SI and CI Difference Formula for 2, 3 and 4 Years (Quick Answer)
The SI and CI difference formula for 3 years is D = P (R/100)² (3 + R/100), or in one line, D = P × R² × (300 + R) ÷ 100³. For 2 years it is simply D = P (R/100)², and for 4 years it is D = P (R/100)² (6 + 4R/100 + R²/100²). Here P is the principal, R is the annual rate in percent, and interest is compounded once a year.
🔥 Key Exam Takeaway: At 10% per annum, CI minus SI is 1% of P for 2 years, 3.1% of P for 3 years and 6.41% of P for 4 years. Learn the percentage table in Section 4 and most SSC and bank questions on this topic take under 30 seconds.
This note is written for SSC CGL, SSC CHSL, IBPS Clerk/PO and SBI Clerk/PO quant. Every formula and answer below has been checked by direct calculation.
CI − SI Formula Table for 2, 3 and 4 Years
Let r = R/100 (the rate as a decimal). Interest is compounded annually.
| Time | CI − SI (in terms of r) | CI − SI (in terms of R) |
|---|---|---|
| 1 year | 0 | 0 |
| 2 years | P r² | P × R² ÷ 100² |
| 3 years | P r² (3 + r) | P × R² × (300 + R) ÷ 100³ |
| 4 years | P r² (6 + 4r + r²) | P × R² × (60000 + 400R + R²) ÷ 100⁴ |
| n years (general) | P[(1+r)ⁿ − 1 − nr] | — |
Two useful ratios that follow from this table:
- 3-year difference ÷ 2-year difference = 3 + r. At 10% it is 3.1; at 20% it is 3.2.
- For 1 year compounded half-yearly, the difference is P(R/200)², because the rate is halved and the time is doubled (it behaves like 2 years at R/2 %).
CI and SI Difference Formula Derivation
You do not need to derive these in the exam, but knowing where they come from stops you from mixing them up.
Difference Between CI and SI for 2 Years Formula
CI = P(1+r)² − P = P(2r + r²) and SI = 2Pr
CI − SI = P r²
Difference Between CI and SI for 3 Years Formula
CI = P(1+r)³ − P = P(3r + 3r² + r³) and SI = 3Pr
CI − SI = P(3r² + r³) = P r² (3 + r)
Difference Between CI and SI for 4 Years Formula
CI = P(1+r)⁴ − P = P(4r + 6r² + 4r³ + r⁴) and SI = 4Pr
CI − SI = P(6r² + 4r³ + r⁴) = P r² (6 + 4r + r²)
Notice the pattern: the difference is always the binomial expansion of (1+r)ⁿ with the first two terms (1 and nr) removed. The coefficients 3, then 6 and 4, come straight from Pascal's triangle.
Year-by-Year CI vs SI Comparison (Why the Gap Grows)
Take ₹1,000 at 10% for 3 years. Simple interest is ₹100 every year. Compound interest earns interest on earlier interest:
| Year | CI earned that year | SI earned that year | Extra from compounding |
|---|---|---|---|
| 1 | ₹100 | ₹100 | ₹0 |
| 2 | ₹110 | ₹100 | ₹10 |
| 3 | ₹121 | ₹100 | ₹21 |
| Total | ₹331 | ₹300 | ₹31 |
So CI − SI = ₹31, which is 3.1% of ₹1,000. The ₹21 in year 3 is "interest on two years of interest" (₹10 + ₹10) plus "interest on interest on interest" (₹1). That is exactly 3Pr² + Pr³ split across years 2 and 3.
How to Solve CI − SI Questions Step by Step
- Identify the time and compounding. Annual for 2, 3 or 4 years? Or half-yearly/quarterly? If half-yearly, halve the rate and double the time first.
- Write the rate as a fraction where possible: 5% = 1/20, 10% = 1/10, 12.5% = 1/8, 20% = 1/5, 25% = 1/4. Fractions make the arithmetic mental.
- Pick the formula from the table: Pr², Pr²(3+r) or Pr²(6+4r+r²).
- Solve for the unknown. If D is given and P is asked, divide D by the multiplier. If the rate is asked, find r² and take the square root.
- Check the options. For 2 and 3 years, D at a "nice" rate gives a clean rupee value. A messy answer usually means the compounding period was misread.
