Profit, loss, discount, simple interest — basic applications

---

1. Profit and Loss

Deriving the percentage formulas (from scratch)

Profit% means: "for every 100 rupees of CP, how many rupees did I gain?" That is a direct proportion.

\;\Rightarrow\; \boxed{\text{Profit}\% = \frac{SP-CP}{CP}\times 100}$$ Rearranging for SP (very useful): $$SP = CP\left(1 + \frac{\text{Profit}\%}{100}\right),\qquad SP = CP\left(1 - \frac{\text{Loss}\%}{100}\right)$$ > [!formula] Summary > $$\text{Profit}\% = \frac{SP-CP}{CP}\times100,\quad \text{Loss}\% = \frac{CP-SP}{CP}\times100$$ > $$SP = CP\Big(1\pm\tfrac{p}{100}\Big),\qquad CP = \frac{SP}{1\pm\frac{p}{100}}$$ > [!example] Basic profit > CP = ₹800, SP = ₹920. Find profit%. > - Profit $= 920-800 = 120$. *Why?* SP > CP so it's a gain. > - Profit% $= \frac{120}{800}\times100 = 15\%$. *Why divide by 800?* CP is the base. > [!example] Find SP from profit% > CP = ₹1200, want 25% profit. Find SP. > - $SP = 1200(1 + 0.25) = 1200\times1.25 = ₹1500$. *Why 1.25?* You keep the whole CP (1) plus 0.25 extra. > [!example] Back out CP (the trap case) > Item sold at ₹1500 giving **25% profit**. Find CP. > - CP is unknown base, so $1500 = CP(1.25)\Rightarrow CP = \frac{1500}{1.25}=₹1200$. *Why not $1500\times0.75$?* The percentage is on CP, **not** on SP — you must divide. --- ## 2. Discount > [!definition] > - ==Marked Price (MP)== (a.k.a. list price / tag price): the price written before any reduction. > - **Discount** $= MP - SP$, and it is a percentage **of MP**. > $$\text{Discount}\% = \frac{MP - SP}{MP}\times100,\qquad SP = MP\Big(1-\tfrac{d}{100}\Big)$$ > [!intuition] WHY is MP the base here (not CP)? > Discount is a *shop marketing decision*: the shopkeeper reduces from the tag. The customer sees MP first, so the reduction is naturally measured against MP. Profit/loss for the shopkeeper is *still* measured against CP. > [!example] Chain: CP → MP → discount → SP → profit > A shopkeeper buys at ₹400 (CP), marks it up to ₹600 (MP), gives 10% discount. > - $SP = 600(1-0.10) = 600\times0.9 = ₹540$. *Why 0.9?* customer pays 90% of the tag. > - Profit $= 540 - 400 = ₹140$. *Why compare to 400?* profit base = CP. > - Profit% $= \frac{140}{400}\times100 = 35\%$. --- ## 3. Simple Interest > [!definition] > - ==Principal (P)==: money borrowed/deposited. > - ==Rate (R)==: percent charged **per year**. > - **Time (T)**: in years. > - **Simple** = interest is charged only on the *original* principal, never on past interest. ### Deriving SI from scratch Interest for **one year** = $R\%$ of $P = P\times\frac{R}{100}$. Since simple interest uses the same $P$ every year, for $T$ years just multiply: $$\boxed{SI = \frac{P\times R\times T}{100}}$$ Amount (what you finally owe/have): $$A = P + SI = P\left(1 + \frac{RT}{100}\right)$$ > [!formula] Rearranged forms > $$P=\frac{100\,SI}{RT},\qquad R=\frac{100\,SI}{PT},\qquad T=\frac{100\,SI}{PR}$$ > [!example] Straight SI > P = ₹5000, R = 8% p.a., T = 3 years. > - $SI = \frac{5000\times8\times3}{100} = ₹1200$. *Why /100?* R is a percent. > - $A = 5000 + 1200 = ₹6200$. > [!example] Find rate > ₹2000 earns ₹360 SI in 2 years. Find R. > - $R = \frac{100\times360}{2000\times2} = \frac{36000}{4000} = 9\%$. *Why this form?* isolate R from $SI=\frac{PRT}{100}$. > [!example] Time as a fraction > P = ₹4000, R = 6%, T = 9 months. > - 9 months $= \frac{9}{12} = 0.75$ years. *Why convert?* R is **per year**. > - $SI = \frac{4000\times6\times0.75}{100} = ₹180$. ![[1.1.21-Profit,-loss,-discount,-simple-interest-—-basic-applications.png]] --- ## Common Mistakes (Steel-manned) > [!mistake] Dividing profit by SP instead of CP > **Why it feels right:** SP is the "final" number, so it seems like the reference. > **Fix:** you invested CP — gain is measured per rupee *invested*. Base = CP always for profit/loss. > [!mistake] Reversing percentages by multiplying instead of dividing > "SP = ₹1500 at 25% profit, so CP = $1500\times0.75$." **Wrong.** > **Why it feels right:** 25% off 1500 looks symmetric. > **Fix:** the 25% is on the *unknown* CP: $1500 = CP\times1.25\Rightarrow CP = 1500/1.25 = 1200$. Percentage-up and percentage-down are **not** inverses. > [!mistake] Mixing MP and CP bases > Applying discount% to CP, or profit% to MP. > **Fix:** Discount lives on **MP**, profit/loss lives on **CP**. Keep a chain: CP → MP → discount → SP. > [!mistake] Forgetting to convert months to years in SI > Using T = 9 instead of 0.75. > **Fix:** R is *per year*; T must be in years. Divide months by 12. