Simple aur compound interest

Interest, principal aur rate

Bank aur post office apne paas rakhe paise par interest yaani extra paisa dete hain, aur log udhaar lene par interest chukaate hain. ‘FD @ 10% p.a.’ jaisa notice rate of interest batata hai; p.a. ka matlab per annum, yaani har saal. Jama kiya gaya paisa principal kehlaata hai. ₹6000 ko 10% p.a. ki fixed deposit mein rakho, to ek saal baad 6000 + 0.10 × 6000 = ₹6600 ho jaate hain. Doosre shabdon mein jama apne aap ka 110% ban jaata hai: 6000 × 1.1 = 6600. Fixed deposit (FD) mein paisa chune hue samay tak band rehta hai, aur pehle nikaalne par aksar penalty lagti hai.

Bina compounding ke

Maan lo bank har saal ke aakhir mein interest bahar de deta hai aur principal aakhir mein lautaata hai. Tab ₹6000 par 10% p.a. har saal ₹600 deta hai. 3 saal mein ₹1800 interest hua, to kul ₹7800 mile, jama ka 130%, yaani 30% fayda. Aam taur par principal p aur saalana rate r (decimal mein) ho, to har saal ka interest p × r, t saal ka interest prt, aur kul rakam p + prt = p(1 + rt). Ise aksar simple interest kehte hain.

Compounding ke saath

Ab maan lo har saal ka interest waapas jama mein jod diya jaata hai. Pehle saal phir bhi ₹600 milte hain, rakam ₹6600. Doosre saal ₹6600 ka 10% yaani ₹660 milta hai, rakam ₹7260. Teesre saal ₹726, rakam ₹7986. Yahi compounding hai: har saal ka interest bhi interest kamaata hai. Har saal rakam 1.1 se guna hoti hai, isliye 3 saal baad 6000 × 1.1 × 1.1 × 1.1 = 6000 × 1.331, yaani 33.1% fayda. t saal baad rakam p(1 + r)ᵗ hoti hai, aur compound interest p(1 + r)ᵗ − p.

Lambe samay mein compounding kyon aage nikalti hai

Shuru mein dono tareeke paas-paas hote hain. ₹20,000 par 10% p.a., 2 saal: bina compounding ₹24,000 aur compounding ke saath ₹24,200, sirf ₹200 ka farq. Par samay aur avadhiyan badhne ke saath farq badhta jaata hai. Bina compounding ₹1000 par 10% har saal wahi ₹100 judte hain, seedhi line, aur 10 saal mein rakam doguni hoti hai. Compounding ke saath har saal 1.1 se guna hota hai, upar mudta hua curve, aur rakam 7 saal se thode zyada mein doguni hoti hai. Pehla linear growth hai; doosra exponential growth. Yahi niyam 3% saalana badhte shehar ya 2.5% har ghante badhte bacteria par bhi lagta hai.

Udhaar ki tulna

Loan par interest bhi aise hi chalta hai. Giridhar ₹12,500 ko 12% p.a. par 3 saal ke liye bina compounding udhaar lete hain aur 12,500 × 0.12 × 3 = ₹4500 interest chukaate hain. Raghava utni hi rakam 10% p.a. saalana compounding par lete hain aur 12,500 × (1.1³ − 1) = ₹4137.50 chukaate hain. Compounding tez badhti hai, phir bhi kam rate ki wajah se in 3 saalon mein Raghava ka interest ₹362.50 kam hai.

Girawat aur depreciation

Compounding neeche ki taraf bhi chal sakti hai. Kai cheezein umar aur istemaal ke saath keemat khoti hain; ise depreciation kehte hain. ₹21,000 ka TV ek saal mein 5% keemat kho de to 0.95 × 21,000 = ₹19,950 ka reh jaata hai. 1250 logon ka gaon jo har dashak lagbhag 10% ghatta hai, teen dashak baad 1250 × 0.9 × 0.9 × 0.9 ≈ 911, yaani lagbhag 910 logon ka reh jaata hai. Har dashak 1.1 ki jagah 0.9 se guna hota hai. Baaki chapter ke liye dekho Chapter 1 ke notes, pratishat vriddhi aur kami, aur NCERT solutions.