CI and SI Difference Short Trick (Percentage of Principal)
Memorise the 10% row and the rows for 5%, 20% and 25%. The rest can be built in seconds from the formulas.
| Rate (R%) | 2 years | 3 years | 4 years | D(3 yrs) ÷ D(2 yrs) |
|---|---|---|---|---|
| 4% | 0.16% | 0.4864% | 0.985856% | 3.04 |
| 5% | 0.25% | 0.7625% | 1.550625% | 3.05 |
| 6% | 0.36% | 1.1016% | 2.247696% | 3.06 |
| 8% | 0.64% | 1.9712% | 4.048896% | 3.08 |
| 10% | 1% | 3.1% | 6.41% | 3.1 |
| 12% | 1.44% | 4.4928% | 9.351936% | 3.12 |
| 12.5% | 1.5625% | 4.8828125% | 10.1806640625% | 3.125 |
| 15% | 2.25% | 7.0875% | 14.900625% | 3.15 |
| 20% | 4% | 12.8% | 27.36% | 3.2 |
| 25% | 6.25% | 20.3125% | 44.140625% | 3.25 |
How to use it: "CI − SI for 3 years at 10% is ₹93. Find P." → 3.1% of P = 93 → P = 9300 ÷ 3.1 = ₹3,000. Done in one line.
Difference Between CI and SI Questions with Solutions (SSC CGL and Bank Exams)
Example 1: 2 years (SSC CHSL level)
Find the difference between CI and SI on ₹12,000 for 2 years at 10% per annum.
Solution: D = 12000 × (1/10)² = 120. Check: CI = ₹2,520 and SI = ₹2,400, so the difference is ₹120.
Example 2: 3 years (SSC CGL Tier 1 level)
Find CI − SI on ₹8,000 for 3 years at 5% per annum.
Solution: D = 8000 × (1/20)² × (3 + 1/20) = 20 × 3.05 = 61. Check: CI = ₹1,261 and SI = ₹1,200. Answer: ₹61.
Example 3: Find the principal, 3 years
The difference between CI and SI on a sum for 3 years at 10% per annum is ₹93. Find the sum.
Solution: At 10%, D = 3.1% of P. So P = (93 × 100) ÷ 3.1 = 3000. Answer: ₹3,000.
Example 4: Find the principal, 2 years (IBPS Clerk level)
CI − SI for 2 years at 8% per annum is ₹160. Find the principal.
Solution: P = 160 ÷ (0.08)² = 160 ÷ 0.0064 = 25000. Answer: ₹25,000.
Example 5: Find the rate
On ₹7,500 the difference between CI and SI for 2 years is ₹48. Find the rate.
Solution: r² = 48/7500 = 0.0064, so r = 0.08. Answer: 8% per annum.
Example 6: 4 years
Find CI − SI on ₹10,000 for 4 years at 10% per annum.
Solution: D = 10000 × 0.01 × (6 + 0.4 + 0.01) = 100 × 6.41 = 641. Check: amount = ₹14,641, so CI = ₹4,641 and SI = ₹4,000. Answer: ₹641.
Example 7: Half-yearly compounding (SBI Clerk level)
Find CI − SI on ₹16,000 for 1 year at 10% per annum, interest compounded half-yearly.
Solution: Half-yearly means 5% per half-year for 2 half-years. D = 16000 × (0.05)² = 40. Check: CI = ₹1,640 and SI for 1 year = ₹1,600. Answer: ₹40.
Example 8: Fractional rate, 3 years (SSC CGL Tier 2 level)
Find CI − SI on ₹5,120 for 3 years at 12½% per annum.
Solution: 12½% = 1/8. D = 5120 × 1/64 × (3 + 1/8) = 80 × 25/8 = 250. Check: CI = ₹2,170 and SI = ₹1,920. Answer: ₹250.
Example 9: From difference to SI (bank PO style)
The difference between CI and SI on a sum for 2 years at 12% per annum is ₹216. What is the SI on the same sum for 3 years at the same rate?
Solution: P = 216 ÷ (0.12)² = 15000. SI for 3 years = 15000 × 0.12 × 3 = 5400. Answer: ₹5,400.
Example 10: Different rates in each year
A sum of ₹5,000 earns 10% in the first year and 20% in the second year. Find CI − SI for the 2 years.
Solution: For rates r₁ and r₂, CI − SI = P r₁ r₂. So D = 5000 × 0.1 × 0.2 = 100. Check: amount = 5000 × 1.1 × 1.2 = ₹6,600, CI = ₹1,600, SI = ₹1,500. Answer: ₹100.
Example 11: Linking 2-year and 3-year differences
CI − SI on a sum for 2 years at 5% is ₹25. Find CI − SI on the same sum for 3 years at 5%.
Solution: D(3) = D(2) × (3 + r) = 25 × 3.05 = 76.25. (The sum is ₹10,000.) Answer: ₹76.25.
Example 12: 3 years at 15%
Find CI − SI on ₹40,000 for 3 years at 15% per annum.
Solution: 7.0875% of 40,000 = 2,835. Check: CI = ₹20,835, SI = ₹18,000. Answer: ₹2,835.
Common Mistakes in CI − SI Difference Questions
- Using the 2-year formula for 3 years. For 3 years the difference is not Pr² times 1.5 or times 3. It is Pr²(3 + r). At 10% that is 3.1 times the 2-year value, not 3 times.
- Forgetting half-yearly or quarterly compounding. "1 year, compounded half-yearly" is a 2-period question at R/2. "1 year, compounded quarterly" is a 4-period question at R/4, so use the 4-period formula.
