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Compound interest

Lesson 23 of 113

1. Ye topic hai kya?

Compound interest = pehle saal ke interest ko asli mein jod do, agla saal us nayi rashi pe. Simple interest P10 / SI page: 1000, 10%, 2 saal pe SI=200, A=1200. Yahan year-2 1100 pe 10% = 110, CI=210, A=1210. Simple-interest P9 yahi bheja tha.

Paper yahan kya maangta

A = P(1+R/100)n, CI = A−P, saal-saal table, 2 saal ka extra P(R/100)2, half-yearly R/2 aur 2n, quarterly R/4, alag-alag saal ke rate guna, population badhe / machine ghate, 1½ saal (1 compound + aadha SI), do equal installment.

Yahan kya nahi

PRT/100 seedha, T=100/R double, 18 months=120, do rate jodna 5%×2+8%×3 — Simple interest. CP 200 SP 250, 10%+20% off 100→90→72 — Profit/loss. 25% of 80 — Percentage. Capital×months share — Partnership.

A = P (1 + R/100)n · CI = A − P
Pehla saal: CI = SI hamesha. Doosra saal se farq. n = kitni baar joda — saal, ya half, ya quarter.

2. Kyun 1210, 1200 nahi

1000 pe 10%. Saal 1: +100 → 1100 (SI jaisa). Saal 2: 10% ab 1100 pe = 110, total 1210. SI mein saal 2 bhi 100, 1200. Extra 10 = 10% of pehle saal ka 100 = P(R/100)2.

1000 1100 SI 1200 CI 1210 0 1 yr 2 yr

Dashed = simple 1200. Upar = compound 1210. Year-1 same; year-2 +10 extra.

Saal 1Saal 2Saal 3
Us saal ka interest100110121
Amount (end)110012101331
SI amount (same P,R)110012001300
100 110 121 yr 1 yr 2 yr 3

Har saal interest badhta hai. SI mein teeno bars 100-100-100.

2 saal bina calculator: A = P + 2·(PR/100) + P(R/100)2
1000, 10%: 1000+200+10 = 1210. 3 saal 10% yaad: 1331 (11³).
Example

P=1000, R=10%, n=2 (annual). CI aur A?

Solution

  1. ×1.1×1.1

    1100, phir 1210. CI=210. SI 200 nahi. Year-2 = 110, 100 nahi.

AnswerCI 210, A 1210
Example

Usi pe year-2 ka interest? (Simple-interest P9.)

Solution

  1. 1100 ka 10%

    110. Pehle saal 100. Total CI 210.

Answer110
Example

8000, 5%, 2 saal annual. A?

Solution

  1. 1.05²=1.1025

    8000×1.1025=8820. CI=820. SI hota 800.

Answer8820
Example

1600, 2 saal, A=1764. R?

Solution

  1. A/P

    1764/1600=1.1025=1.05² → 5%.

Answer5%

3. Half-yearly, quarterly, 1½ saal

Bank jitni baar jode, utna zyada. Annual: n saal, rate R. Half-yearly: har 6 mahine, rate R/2, periods 2n. Quarterly: R/4, 4n.

Half: A = P (1 + R/200)2n
Quarter: A = P (1 + R/400)4n

1000, 10% p.a., 1 saal half-yearly: 5% × 2 dafa. 1000×1.05×1.05 = 1102.50. Annual hota 1100. Extra 2.50.

12% p.a. quarterly, 1 saal, P=1000: 3% × 4. (1.03)4 = 1.12550881 → A ≈ 1125.51.

1½ saal, annual compound: pehle poora saal compound, bache 6 mahine us nayi rashi pe simple (aadha rate). 8000, 10%, 1½ yr: saal 1 → 8800; 6 mahine 5% of 8800 = 440; A=9240. (Poora 1.5 ko (1.1)1.5 mat — paper aisa hi todta hai jab “compounded annually” likha ho.)

Example

1000, 10% p.a., 1 saal, half-yearly. A?

Solution

  1. 1.05²

    1102.50. Annual 1100 trap.

Answer1102.50
Example

8000, 10% annual compound, 1½ saal. A?

Solution

  1. 8800 + 440

    9240. SI 1½ yr = 1200, A=9200 — alag.

Answer9240

4. CI−SI, alag rates, ghatna, installment

2 saal: CI − SI = P (R/100)2. 1000, 10% → 10. 2500, 8% → 2500×0.0064 = 16, phir SI 2 yr = 400, CI = 416.

Alag saal, alag rate — guna, jodo mat. 5000 pe pehle 8% phir 10%: 5000×1.08×1.10 = 5940. SI-add trap 400+500=900, A=5900. (Simple page pe 5%×2 + 8%×3 jodna tha kyunki wahan principal nahi badla.)

Population / growth: same formula, R positive. 20000, 5%, 2 saal → 20000×1.1025 = 22050. (Percentage wala 8000×1.1×1.2 yahan copy nahi.)

Depreciation / machine: A = P (1 − R/100)n. 10000, 10%, 2 saal → 10000×0.9×0.9 = 8100.

Do equal annual installment (loan aaj, two payments year-end): x abhi ka value nahi — har payment ko peeche discount karo.
Loan 2100, 10% CI, 2 saal: pehla x 1 saal compound se aaya (divide 1.1), dusra 2 saal (divide 1.21).
x/1.1 + x/1.21 = 2100. LCD 1.21: 1.1x + x = 2100×1.21 → 2.1x = 2541 → x = 1210. Trap: 2100/2=1050 (interest zero); 2100×1.1=2310 (ek lump, 2 equal nahi).

