on Rs. 31,250 for 3 years at 8% per annum compounded annually.
   \(A = 31250 \times (1.08)^3 = 31250 \times 1.259712 = Rs. 39366\)
   CI = 39366 - 31250 = Rs. 8126
10. On what sum will the CI at 5% per annum for 2 years compounded annually be Rs. 164?
    Let principal = P.
    \( CI = P[(1.05)^2 - 1] = P[1.1025 - 1] = 0.1025P \)
    \( 0.1025P = 164 \rightarrow P = 164/0.1025 = Rs. 1600 \)
11. Difference between CI and SI on Rs. 20,000 at 12% per annum for 2 years?
    SI = \( 20000 \times 0.12 \times 2 = Rs. 4800 \)
    CI = \( 20000 \times (1.12)^2 - 20000 = 20000 \times 1.2544 - 20000 = Rs. 5088 \)
    Difference = Rs. 288
12. CI on Rs. 8,000 for 1 year at 16% p.a. compounded half-yearly. (Take (1.08)^2 = 1.1664)
    Rate per half-year = 8%; periods = 2
    \(A = 8000 \times (1.08)^2 = 8000 \times 1.1664 = Rs. 9331.20\)
    CI = Rs. 1331.20
13. Rs. 31,250 becomes Rs. 39,365.625 in 3 years compounded annually. Find the rate.
    \(39365.625 = 31250 \times (1+r/100)^3\)
    \(1+r/100 = (39365.625/31250)^{1/3} = 1.08\)
    \(r = 8\%\)
14. Difference between CI and SI on Rs. 15,000 for 3 years at 10% per annum?
    SI = \(15000 \times 0.1 \times 3 = Rs. 4500\)
    \(A = 15000 \times (1.1)^3 = 15000 \times 1.331 = Rs. 19965\),
    CI = Rs. 4965. Difference = Rs. 465
15. CI on Rs. 7,800 at 5% p.a. for 3 years compounded yearly?
    \(A = 7800 \times (1.05)^3 = 7800 \times 1.157625 = Rs. 9039.47\)
    CI = Rs. 1239.47
16. Find the CI (to nearest rupee) on Rs. 29,000 at 16% p.a. for 2 years, compounded annually.
    \(A = 29000 \times (1.16)^2 = 29000 \times 1.3456 = Rs. 39022.4\)
    CI = Rs. 10022.4 ≈ Rs. 10022
17. CI on Rs. 13,000 for 3 years at 7% p.a. compounded yearly.
    \(A = 13000 \times (1.07)^3 = 13000 \times 1.225043 = Rs. 15925.56\)
    CI = Rs. 2925.56
18. Find the CI on Rs. 9,600 at 9% p.a. for 2 years, compounded yearly.
    \(A = 9600 \times (1.09)^2 = 9600 \times 1.1881 = Rs. 11406.04\)
    CI = Rs. 1806.04
19. What will Rs. 12,500 amount to in 2 years at 8% p.a. compounded half-yearly?
    Half-yearly rate = 4%; periods = 4.
    \(A = 12500 \times (1.04)^4 = 12500 \times 1.16985856 = Rs. 14623.23\)
    CI = Rs. 2123.23
20. A sum of Rs. 700 is borrowed for 1.5 years at 5% p.a. compounded half-yearly. Find CI.
    Rate/period = 2.5%; periods = 3
    \(A = 700 \times (1.025)^3 = 700 \times 1.076890625 = Rs. 753.82\)
    CI = Rs. 53.82
21. At what annual rate will Rs. 5,000 double in 8 years if interest is compounded annually?
    \(2 = (1 + r/100)^8 \implies (1 + r/100) = 2^{1/8} \approx 1.0905\)
    \(r = 9.05\%\)
22. Find CI on Rs. 18,000 at 12% p.a. compounded half-yearly for 1.5 years.
    Rate/half-year = 6%; periods = 3
    \(A = 18000 \times (1.06)^3 = 18000 \times 1.191016 = Rs. 21438.29\)
    CI = Rs. 3438.29
23. If CI for 2 years on a sum is Rs. 328 at 8% p.a., compounded annually, what is the sum?
    CI = \( P \times [1.08^2 - 1] = P \times 0.1664 = 328 \implies P = 328 / 0.1664 = Rs. 1971 \)
24. In how much time will an amount double at 12.5% p.a. compounded annually (to nearest year)?