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine you buy a toy for ₹100 (that's your *cost*). If you sell it for ₹120, you made ₹20 extra — and 20 out of 100 is 20%, so 20% profit. A **discount** is like a shop tearing off part of the price sticker: the sticker (marked price) is the starting point. **Simple interest** is like a plant that gives you the same amount of fruit every year based only on the first seed you planted — it never counts last year's fruit. Every case is just: *how big is the change compared to the starting number?* > [!mnemonic] Who is the boss (base)? > **"CP for Cash you Paid, MP for Marked tag, P for Principal seed."** > Profit% → CP · Discount% → MP · Interest → P. Pick the boss, divide by it. > [!recall] Active recall — cover the answers > - Base for profit%? ::: Cost Price (CP) > - Base for discount%? ::: Marked Price (MP) > - Why divide (not multiply) to reverse a % increase? ::: because the % is on the unknown base, so SP = base×(1+p/100) ⇒ base = SP/(1+p/100) --- ## Connections - [[Percentages]] — the parent tool; everything here is a percentage of a base. - [[Ratio and Proportion]] — profit% is the proportion Profit : CP. - [[Compound Interest]] — SI's sibling where interest *does* stack on interest. - [[Marked Price and Successive Discounts]] — extends discount to multiple reductions. - [[Linear Equations]] — "back out CP/P" problems are one-variable equations. #flashcards/maths Profit formula in terms of SP and CP ::: Profit = SP − CP Profit% formula ::: (SP − CP)/CP × 100 SP given CP and profit p% ::: SP = CP(1 + p/100) CP given SP and profit p% ::: CP = SP / (1 + p/100) Discount% formula ::: (MP − SP)/MP × 100 SP given MP and discount d% ::: SP = MP(1 − d/100) Simple Interest formula ::: SI = P·R·T / 100 Amount with SI ::: A = P(1 + RT/100) Rate from SI ::: R = 100·SI/(P·T) Convert 9 months to years for SI ::: 9/12 = 0.75 years Base for profit/loss vs base for discount ::: CP for profit/loss, MP for discount ## 🖼️ Concept Map ```mermaid flowchart TD BASE[Compare final to base as percent] CP[Cost Price] SP[Selling Price] MP[Marked Price] P[Principal] PL[Profit or Loss] DISC[Discount] SI[Simple Interest] BASE -->|core idea| PL BASE -->|core idea| DISC BASE -->|core idea| SI CP -->|base for| PL SP -->|compared to CP| PL PL -->|percent of CP| CP MP -->|base for| DISC SP -->|compared to MP| DISC DISC -->|SP = MP x 1 minus d| SP P -->|base for| SI SI -->|scaled by time| P ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, in chaaron topics ka asli funda ek hi hai: **kisi final amount ko base amount se compare karo, aur percentage me bolo**. Bas har baar "base kaun hai?" yeh pehchan lo. Profit/loss me base hamesha **Cost Price (CP)** hota hai — kyunki tumne CP invest kiya tha, to gain/loss usi ke against naapa jaata hai. Discount me base **Marked Price (MP)** hota hai — dukaandaar sticker se kaatta hai. Aur interest me base **Principal (P)** hota hai. > > Sabse common galti: SP se profit% nikalna. Nahi! Profit% = (SP−CP)/CP × 100 — divide by CP. Aur ek bada trap — agar "25% profit pe SP = 1500" diya ho aur CP maangi ho, to log 1500×0.75 kar dete hain, jo **galat** hai. Sahi: 1500 = CP × 1.25, isliye CP = 1500/1.25 = 1200. Yaad rakho, percentage-badhana aur percentage-ghatana ek dusre ke ulta (inverse) nahi hote, isliye reverse karne ke liye **divide** karo. > > Simple Interest me formula khud banana easy hai: ek saal ka interest = P ka R% = P×R/100. Simple interest me har saal wahi P use hota hai (purane interest pe interest nahi lagta), to T saal ke liye bas multiply kar do: SI = PRT/100. Yaad rakho R "per year" hai, isliye agar time months me diya ho (jaise 9 months) to pehle 9/12 = 0.75 year me convert karo, warna answer galat aayega. > > Exam me speed ke liye seedha SP = CP(1 ± p/100) aur SP = MP(1 − d/100) use karo — yeh dono forms 80% problems solve kar dete hain. Base pehchano, formula khud reconstruct ho jayega, ratta maarne ki zaroorat nahi. ![[audio/1.1.21-Profit,-loss,-discount,-simple-interest-—-basic-applications.mp3]]