Pehla saal CI≠SI mat socho — equal hote hain. Half-yearly pe R ko R hi mat chhodna. Alag rates ko 8+10=18 mat. 10%+20% shop discount 28% Profit/loss — yahan growth 100→110→121. Double ka T=100/R Simple pe; yahan (1+R/100)n=2.
Example

1000, 10%, 2 saal. CI − SI?

Solution

  1. (0.1)² × P

    10. 210−200.

Answer10
Example

5000, year-1 8%, year-2 10%. A?

Solution

  1. ×1.08×1.10

    5940. 5900 SI-add.

Answer5940
Example

Machine 10000, 10% p.a. fall, 2 saal. Value?

Solution

  1. ×0.9×0.9

    8100. 8000 mat (10+10).

Answer8100
Example

Loan 2100, 10% CI, 2 equal annual installments. Har ek?

Solution

  1. PV

    Year-1 payment aaj = x/1.1. Year-2 = x/1.21. Sum = 2100.

  2. Solve

    x(1/1.1 + 1/1.21)=2100 → x(2.1)/1.21=2100 → x=1210. 1050 half nahi; partnership C×T nahi.

Answer1210

5. Sawal — basic se pro

Q1 · Basic

P=1000, R=10%, n=2 annual. CI?

  1. 210
  2. 200
  3. 100
  4. 1210

Solution

  1. 210

    B SI. D amount.

Answer210
Q2 · Basic

Uska amount?

  1. 1210
  2. 1200
  3. 1100
  4. 210

Solution

  1. 1210

    B SI amount.

Answer1210
Q3 · Basic

Year-2 ka interest usi sum pe?

  1. 110
  2. 100
  3. 210
  4. 10

Solution

  1. 1100×10%

    110. D extra-only.

Answer110
Q4 · Medium

Pehle saal CI vs SI (same P,R)?

  1. Barabar
  2. CI zyada
  3. SI zyada
  4. CI double

Solution

  1. Year-1

    Abhi compound hone ko kuch nahi. Equal.

AnswerBarabar
Q5 · Medium

1000, 10%, 3 saal annual. A?

  1. 1331
  2. 1300
  3. 1210
  4. 331

Solution

  1. ×1.1³

    1331. B SI 3 yr.

Answer1331
Q6 · Medium

1000, 10% p.a., 1 saal half-yearly. A?

  1. 1102.50
  2. 1100
  3. 1210
  4. 1050

Solution

  1. 1.05²

    1102.50. B annual.

Answer1102.50
Q7 · Medium

1000, 10%, 2 saal. CI − SI?

  1. 10
  2. 210
  3. 20
  4. 100

Solution

  1. P(R/100)²

    10.

Answer10
Q8 · Pro

5000, 8% phir 10%. A?

  1. 5940
  2. 5900
  3. 6000
  4. 5400

Solution

  1. ×1.08×1.1

    5940. B SI-add.

Answer5940
Q9 · Pro

10000, 10% p.a. depreciation, 2 saal. Value?

  1. 8100
  2. 8000
  3. 9000
  4. 12100

Solution

  1. 0.9²

    8100. B 10+10.

Answer8100
Q10 · Pro

1000, 10%, 2 saal, sirf PRT/100 se 200. Is chapter?

  1. Nahi — Simple interest
  2. Haan, CI=200
  3. A=1200 compound
  4. Year-2=100

Solution

  1. Seedha P

    A. Yahan 210 / 1210.

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

8000, 5%, 2 saal annual. A?

  1. 8820
  2. 8800
  3. 8400
  4. 8200

Solution

  1. 1.1025

    8820. B SI A.

Answer8820
P2

1600 → 1764 in 2 saal. R?

  1. 5%
  2. 10%
  3. 8%
  4. 5.25%

Solution

  1. √1.1025

    5%.

Answer5%
P3

2000, 10%, 3 saal. CI?

  1. 662
  2. 600
  3. 2662
  4. 331

Solution

  1. 2000×1.331

    A=2662, CI=662. B SI.

Answer662
P4

1000, 12% p.a., 1 saal quarterly. A ≈?

  1. 1125.51
  2. 1120
  3. 1124
  4. 1030

Solution

  1. (1.03)⁴

    1125.51. B SI 120, A=1120.

Answer1125.51
P5

8000, 10% annual, 1½ saal. A?

  1. 9240
  2. 9200
  3. 8800
  4. 9680

Solution

  1. 8800+440

    9240. B SI 1½.

Answer9240
P6

20000, 5% p.a. growth, 2 saal. Population?

  1. 22050
  2. 22000
  3. 21000
  4. 24000

Solution

  1. ×1.1025

    22050. B SI-add 5+5.

Answer22050
P7

Loan 2100, 10% CI, 2 equal yearly. Installment?

  1. 1210
  2. 1050
  3. 1100
  4. 1320

Solution

  1. PV

    x/1.1 + x/1.21 = 2100 → x=1210. B=1050 half (interest 0). Partnership C×T nahi.

Answer1210
P8

1000, 8%, 18 months, sirf PRT/100. Is chapter?

  1. Nahi — Simple interest
  2. Haan, A=1102.50
  3. Half-yearly 2 yr
  4. CI=120

Solution

  1. T=3/2 seedha

    A. SI=120 wahan.

AnswerA
P9

CP 200, SP 250. Is chapter?

  1. Nahi — Profit / loss
  2. Haan, CI 25%
  3. R=25% n=1
  4. Amount 250

Solution

  1. Shop

    A.

AnswerA
P10

Profit share capital × time. Is chapter?

  1. Nahi — Partnership
  2. Haan, installment
  3. CI split
  4. Ratio 8:12:15

Solution

  1. Share

    A. Installment yahan loan-PV; partner-time wahan.

AnswerA

← Index · Simple interest · Profit / loss

Agle topic: Partnership.