- Mixing up "CI of year 3 minus CI of year 2" with CI − SI. These are different questions. The interest earned in year 3 minus the interest earned in year 2 equals P r² (1 + r), i.e. the year-2 interest multiplied by r.
- Taking the wrong square root. When finding the rate, r² = D/P. If D/P = 0.0016, r = 0.04, i.e. 4%, not 0.4%.
- Adding SI over 3 years wrongly. SI is the same every year: 3 × P × r. Only CI changes year to year.
- Ignoring the year-wise rates rule. With different rates in each year, the 2-year difference is P r₁ r₂, not P r² with an average rate.
CI SI Difference Formula: Quick Revision Table
| What is asked | What to do |
|---|---|
| D for 2 years | P r² |
| D for 3 years | P r² (3 + r) |
| D for 4 years | P r² (6 + 4r + r²) |
| P from D (2 years) | P = D / r² |
| P from D (3 years) | P = D / [r²(3 + r)] |
| Rate from D (2 years) | r = √(D/P) |
| D(3 yrs) from D(2 yrs) | multiply by (3 + r) |
| Half-yearly, 1 year | P (R/200)² |
| Different rates, 2 years | P r₁ r₂ |
| At 10% | 1%, 3.1%, 6.41% of P (2, 3, 4 years) |
CI SI Difference Practice Questions with Answers
Q1. Find CI − SI on ₹6,250 for 2 years at 4% per annum. Answer: ₹10. 6250 × (0.04)² = 6250 × 0.0016 = 10.
Q2. Find CI − SI on ₹15,000 for 3 years at 10% per annum. Answer: ₹465. 3.1% of 15,000 = 465. (CI = ₹4,965, SI = ₹4,500.)
Q3. The difference between CI and SI on a sum for 3 years at 5% per annum is ₹122. Find the sum. Answer: ₹16,000. At 5%, D = 0.7625% of P, so P = 122 ÷ 0.007625 = 16,000.
Q4. CI − SI on a sum for 2 years at 15% per annum is ₹63. Find the sum. Answer: ₹2,800. P = 63 ÷ 0.0225 = 2,800.
Q5. On ₹20,000 the difference between CI and SI for 2 years is ₹200. Find the rate. Answer: 10%. r² = 200/20000 = 0.01, so r = 0.1.
Q6. Find CI − SI on ₹16,000 for 4 years at 5% per annum. Answer: ₹248.10. 16000 × 0.0025 × (6 + 0.2 + 0.0025) = 40 × 6.2025 = 248.1. (CI = ₹3,448.10, SI = ₹3,200.)
Q7. For a certain sum and rate, D(3 years) ÷ D(2 years) = 3.08. Find the rate. Answer: 8%. The ratio is 3 + r, so r = 0.08.
Q8. Find CI − SI on ₹40,000 for 1 year at 8% per annum, compounded half-yearly. Answer: ₹64. Half-yearly rate 4% for 2 periods: 40000 × (0.04)² = 64.
Q9. SI on a sum for 2 years at 10% per annum is ₹1,800. Find CI − SI for the same sum, time and rate. Answer: ₹90. P = 1800 ÷ 0.2 = 9,000; D = 9000 × 0.01 = 90. Shortcut: for 2 years, D = SI × r ÷ 2 = 1800 × 0.1 ÷ 2 = 90.
Q10. Find CI − SI on ₹12,800 for 3 years at 25% per annum. Answer: ₹2,600. 12800 × 1/16 × (3 + 1/4) = 800 × 3.25 = 2600. (CI = ₹12,200, SI = ₹9,600.)
SI and CI Difference Formula FAQs
What is the formula for the difference between CI and SI for 3 years?
D = P (R/100)² (3 + R/100), which is the same as P × R² × (300 + R) ÷ 100³. It assumes interest is compounded annually.
What is the difference between CI and SI for 2 years?
D = P (R/100)². It equals the interest earned on the first year's interest.
What is the CI and SI difference formula for 4 years?
D = P r² (6 + 4r + r²) with r = R/100. At 10% it is 6.41% of the principal.
Is the 3-year difference three times the 2-year difference?
No. It is (3 + r) times the 2-year difference. At 10% that is 3.1 times, at 20% it is 3.2 times.
How do you find the principal when CI − SI is given?
Divide the difference by the multiplier: P = D / r² for 2 years and P = D / [r²(3 + r)] for 3 years.
Does the formula change for half-yearly compounding?
Yes. Halve the rate and double the number of periods, then apply the same formula. For 1 year compounded half-yearly, D = P (R/200)².
Is this topic asked in SSC CGL and bank exams?
Yes. CI − SI difference questions appear regularly in SSC CGL and CHSL Tier 1 quant and in the arithmetic part of IBPS and SBI Clerk and PO prelims, usually as find-the-difference, find-the-principal or find-the-rate questions.
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