Intuition What this page isThe parent note gave you the tools . Here we throw every kind of problem at those tools, one at a time, so you never meet a case you haven't already beaten.
Before each answer you'll see "Forecast:" — pause and guess the answer yourself. Getting it wrong first is how the idea sticks.
Everything below rests on one habit from the parent: pick the base, then divide by it . If percentages themselves feel shaky, revisit Percentages first.
Every problem in this topic is one of these cells. We will hit all of them .
#
Cell (case class)
What makes it different
Example
A
Profit forward — CP & SP givendirect, base = CP
Ex 1
B
Loss forward — SP < CPthe sign flips
Ex 2
C
Back out the base (the trap)CP unknown, must divide not multiply
Ex 3
D
Zero / break-even SP = CP, profit = 0
Ex 4
E
Full chain CP→MP→discount→SP→profittwo different bases in one problem
Ex 5
F
SI forward — find interest & amountbase = P, scaled by T
Ex 6
G
SI reverse — find R (or P or T)isolate one variable
Ex 7
H
Fractional time — months, not yearsconvert T
Ex 8
I
Exam twist — combine ideas / limiting valueword problem stitching cells
Ex 9
The picture below is the map every example lives on: which base am I dividing by?
Worked example Ex 1 · CP and SP given, find profit%A trader buys a fan for CP = ₹1250 and sells it for SP = ₹1500 .
Find the profit and profit%.
Forecast: more or less than 20%? Guess before reading.
Profit = S P − C P = 1500 − 1250 = ₹250
Why this step? SP is bigger than CP, so money grew — the raw gain is just the difference.Profit% = C P Profit × 100 = 1250 250 × 100 = 20%
Why divide by 1250? CP is the money you put in ; profit is measured per rupee invested (see the blue node in the map). Base = CP.
Verify: rebuild SP from the answer: S P = C P ( 1 + 100 20 ) = 1250 × 1.2 = 1500.
Worked example Ex 2 · SP below CPA phone bought at CP = ₹8000 is sold for SP = ₹6800 .
Find the loss and loss%.
Forecast: the answer is a loss , so expect the percentage to be smaller than 20% here.
Loss = C P − S P = 8000 − 6800 = ₹1200
Why C P − S P When C P > S P S P − C P negative . We name that shortfall a loss and write it as a positive amount by subtracting the smaller from the larger.Loss% = 8000 1200 × 100 = 15%
Why still divide by CP? Base never changes for profit/loss — it's what you paid.
Verify: S P = C P ( 1 − 100 15 ) = 8000 × 0.85 = 6800.
Loss% is not negative profit% you invent — it's a positive number with the word "loss". Same base, opposite direction.
Worked example Ex 3 · SP and profit% given, find CPAn item sold for SP = ₹1500 gave a 25% profit . Find CP.
Forecast: many people compute 1500 × 0.75 = 1125
Write what the percentage acts on: the 25% is a profit on CP , so
S P = C P ( 1 + 100 25 ) = C P × 1.25. Why this form? The base of the percentage is the unknown CP , not the known SP. The extra 25% sits on top of CP.Solve the one-variable equation (this is just Linear Equations ):
1500 = C P × 1.25 ⇒ C P = 1.25 1500 = ₹1200. Why divide, not multiply? To undo "×1.25" you divide by 1.25. Taking 1500 × 0.75 wrong base (SP).
Verify: 1200 × 1.25 = 1500. 1125 =
Intuition Why up and down aren't inversesGoing up 25% multiplies by 1.25. Going down 25% multiplies by 0.75. Since 1.25 × 0.75 = 0.9375 = 1
Worked example Ex 4 · Selling at exactly costA shopkeeper buys goods for CP = ₹640 and, to clear stock, sells at SP = ₹640 .
Find profit%, then find profit% if he sells at ₹640 but had marked it ₹800 with a discount.
Forecast: what's a profit% of zero rupees gain?
Profit = 640 − 640 = ₹0
Why this matters? This is the boundary between profit and loss — the "zero input". Everything is well-defined here; nothing breaks.Profit% = 640 0 × 100 = 0%
Why not undefined? We divide by CP (640) , which is nonzero — so the fraction is a perfectly fine 0 by profit would be nonsense, but we never do that.)The discount side: MP = 800, SP = 640, so
Discount % = 800 800 − 640 × 100 = 800 160 × 100 = 20%. Why base = MP now? Discount always sits on the marked price — a different base from profit. Break-even on profit and a 20% discount coexist happily.
Verify: S P = M P ( 1 − 100 20 ) = 800 × 0.8 = 640 S P = C P ( 1 + 0 ) = 640
Worked example Ex 5 · CP → MP → discount → SP → profit%A shopkeeper buys a bag at CP = ₹400 , marks it up to MP = ₹600 , and offers a 10% discount .
Find the final profit%.
Forecast: is the final profit% closer to 35% or 50%?
