Simple & Compound Interest
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Simple & Compound Interest — Key Facts for MAT
Simple Interest (SI) is interest calculated only on the original principal amount. The formula:
$$SI = \frac{P \times R \times T}{100}$$
Where P = Principal, R = Rate per annum, T = Time in years.
Amount = Principal + Simple Interest = $P + \frac{PRT}{100}$
Compound Interest (CI) is interest calculated on the principal + accumulated interest. The standard formula:
$$A = P\left(1 + \frac{R}{100}\right)^T$$
Where A = Amount after T years, and CI = A − P.
⚡ Exam shortcut — Rule of 72: To find years to double your money at rate R%, divide 72 by R. At 8% p.a., money doubles in 9 years exactly (72 ÷ 8 = 9).
⚡ Exam shortcut — Difference SI vs CI: For 2 years, the difference between CI and SI at rate R% is: $\frac{P \times R^2}{100^2}$. At 10% on ₹10,000 for 2 years: CI = ₹2,100, SI = ₹2,000, difference = ₹100.
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Simple & Compound Interest — MAT Study Guide
Simple Interest: Deep Dive
Simple Interest is the most basic form of interest in financial mathematics. It is commonly used in short-term loans, fixed deposits, and certain government schemes in India.
Key derivations:
- If time is in months: $SI = \frac{P \times R \times M}{12 \times 100}$
- If time is in days: $SI = \frac{P \times R \times D}{365 \times 100}$
Example (MAT 2022 pattern): A sum of ₹8,000 is lent at 5% p.a. for 3 years. Find the total interest. $$SI = \frac{8000 \times 5 \times 3}{100} = ₹1,200$$ Amount = ₹8,000 + ₹1,200 = ₹9,200
Compound Interest: Deep Dive
Compound interest is what banks and most financial institutions actually use. Interest is added to the principal at regular intervals (annually, half-yearly, or quarterly), and subsequent interest is calculated on this new total.
Half-yearly compounding: Rate becomes R/2, time becomes 2T $$A = P\left(1 + \frac{R}{200}\right)^{2T}$$
Quarterly compounding: Rate becomes R/4, time becomes 4T $$A = P\left(1 + \frac{R}{400}\right)^{4T}$$
Example (MAT 2023 pattern): Find the compound interest on ₹6,000 at 10% p.a. for 1.5 years, compounded half-yearly.
Since compounding is half-yearly: Time = 3 periods, Rate = 5% per half-year: $$A = 6000 \times \left(1 + \frac{5}{100}\right)^3 = 6000 \times (1.05)^3 = 6000 \times 1.157625 = ₹6,945.75$$ CI = ₹6,945.75 − ₹6,000 = ₹945.75
Population Growth Formula (Common in MAT)
$$P_t = P_0\left(1 + \frac{R}{100}\right)^t$$
This is identical to compound interest formula — just applied to population, bacteria, or value appreciation.
Equated Monthly Installments (EMI)
EMI for a loan of principal P at annual rate R% for N months: $$EMI = \frac{P \times R \times (1+R)^N}{12 \times ((1+R)^N - 1)}$$
⚡ MAT shortcut for installment problems: When a sum X is divided into two parts at rates R1% and R2% and gives the same annual interest, use: Part₁ : Part₂ = (100 − R₂) : (100 − R₁).
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Simple & Compound Interest — Comprehensive MAT Notes
Derivation: Why Compound Interest Grows Faster
Simple interest treats each year’s principal as unchanged. Compound interest treats it as growing. For 3 years at rate R%:
| Year | SI on ₹P | CI on ₹P |
|---|---|---|
| 1 | PR/100 | PR/100 |
| 2 | PR/100 | P[(R/100) + (R/100)²] |
| 3 | PR/100 | P[(R/100) + 2(R/100)² + (R/100)³] |
The CI advantage compounds over time. At 10% p.a., ₹1,00,000 grows to:
- SI (10 years): ₹2,00,000
- CI (10 years): ₹2,59,374
That’s 59.4% more than simple interest.
Fractional Year Compound Interest
When time is not a whole number, apply the integer part with full compounding, then calculate simple interest on the fractional part from that amount.
Example: ₹5,000 at 12% p.a. for 1.5 years compounded annually. $$A = 5000 \times (1.12)^1 = ₹5,600 \text{ (after 1 year)}$$ For 0.5 year at 12%: $SI = \frac{5600 \times 12 \times 0.5}{100} = ₹336$ Total Amount = ₹5,600 + ₹336 = ₹5,936
Present Value Formula
What amount today equals ₹X in T years at rate R%? $$PV = \frac{X}{\left(1 + \frac{R}{100}\right)^T}$$
This is the reverse of compound interest accumulation.
Interest Calculation When Rates Differ Each Year
Sometimes MAT questions specify different rates for different years:
Example: ₹10,000 at 10% for year 1, 15% for year 2, and 20% for year 3 (CI). $$A = 10000 \times 1.10 \times 1.15 \times 1.20 = ₹15,156$$
Net Fractional Change Problems
Example (MAT 2021): A product’s price increases by 20% in January, decreases by 20% in February, and increases by 10% in March. What is the net change from the original price?
After January: $P \times 1.20$ After February: $P \times 1.20 \times 0.80 = P \times 0.96$ After March: $P \times 0.96 \times 1.10 = P \times 1.056$
Net increase = 5.6%
⚡ Key insight: A 20% increase followed by a 20% decrease does NOT return to the original value. It leaves you at 96% — a 4% net loss.
Common MAT Question Patterns
Pattern 1 — Find the rate: “A sum becomes 5 times in 8 years at CI. Find the rate.” Solution: $(1 + R/100)^8 = 5$ → $R/100 = 5^{1/8} - 1$ → $R \approx 22.5%$
Pattern 2 — Two rates successive: “A sum at 20% p.a. for 2 years gives the same CI as at x% p.a. for 3 years. Find x.” Set up: $(1.2)^2 = (1 + x/100)^3$ → $x \approx 12.7%$
Pattern 3 — Difference SI/CI for 3 years: $$CI - SI = P\left[\left(1+\frac{R}{100}\right)^3 - 1 - \frac{3R}{100}\right]$$
Pattern 4 — Instalment problems: “A TV priced at ₹30,000 is bought for ₹x cash or ₹y per month for 12 months at 10% CI. Find y.” Formula: $x = \frac{y}{1+R/100} + \frac{y}{(1+R/100)^2} + … + \frac{y}{(1+R/100)^{12}}$
⚡ Common student mistake: Confusing “rate per annum” with “rate per half-year” when compounding frequency changes. Always adjust both R and T proportionally. Also forgetting that CI is calculated on the new principal (principal + previous interest), not the original.
Previous Year MAT Questions (Key Types)
- Two sums invested: At different rates, find when they become equal.
- Installment problems: Buying a refrigerator on EMI — find the marked price or rate.
- Banker’s gain: Difference between true discount and banker’s discount at same rate.
- Population: Village population grows at 5% per annum, find after 3 years.
- Depreciation: Machine value decreases at 10% p.a., find value after 5 years.
- Mixed SI/CI: A sum earns SI for 2 years and CI for next 3 years — find total amount.
- Equal installments: Find the present worth of a future payment or vice versa.
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