    \( (1 + 0.125)^n = 2 \implies n \log 1.125 = \log 2 \)\ \( n = \log 2 / \log 1.125 = 0.3010/0.0512 = 5.88 \approx 6 \) years.
25. CI & amount on Rs. 25,000 for 2 years at 12% p.a. compounded half-yearly.
    Rate per half-year = 6%; periods = 4
    \(A = 25000 \times (1.06)^4 = 25000 \times 1.262477 = Rs. 31561.93\)
    CI = Rs. 6561.93
26. A sum is invested at 12% p.a. CI, doubled in 6 years. Find the compounding period per year used.
    If compounded annually: \((1+0.12)^6 \approx 1.9738\), not doubled.
    If compounded half-yearly: \(r = 6\%, n = 12: 1.06^{12} \approx 2.0122 \), thus compounded half-yearly.
27. Which is greater: Rs. 6,000 at 12% for 2 years compounded annually or half-yearly?
    Annually: \(A = 6000 \times (1.12)^2 = 6000 \times 1.2544 = Rs. 7526.4\)
    Half-yearly: Rate = 6%, periods = 4: \(A = 6000 \times (1.06)^4 = 6000 \times 1.2625 = Rs. 7575\)
    Half-yearly is greater.
28. A loan of Rs. 1,000 at 10% pa for 1 year compounded semi-annually. Find amount to be repaid.
    Rate/period = 5%, periods = 2: \(A = 1000 \times (1.05)^2 = 1000 \times 1.1025 = Rs. 1102.5\)
29. Find the principal if CI at 5% for 2 years compounded yearly is Rs. 410.
    \(CI = P[(1.05)^2 - 1] = P \times 0.1025, CI = 410\)
    \(P = 410 / 0.1025 = Rs. 4000\)
30. If Rs. 6,500 becomes Rs. 7,581 at CI in 2 years, compounded yearly, find the rate.
    \(7581 = 6500 \times (1 + r/100)^2\)
    \(1 + r/100 = (7581/6500)^{1/2} = 1.08\), so r = 8%
31. Find the principal which amounts to Rs. 22,680 at interest 12% first year, 12.5% second year.
    Let principal \(= P\). After 1 year: \(A_1 = P \times 1.12\)
    After 2nd year: \(A_2 = A_1 \times 1.125\)
    \(P \times 1.12 \times 1.125 = 22,680 \implies P = 22,680 / 1.26 = Rs. 18,000\)
32. What sum amounts to Rs. 62,208 in 4 years at 20% p.a. compounded yearly?
    \(A = P \times (1.2)^4\), so \( P = 62208 / (1.2^4) = 62208 / 2.0736 = Rs. 30,000\)
33. If Rs. 12,100 amounts to Rs. 13,310 in 1 year compounded quarterly, find the rate.
    \(13,310 = 12,100 \times (1 + r/400)^4\)
    \( (1 + r/400)^4 = 13310/12100 = 1.1 \implies 1 + r/400 = 1.0241 \rightarrow r = 9.64\% \) approx.
34. In how many years will Rs. 8000 amount to Rs. 9261 at 5% p.a. compounded yearly?
    \(9261 = 8000(1.05)^n \implies (1.05)^n = 1.57625\)
    Take logs: \(n \log 1.05 = \log 1.57625 \implies n = 0.1973/0.0212 \approx 9.3 = 9\) years (nearest integer).
35. What is the difference between SI and CI on Rs. 5,000 for 2 years at 10% p.a.?
    SI = \(5000 \times 0.1 \times 2 = Rs. 1000\)
    CI = \(5000 \times 1.21 - 5000 = Rs. 1050\)
    Difference = Rs. 50
36. What is the total CI on Rs. 11,000 for 2 years at 10% p.a. compounded yearly?
    \(A = 11000 \times (1.10)^2 = 11000 \times 1.21 = Rs. 13,310\)
    CI = Rs. 2310
37. Find the amount of Rs. 8000 at 10.5% p.a. for 3 years, compounded yearly.
    \(A = 8000 \times (1.105)^3 = 8000 \times 1.364366 = Rs. 10,914.93\)
    CI = Rs. 2914.93
38. Find the CI and amount for Rs. 5,000 for 3 years at 6.25% p.a.