Apply discount to MP (base = MP):
S P = M P ( 1 − 100 10 ) = 600 × 0.9 = ₹540. Why 0.9? The customer pays 90% of the tag.Now switch base to CP for profit:
Profit = S P − C P = 540 − 400 = ₹140. Why change base? Discount lived on MP; profit lives on CP. Same problem, two bosses (see the map's two colours).Profit% = 400 140 × 100 = 35%
Verify: C P ( 1 + 100 35 ) = 400 × 1.35 = 540 = S P .
This case extends to several discounts in Marked Price and Successive Discounts — each discount multiplies, never adds.
Worked example Ex 6 · Find SI and amountP = ₹5000 , R = 8% per year, T = 3 years . Find SI and the amount A.
Forecast: three years at 8% — roughly a quarter of P as interest?
Interest for one year = P × 100 R = 5000 × 100 8 = ₹400
Why /100? R is a percent — "8 per hundred" means multiply by 8/100 Multiply by T (simple interest re-uses the same P each year):
S I = 100 P R T = 100 5000 × 8 × 3 = ₹1200. Why just multiply by 3? "Simple" means no stacking — each year earns the identical ₹400. (Stacking would be Compound Interest .)Amount = P + S I = 5000 + 1200 = ₹6200
Verify: A = P ( 1 + 100 R T ) = 5000 ( 1 + 100 8 × 3 ) = 5000 × 1.24 = 6200.
The figure below shows why simple interest is a straight line — equal steps every year.
Worked example Ex 7 · Find the rate₹2000 earns ₹360 simple interest in 2 years . Find R.
Forecast: ₹360 on ₹2000 over 2 years — single-digit rate?
Start from S I = 100 P R T
R = P T 100 S I . Why this rearrangement? R is the only unknown; multiply both sides by P T 100 Linear Equations .)Substitute:
R = 2000 × 2 100 × 360 = 4000 36000 = 9%.
Verify (units too): 100 2000 × 9 × 2 = 100 36000 = ₹360.
Worked example Ex 8 · Time given in monthsP = ₹4000 , R = 6% per year, T = 9 months . Find SI.
Forecast: less than a full year, so interest is under ₹240 (the 1-year figure). Guess.
Convert months to years:
T = 12 9 = 0.75 years . Why convert? R is per year . Feeding "9" as if it were years would give 12× too much interest.Apply the formula:
S I = 100 4000 × 6 × 0.75 = 100 18000 = ₹180.
Verify: one full year would give 100 4000 × 6 × 1 = ₹240 0.75 × 240 = ₹180.
Common mistake The forgotten /12Using T = 9 S I = 100 4000 × 6 × 9 = ₹2160 huge , check the time unit first.
Worked example Ex 9 · Deposit that must reach a targetRiya wants an amount of ₹6300 back from a simple-interest deposit of P = ₹5000 at R = ? over T = 4 years . She also asks: what happens to the required rate as T grows very large?
Forecast: she needs ₹1300 of interest — will R be above or below 8%?
Interest needed = A − P = 6300 − 5000 = ₹1300
Why? Amount = principal + interest, so interest is the target minus what she started with.Solve for R with R = P T 100 S I R = 5000 × 4 100 × 1300 = 20000 130000 = 6.5%. Why this form? Same reverse move as Cell G — isolate R.Limiting behaviour (the twist): for a fixed target interest ₹1300,
R = 5000 × T 100 × 1300 = T 26 . T 0 smaller the rate needed to reach the same interest. It never hits exactly 0 (that would need infinite time).
Why care? This is the "limiting input" cell of the matrix: the formula stays sensible as T → ∞, approaching but never reaching zero.
Verify: at R = 6.5%, S I = 100 5000 × 6.5 × 4 = ₹1300 A = 5000 + 1300 = ₹6300. R ( T = 4 ) = 26/4 = 6.5%
Recall Which cell was which? (cover the answers)
Ex 3 "SP=1500 at 25% profit, find CP" is which trap? ::: Cell C — must divide by 1.25, never multiply by 0.75
Ex 4 profit% at break-even? ::: 0% — well-defined because we divide by CP (nonzero)
Ex 5 which base for the discount, which for the profit? ::: discount on MP (600), profit on CP (400)
Ex 8 why convert 9 months? ::: R is per year, so T must be in years → 9/12 = 0.75
Ex 9 required rate as T → ∞ for fixed interest? ::: it shrinks toward 0 like 26/T, never reaching 0
Mnemonic One-line survival kit"Name the base, divide by it; to reverse a %, divide — never subtract."
Percentages — every cell above is one percentage of one base.Ratio and Proportion — profit% is the ratio Profit : CP written per 100.Linear Equations — Cells C, G, I are single-variable equations in disguise.Marked Price and Successive Discounts — extends Cell E to many discounts.Compound Interest — the "stacking" sibling of Cells F–I.Back to parent topic 
