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Simple Interest

Learning Goals

By the end of this chapter, you will be able to:

• Compute Simple Interest (SI) for any principal, rate, and time — including fractional years and months.
• Find any one unknown (P, R, or T) when the other three are given.
• Handle "amount becomes n times" and "sum doubles/triples" problems in seconds.
• Solve instalment, part-investment, and rate-change problems that appear in SSC CGL Tier-II, IBPS PO Mains, and CSAT.
• Spot the traps examiners set between Simple Interest and Compound Interest.

The Hook: The ₹10,000 Lesson

Two friends each borrow ₹10,000 for 3 years at 10% per year. One borrows on simple interest, the other on compound interest. At the end, the first friend pays back ₹13,000. The second pays back ₹13,310. Same money, same rate, same time — yet ₹310 apart.

That ₹310 gap is exactly why examiners love this chapter. Simple Interest looks trivially easy, and that is precisely the trap: under exam pressure, students mix up SI and CI, forget to convert months into years, or misread "amount" as "interest." Master SI deeply, and you not only secure these marks — you build the foundation for Compound Interest, which is just SI applied repeatedly.

1. What Simple Interest Really Means

When you borrow money, you pay a "rent" for using it. That rent is called interest. In simple interest, the rent is calculated only on the original sum (the principal), every single year. It never grows, never compounds.

• Principal (P): the original sum borrowed or invested.
• Rate (R): interest per ₹100 per year, written as R% p.a.
• Time (T): the period, always converted into years.
• Simple Interest (SI): the total rent paid.
• Amount (A): Principal + Interest = P + SI.

The core formula:

SI = (P × R × T) / 100

Amount A = P + SI = P(1 + RT/100)

🧠 Active Recall: Without looking — if P = ₹5,000, R = 8%, T = 2 years, what is the SI? What is the Amount? (SI = ₹800, A = ₹5,800. Check yourself.)

2. Why the Formula Works (Don't Just Memorise It)

R% means: every year, you pay ₹R for every ₹100 of principal. So in one year, interest on ₹P is (P × R)/100. Over T years, multiply by T. That's it — the whole formula is just "rent per year × number of years."

This understanding unlocks the rearrangements instantly:

• P = (SI × 100) / (R × T)
• R = (SI × 100) / (P × T)
• T = (SI × 100) / (P × R)

Notice the pattern: whatever you want sits alone; SI × 100 is always on top; the other two variables multiply below.

🧠 Active Recall: A sum earns ₹1,200 as SI in 4 years at 5%. Can you find the principal using the rearranged formula? (P = 1200 × 100 / (5 × 4) = ₹6,000.)

3. The Time Trap: Months and Days

Examiners rarely give clean years. Convert carefully:

• Months → years: divide by 12. (T = 8 months = 8/12 = 2/3 year)
• Days → years: divide by 365 (unless the question says 366 or 360).

Example: SI on ₹7,300 at 6% for 73 days.
T = 73/365 = 1/5 year. SI = (7300 × 6 × 1/5)/100 = ₹87.60.

A classic trap: "2 years 6 months" must become 2.5 years before you apply the formula — students who plug in 2.6 lose the mark.

4. The "Becomes n Times" Family

This is the most repeated advanced pattern. If a sum becomes n times itself (i.e., Amount = nP), then the interest earned is (n − 1)P.

So: (n − 1)P = PRT/100 → RT = 100(n − 1)

Example: A sum doubles in 8 years at SI. Find the rate.
Here n = 2, so RT = 100 → R × 8 = 100 → R = 12.5%.

The killer follow-up: "A sum doubles in 8 years. In how many years will it become 4 times (at the same SI rate)?"
Doubling means earning 1P of interest in 8 years. To become 4 times, you must earn 3P — three times the interest — so 3 × 8 = 24 years. In SI, interest grows linearly, so time scales linearly. (In CI it would be 16 years — a favourite confusion trap!)

🧠 Active Recall: A sum triples in 12 years at simple interest. When will it become 5 times? (Tripling = earning 2P in 12 years, so 1P per 6 years. Five times = earning 4P → 24 years.)

5. Split Investments and Rate Changes

Split investment: ₹P is split into two parts at rates R₁ and R₂ for the same time, earning equal interest. Then the parts are in the ratio R₂ : R₁ (inverse ratio — the higher rate needs the smaller part).

Example: ₹10,000 split at 6% and 8% for equal SI in the same time. Parts ratio = 8 : 6 = 4 : 3, so the part at 6% is 4/7 × 10,000 = ₹5,714.29 and the part at 8% is 3/7 × 10,000 = ₹4,285.71. Verify: SI₁ = (5714.29 × 6 × T)/100 = 342.86T and SI₂ = (4285.71 × 8 × T)/100 = 342.86T — equal. ✓

Rate change over time: If the rate is R₁ for T₁ years and R₂ for T₂ years, just add the interests:
SI = P(R₁T₁ + R₂T₂)/100

6. Instalments — The Advanced Favourite

A debt is repaid in equal annual instalments of ₹x under simple interest at rate R. The reliable method: each instalment, paid after a different number of years, clears a different slice of the loan. An instalment of ₹x paid after T years clears a principal slice of:

Principal slice = x / (1 + RT/100)

because that slice, plus T years of simple interest on it, grows to exactly ₹x. Add the slices of all instalments to get the loan:

Loan = Σ x / (1 + RTᵢ/100), where Tᵢ = 1, 2, 3, … years for the 1st, 2nd, 3rd, … instalment.

Example: A loan is repaid in 2 annual instalments of ₹1,050 each at 10% SI. Find the loan.

• First instalment (paid after 1 year): 1050 / (1 + 10×1/100) = 1050/1.1 = ₹954.55
• Second instalment (paid after 2 years): 1050 / (1 + 10×2/100) = 1050/1.2 = ₹875
• Loan = 954.55 + 875 = ₹1,829.55

Lesson: there is no safe one-line shortcut here — divide each instalment by (1 + RT/100) with its own T, then add. Trust the method, not a formula you half-remember.

🧠 Active Recall: Why does each instalment get divided by a *different* factor? (Because each is paid after a different number of years, so each carries a different amount of interest.)

7. SI vs CI — Know the Boundary

For 1 year, SI and CI are identical. They diverge from year 2 onward, because CI pays interest on interest. For 2 years, the difference is P(R/100)². If a question mentions "interest on interest" or "compounded," leave this chapter's formulas behind. If the question says nothing about the type of interest, read it twice before assuming — many exams state "simple interest" explicitly, and when they don't, the wording usually tells you which one is meant.

🧠 Mnemonic: "PeRT-100"

Remember the formula as a person: Mr. PeRT works for 100.
P × R × T over 100 — Principal, Rate, Time, divided by 100. And his wife "A = P + SI" — the Amount is always Principal *plus* his salary (SI), never minus. If you can picture Mr. PeRT handing his paycheque to Mrs. A, you will never swap Amount and Interest again.

Summary Table

📌 Review this chapter after 2 days. Redo the "becomes n times" and instalment examples with a pen, without looking at the solutions — those two families decide most SI questions in Tier-II and PO Mains.