    \(A = 5000 \times (1.0625)^3 = 5000 \times 1.199512 = Rs. 5997.56\)
    CI = Rs. 997.56
39. Find the CI for Rs. 9,000 at 14% p.a. for 2 years, interest added every 6 months.
    Rate per half = 7%, periods = 4.
    \(A = 9000 \times (1.07)^4 = 9000 \times 1.310796 = Rs. 11,797.16\)
    CI = Rs. 2,797.16
40. A sum triples itself in 8 years compounded annually. Find the rate (nearest integer).
    \(3P = P(1+r/100)^8 \implies (1+r/100)^8 = 3 \implies (1+r/100) = 3^{1/8} \approx 1.147 \implies r = 14.7\%\)
41. The difference between CI and SI on Rs. 12000 for 2 years at 15% p.a.?
    SI = Rs. 12000 × 0.15 × 2 = Rs. 3600.
    CI = 12000 × (1.15)^2 - 12000 = 12000 × 1.3225 - 12000 = Rs. 3870
    Diff = Rs. 270
42. If CI on Rs. 3,000 in 3 years is Rs. 692.16, find the rate % compounded annually.
    \(A = 3000 + 692.16 = 3692.16\)
    \( 3692.16 = 3000 \times (1+r/100)^3 \implies (1+r/100)^3 = 1.23072 \implies (1+r/100) = 1.071999 \implies r = 7.2\% \)
43. CI on Rs. 2,400 at 7.5% p.a. for 2 years, compounded yearly?
    \(A = 2400 \times (1.075)^2 = 2400 \times 1.155625 = Rs. 2773.5\)
    CI = Rs. 373.5
44. Principal is Rs. 1,800 compounded half-yearly at 6% for 1 year, CI =?
    Rate = 3%, periods = 2
    \(A = 1800 \times (1.03)^2 = 1800 \times 1.0609 = Rs. 1909.62\)
    CI = Rs. 109.62
45. A man deposits Rs. 25,000 for 2 years at 12% p.a. compounded half-yearly. Find amount.
    Rate/half = 6%, periods = 4
    \(A = 25000 \times (1.06)^4 = 25000 \times 1.262477 = Rs. 31561.93\)
46. Amount of Rs. 1,200 at 11% compounded half-yearly for 1.5 years?
    Rate/half = 5.5%, periods = 3
    \(A = 1200 \times (1.055)^3 = 1200 \times 1.1741 = Rs. 1408.95\)
47. Find the C.I. for Rs. 5,000 at 6% p.a. for 2 years when interest is compounded annually.
    \(A = 5000 \times (1.06)^2 = 5000 \times 1.1236 = Rs. 5618\)
    CI = Rs. 618
48. CI for Rs. 10,000 at 5% p.a. for 3 years, compounded annually?
    \(A = 10000 \times (1.05)^3 = 10000 \times 1.157625 = Rs. 11576.25\)
    CI = Rs. 1576.25
49. If a man lends Rs. 1,25,000 at 8% CI p.a., compounded annually for 3 years, what will the amount become?
    \(A = 125000 \times (1.08)^3 = 125000 \times 1.2597 = Rs. 157,462.5\)

Tips & Tricks for Compound Interest

• Always check if the compounding is annual, half-yearly, quarterly, or with varying rates.
• For mixed rates or variable years, split calculations stepwise for each period.
• If calculating time/rate needed to double/triple, use logarithms.
• For difference CI – SI (for 2 years): \( P \times (r/100)^2 \).
• For effective rate when interest compounded more than once a year: Use equivalent annual rate formula.

Extra Practice (solve with formulas above)

• Find CI for Rs. 3,500 at 8% p.a. for 2 years compounded yearly.
• A sum amounts to Rs. 8,800 in 2 years at 10% p.a. compounded yearly. Find the principal.
• Find the difference between CI and SI on Rs. 25,000 for 2 years at 12% per annum.
• A sum of Rs. 2,500 earns Rs. 525 as CI in 2 years compounded annually. Find the rate.

By Abhinav Sir