Simple & Compound Interest

Interest questions are a certainty in CDS and follow fixed formulae, so they are among the safest marks. The whole topic sits on one contrast: simple interest is on the principal only, compound interest is on principal plus accumulated interest.

Simple interest (SI)

• SI = (P x R x T)/100, where P = principal, R = rate% per annum, T = time in years.
• Amount A = P + SI = P(1 + RT/100).

Compound interest (CI)

• Amount A = P(1 + R/100)^n for n years; CI = A - P.
• Compounded half-yearly: use rate R/2 and time 2n. Quarterly: R/4 and 4n.

Shortcut differences (CI - SI)

• For 2 years: difference = P(R/100)^2.
• For 3 years: difference = P(R/100)^2 x (3 + R/100).

Exam Tricks & Tips

• 🎯 SI grows by the same amount each year; CI grows faster because interest earns interest.
• 🎯 For a 2-year gap between CI and SI, jump straight to P(R/100)^2 instead of computing both.
• 🎯 Half-yearly compounding: halve the rate and double the periods (R/2, 2n).
• 🎯 If a sum doubles in T years at simple interest, the rate is R = 100/T percent.
• 🎯 Under CI, if money becomes x times in n years, it becomes x^2 times in 2n years (square the growth).
• ❌ Common mistake: using the annual rate directly for half-yearly compounding instead of halving the rate and doubling the time.

Expected exam pattern

Direct SI/CI computation, find-the-rate/time, and CI-minus-SI difference questions; 6-10 formula-driven marks.

Quick recap

SI = PRT/100; A(CI) = P(1 + R/100)^n; 2-year difference = P(R/100)^2; adjust rate and time for non-annual compounding.

Cover the answer, recall, then check. 11 cards on interest.

Q1. Formula for simple interest?
A1. SI = (P x R x T)/100.

Q2. Amount under compound interest for n years?
A2. A = P(1 + R/100)^n.

Q3. CI - SI for 2 years?
A3. P(R/100)^2.

Q4. CI - SI for 3 years?
A4. P(R/100)^2 x (3 + R/100).

Q5. How do you compound half-yearly?
A5. Use rate R/2 and time 2n periods.

Q6. SI on 2000 at 5% for 3 years?
A6. (2000 x 5 x 3)/100 = 300.

Q7. A sum doubles in 10 years at SI; find the rate.
A7. R = 100/T = 100/10 = 10%.

Q8. CI on 1000 at 10% for 2 years?
A8. 1000(1.1)^2 - 1000 = 1210 - 1000 = 210.

Q9. Difference between CI and SI on 5000 at 4% for 2 years?
A9. 5000 x (4/100)^2 = 5000 x 0.0016 = 8.

Q10. Under CI, if a sum triples in 5 years, what happens in 10 years?
A10. It becomes 3^2 = 9 times.

Q11. Amount under SI in terms of P, R, T?
A11. A = P(1 + RT/100).

Interest problems model how money grows over time and are a staple of CDS arithmetic. The topic hinges on two formulas — simple and compound interest — and understanding when each applies.

Core idea / what this tests

This tests calculation of interest and amount under simple interest (interest on the original principal only) and compound interest (interest on principal plus accumulated interest). CDS asks for interest, amount, rate, time, or the difference between the two schemes.

Deep explanation

Beginner — simple interest

With principal P, rate R% per annum, and time T years:

• Simple Interest (SI) = (P × R × T) / 100.
• Amount = P + SI.
  Interest is the same each year because it is always computed on the original P.

Intermediate — compound interest

Compound interest adds each period's interest to the principal, so the next period earns "interest on interest":

• Amount A = P (1 + R/100)ᵀ (compounded annually); CI = A − P.
• If compounded half-yearly, use rate R/2 and time 2T; quarterly, R/4 and 4T.
  For 2 years, CI can be found directly as CI = P[(1 + R/100)² − 1].

Advanced — the SI–CI difference and shortcuts

• Difference for 2 years: CI − SI = P(R/100)². This one-line formula is a huge time-saver.
• Difference for 3 years: CI − SI = P(R/100)² × (3 + R/100), or equivalently P·R²(300+R)/100³.
• When only the SI for successive equal periods is compared, note CI grows because each year's interest is larger than the last. Also, "the sum doubles in T years at SI" means the interest equalled the principal, so R × T = 100.

Worked example

Find the compound interest on ₹8,000 at 5% per annum for 2 years, compounded annually.

Step 1 — write the amount formula: A = P(1 + R/100)ᵀ = 8000 × (1 + 5/100)² = 8000 × (1.05)².
Step 2 — compute (1.05)² = 1.1025, so A = 8000 × 1.1025 = 8,820.
Step 3 — CI = A − P = 8820 − 8000 = ₹820.
Cross-check with the difference formula: SI = (8000 × 5 × 2)/100 = 800; CI − SI = P(R/100)² = 8000 × (0.05)² = 8000 × 0.0025 = 20, so CI = 800 + 20 = 820 ✓.
Answer: CI = ₹820. Two methods agree.

CDS relevance

Interest questions appear reliably in CDS and are quick with the right formula. The 2-year difference shortcut CI − SI = P(R/100)² is a favourite CDS scorer, and understanding compounding frequency (half-yearly/quarterly) prevents common setup mistakes.

Exam tricks & shortcuts

• SI = PRT/100; CI amount = P(1 + R/100)ᵀ.
• 2-year difference: CI − SI = P(R/100)² — memorise it.
• Half-yearly: halve the rate, double the time; quarterly: quarter the rate, quadruple the time.
• "Sum doubles at SI in T years" ⇒ RT = 100; "triples" ⇒ RT = 200.
• For small rates and 2 years, CI ≈ SI + a small correction (the difference formula).

Key formulas

• Simple interest: SI = P·R·T/100; Amount = P(1 + RT/100).
• Compound interest: A = P(1 + R/100)ⁿ; CI = A − P.
• Compounded k times/year: A = P(1 + R/(100k))^(kn).
• CI − SI for 2 years = P(R/100)²; for 3 years = P(R/100)²(3 + R/100).
• Sum doubles at SI in T years ⇒ R = 100/T.
• CI exceeds SI whenever time > 1 period.

Usage: SI grows linearly; CI grows by a fixed ratio each period.

Worked Example

Problem: Find the compound interest on ₹10,000 at 10% per annum for 2 years, compounded annually. Also find how much more this is than the simple interest for the same period.

Solution:
Step 1 — Compound amount using A = P(1 + r/100)ⁿ:
A = 10000 × (1 + 10/100)² = 10000 × (1.1)² = 10000 × 1.21 = ₹12,100.
Compound interest (CI) = A − P = 12,100 − 10,000 = ₹2,100.

Step 2 — Simple interest using SI = P × r × t / 100:
SI = 10000 × 10 × 2 / 100 = ₹2,000.

Step 3 — Difference:
CI − SI = 2,100 − 2,000 = ₹100.

This extra ₹100 is the "interest on the first year's interest" (10% of ₹1,000), which compounding adds but simple interest does not.

Answer: CI = ₹2,100; it is ₹100 more than the simple interest of ₹2,000.