2.5 Difference Between CI and SI for 2 Years
Free study material · concepts, shortcuts & solved questions
This is one of the single most exam-useful shortcut formulas in the entire chapter. For a sum P at rate R% for 2 years:
SI = (2PR)/(100), CI = P[(1+(R)/(100))^2 - 1] = (2PR)/(100) + P((R)/(100))^2
Subtracting, the linear terms cancel and we are left with:
[CI - SI (2 years) = P((R)/(100))^2 = (PR^2)/(10000)]
Intuition: The difference is simply the interest earned on the first year’s interest during the second year — that is the only extra amount CI generates that SI does not. This single formula is worth more exam marks per second of effort than almost anything else in the chapter, because “difference between CI and SI for 2 years” is one of the most frequently repeated question stems across SSC CGL, CHSL, MTS, and RRB NTPC papers, appearing in some form in nearly every attempt cycle.
Solved Example 2.5.1: Find the difference between CI and SI on Rs. 25,000 for 2 years at 4% p.a.
Solution: Diff = PR²/10000 = 25000 × 16/10000 = Rs. 40.
Solved Example 2.5.2: The difference between CI and SI on a sum for 2 years at 6% p.a. is Rs. 63. Find the sum.
Solution: 63 = P × 6²/10000 = 36P/10000, so P = 63 × 10000/36 = Rs. 17,500.
Solved Example 2.5.3: The difference between CI and SI on Rs. 8,000 for 2 years is Rs. 20. Find the rate of interest.
Solution: 20 = 8000 × R²/10000 → R² = 200000/8000 = 25 → R = 5% p.a.
Solved Example 2.5.4: The difference between CI and SI on a sum for 2 years at 2.5% p.a. is Rs. 6.25. Find the sum (edge case: a decimal rate).
Solution: Diff = PR²/10000. 6.25 = P × (2.5)²/10000 = P × 6.25/10000. So P = 6.25 × 10000/6.25 = Rs. 10,000. Even with a decimal rate, the formula behaves identically — just square the decimal rate carefully.
Solved Example 2.5.5: The difference between CI and SI on a certain sum for 2 years at 12.5% p.a. is Rs. 140. Find the sum, and also find the SI earned over these 2 years.
Solution: Diff = PR²/10000. 140 = P × (12.5)²/10000 = P × 156.25/10000. So P = 140 × 10000/156.25 = Rs. 8,960. SI for 2 years = 2PR/100 = (2 × 8960 × 12.5)/100 = 224000/100 = Rs. 2,240 (and hence CI = 2240 + 140 = Rs. 2,380, as a self-check).