Profit, loss, discount, simple interest — basic applications

1.1.21 · Maths › Arithmetic & Number Systems

1. Profit and Loss

Percentage formulas derive karna (scratch se)

Profit% ka matlab hai: "CP ke har 100 rupaye pe mujhe kitne rupaye mila?" Ye ek direct proportion hai.

\;\Rightarrow\; \boxed{\text{Profit}\% = \frac{SP-CP}{CP}\times 100}$$ SP ke liye rearrange karna (bahut useful): $$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}}$$ > [!example] Basic profit > CP = ₹800, SP = ₹920. Profit% nikalo. > - Profit $= 920-800 = 120$. *Kyun?* SP > CP hai isliye gain hai. > - Profit% $= \frac{120}{800}\times100 = 15\%$. *800 se kyun divide kiya?* CP base hai. > [!example] Profit% se SP nikalna > CP = ₹1200, 25% profit chahiye. SP nikalo. > - $SP = 1200(1 + 0.25) = 1200\times1.25 = ₹1500$. *1.25 kyun?* Poora CP (1) rakhte ho usmein 0.25 extra jod do. > [!example] CP wapas nikalna (trap wala case) > Item ₹1500 mein becha aur **25% profit** mila. CP nikalo. > - CP unknown base hai, isliye $1500 = CP(1.25)\Rightarrow CP = \frac{1500}{1.25}=₹1200$. *$1500\times0.75$ kyun nahi?* Percentage CP pe hai, **SP pe nahi** — divide karna padega. --- ## 2. Discount > [!definition] > - ==Marked Price (MP)== (a.k.a. list price / tag price): wo price jo kisi bhi reduction se pehle likha hota hai. > - **Discount** $= MP - SP$, aur ye **MP** ka percentage hota hai. > $$\text{Discount}\% = \frac{MP - SP}{MP}\times100,\qquad SP = MP\Big(1-\tfrac{d}{100}\Big)$$ > [!intuition] Yahaan base MP kyun hai (CP kyun nahi)? > Discount ek *shop ki marketing decision* hai: shopkeeper tag se price kam karta hai. Customer pehle MP dekhta hai, isliye reduction naturally MP ke against measure hoti hai. Shopkeeper ka profit/loss *phir bhi* CP ke against hi measure hota hai. > [!example] Chain: CP → MP → discount → SP → profit > Ek shopkeeper ₹400 (CP) mein kharidta hai, ₹600 (MP) pe mark karta hai, 10% discount deta hai. > - $SP = 600(1-0.10) = 600\times0.9 = ₹540$. *0.9 kyun?* Customer tag ka 90% pay karta hai. > - Profit $= 540 - 400 = ₹140$. *400 se kyun compare kiya?* profit ka base = CP. > - Profit% $= \frac{140}{400}\times100 = 35\%$. --- ## 3. Simple Interest > [!definition] > - ==Principal (P)==: borrowed/deposit kiya hua paisa. > - ==Rate (R)==: **per year** charge hone wala percent. > - **Time (T)**: years mein. > - **Simple** = interest sirf *original* principal pe lagta hai, past ke interest pe kabhi nahi. ### SI ko scratch se derive karna **Ek saal** ka interest $= P$ ka $R\% = P\times\frac{R}{100}$. Kyunki simple interest har saal same $P$ use karta hai, $T$ saalon ke liye bas multiply karo: $$\boxed{SI = \frac{P\times R\times T}{100}}$$ Amount (jo tum finally dete/lete ho): $$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] Seedha SI > P = ₹5000, R = 8% p.a., T = 3 years. > - $SI = \frac{5000\times8\times3}{100} = ₹1200$. */100 kyun?* R ek percent hai. > - $A = 5000 + 1200 = ₹6200$. > [!example] Rate nikalna > ₹2000 par 2 saal mein ₹360 SI milta hai. R nikalo. > - $R = \frac{100\times360}{2000\times2} = \frac{36000}{4000} = 9\%$. *Ye form kyun?* $SI=\frac{PRT}{100}$ mein se R isolate karo. > [!example] Time fraction ke roop mein > P = ₹4000, R = 6%, T = 9 months. > - 9 months $= \frac{9}{12} = 0.75$ years. *Convert kyun kiya?* R **per year** hai. > - $SI = \frac{4000\times6\times0.75}{100} = ₹180$. ![[1.1.21-Profit,-loss,-discount,-simple-interest-—-basic-applications.png]] --- ## Common Mistakes (Steel-manned) > [!mistake] Profit ko CP ki jagah SP se divide karna > **Kyun sahi lagta hai:** SP "final" number hai, isliye reference lagta hai. > **Fix:** tumne CP invest kiya tha — gain measure hoti hai *invested* rupaye ke per. Profit/loss ke liye base = CP hamesha. > [!mistake] Multiply karke percentage reverse karna (divide ki jagah) > "SP = ₹1500 at 25% profit, toh CP = $1500\times0.75$." **Galat.** > **Kyun sahi lagta hai:** 1500 pe 25% off symmetric lagta hai. > **Fix:** 25% *unknown* CP pe hai: $1500 = CP\times1.25\Rightarrow CP = 1500/1.25 = 1200$. Percentage-up aur percentage-down **inverses nahi hain**. > [!mistake] MP aur CP bases ko mix karna > Discount% ko CP pe apply karna, ya profit% ko MP pe. > **Fix:** Discount **MP** pe hota hai, profit/loss **CP** pe hota hai. Chain yaad rakho: CP → MP → discount → SP. > [!mistake] SI mein months ko years mein convert karna bhool jaana > T = 0.75 ki jagah T = 9 use karna. > **Fix:** R *per year* hai; T years mein hona chahiye. Months ko 12 se divide karo. --- > [!recall]- Feynman: ek 12-saal ke bachche ko samjhao > Socho tumne ek toy ₹100 mein kharida (ye tumhara *cost* hai). Agar tum ise ₹120 mein becho, tumne ₹20 extra kamaya — aur 100 mein se 20 matlab 20% hai, toh 20% profit hua. **Discount** waise hai jaise ek shop price sticker ka kuch hissa fad de: sticker (marked price) starting point hai. **Simple interest** ek aisi plant ki tarah hai jo har saal utna hi phal deti hai based on pehle beej par — pichle saal ke phal ko kabhi count nahi karti. Har case mein bas yehi hai: *change kitna bada hai starting number ke compare mein?* > [!mnemonic] Boss (base) kaun hai? > **"CP for Cash you Paid, MP for Marked tag, P for Principal seed."** > Profit% → CP · Discount% → MP · Interest → P. Boss chunno, usi se divide karo. > [!recall] Active recall — answers cover karo > - Profit% ke liye base? ::: Cost Price (CP) > - Discount% ke liye base? ::: Marked Price (MP) > - % increase reverse karne ke liye multiply kyun nahi karte, divide karte hain? ::: kyunki % unknown base pe hai, isliye SP = base×(1+p/100) ⇒ base = SP/(1+p/100) --- ## Connections - [[Percentages]] — parent tool; yahan sab kuch ek base ka percentage hai. - [[Ratio and Proportion]] — profit% proportion hai Profit : CP ka. - [[Compound Interest]] — SI ka sibling jahan interest *stack* hota hai interest pe. - [[Marked Price and Successive Discounts]] — discount ko multiple reductions tak extend karta hai. - [[Linear Equations]] — "CP/P back out karo" wale problems one-variable equations hain. #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 ```