Ch 8: Simple & Compound Interest | Maths

Dosto, Interest (byaaj) matlab paise ka kiraya. Aap bank mein paisa rakhte ho ya kisi se udhaar lete ho — extra jo dena/milna hota hai wahi byaaj hai.

Do tarike hote hain: Simple Interest (SI) — byaaj har saal sirf asli paise (principal) par lagta hai. Compound Interest (CI) — byaaj par bhi byaaj lagta hai, isliye paisa tezi se badhta hai.

CET Group D mein 1–2 questions pakke hain. Aur khushi ki baat — CI toh Chapter 4 ka successive percentage hi hai naye naam se! 2 saal ka CI = a + b + ab/100 wala hi formula. Chalo shuru! 💪

1. Zaroori Shabd

Short	Naam	Matlab
P	Principal (Mooldhan)	Asli paisa jo lagaya/udhaar liya
R	Rate (Dar)	% per year (saalana)
T / n	Time (Samay)	Kitne saal
SI / CI	Interest (Byaaj)	Extra paisa
A	Amount (Mishradhan)	P + Interest (total wapas)

2. Simple Interest (Sadharan Byaaj) — ⭐

⭐ SI = (P × R × T) / 100 · A = P + SI Isi ek formula ko ghuma ke: P = SI×100/(R×T) · R = SI×100/(P×T) · T = SI×100/(P×R)
Yaani jo cheez nahi pata, use ek taraf kar do — baaki sab teen cheezein pata hoti hain.

✏️ Solved Example 1 — P ₹5000, R 8%, T 3 saal Step 1: SI = (5000 × 8 × 3)/100
Step 2: = 120000/100 = ₹1200 ✓ (Amount = 5000 + 1200 = ₹6200)

✏️ Solved Example 2 — ₹800, 3 saal mein ₹920 ban gaya. Rate? Step 1: SI = 920 − 800 = ₹120
Step 2: R = (120 × 100)/(800 × 3) = 12000/2400 = 5% ✓

📌 SI ki khaas baat SI mein har saal ka byaaj barabar hota hai. Isliye 3 saal ka SI = 1 saal ka byaaj × 3. Isi wajah se SI ke sawal ekdum seedhe hote hain.

3. Compound Interest (Chakravriddhi Byaaj) — ⭐

⭐ A = P (1 + R/100)ⁿ · CI = A − P Yahan har saal ka byaaj principal mein jud jata hai, aur agle saal us badi rakam par byaaj lagta hai.

✏️ Solved Example — P ₹5000, R 10%, 2 saal ka CI Step 1: A = 5000 × (1.1)² = 5000 × 1.21 = ₹6050
Step 2: CI = 6050 − 5000 = ₹1050 ✓
Saal-dar-saal soch: 5000 → +500 = 5500 → +550 = 6050. Dusre saal ka byaaj bada (550), kyunki 500 par bhi byaaj laga!

⚡ CI ka Chapter-4 wala shortcut (2 saal) 2 saal ka effective CI rate = R + R + R²/100 (successive change!)
10% par → 10 + 10 + 1 = 21% → 5000 ka 21% = ₹1050 ✓ (ek line mein answer!)
20% par → 20 + 20 + 4 = 44% · 5% par → 5 + 5 + 0.25 = 10.25%

4. Half-yearly aur Quarterly CI

Kaise lagta hai	Rate	Time
Yearly (saalana)	R	n
Half-yearly (chhamahi)	R/2	2n
Quarterly (timahi)	R/4	4n

✏️ Solved Example — ₹8000, 10% saalana, half-yearly, 1 saal Step 1: Rate = 10/2 = 5%, time = 1 × 2 = 2 baar
Step 2: A = 8000 × (1.05)² = 8000 × 1.1025 = ₹8820
Step 3: CI = 8820 − 8000 = ₹820 ✓ (yearly hota to sirf ₹800 — half-yearly zyada deta hai!)

5. CI aur SI ka Antar (Difference) — ⭐ Exam favourite

⭐ 2 saal ka antar = P × (R/100)² 3 saal ka antar = P × (R/100)² × (3 + R/100)
Solved Example: ₹8000 par 5% se 2 saal mein CI aur SI ka antar?
Step 1: 8000 × (5/100)² = 8000 × 0.0025 = ₹20 ✓
Ulta sawal: "Antar ₹50 hai, rate 10%, 2 saal → P?" → P × 0.01 = 50 → P = ₹5000

📌 Kyun hota hai antar? Kyunki CI mein dusre saal byaaj par bhi byaaj lagta hai. Isliye CI hamesha SI se zyada (ya barabar) hota hai. 1 saal mein dono barabar hote hain (yearly compounding par)!

6. Paisa Double/Triple hone wale sawal (SI)

⭐ SI mein: double hone ka matlab SI = P, triple ka matlab SI = 2P Solved Example 1: Koi rakam 8 saal mein double ho jati hai. Rate?
→ SI = P → R = (P × 100)/(P × 8) = 12.5% ✓
Solved Example 2: Rakam 5 saal mein double hoti hai, to triple kab hogi?
→ R = 100/5 = 20% → triple ke liye SI = 2P → T = (2P × 100)/(P × 20) = 10 saal ✓

✏️ Type 1 — Seedha SI (P, R, T mein se koi ek gayab) "3 saal mein 6% se SI ₹720 mila. Rakam?"
Step 1: P = (SI × 100)/(R × T) = (720 × 100)/(6 × 3)
Step 2: = 72000/18 = ₹4000 ✓

✏️ Type 2 — Amount diya hai, rate ya time poocha "₹1200, 3 saal mein ₹1560 ho gaya. Rate?"
Step 1: SI = 1560 − 1200 = 360 (pehle SI nikalo, amount se seedha formula mat lagao!)
Step 2: R = (360 × 100)/(1200 × 3) = 10% ✓

✏️ Type 3 — CI se principal wapas nikalna "10% CI se 2 saal baad rakam ₹4840 ho gayi. Mooldhan?"
Step 1: P × (1.1)² = 4840 → P × 1.21 = 4840
Step 2: P = 4840/1.21 = ₹4000 ✓ (shortcut: 21% badha hai → P = 4840 × 100/121)

✏️ Type 4 — CI − SI ka antar Antar diya ho to P × (R/100)² se seedha nikaalo. Ye 2-saal wala antar asal mein "pehle saal ke byaaj par laga byaaj" hi hai.
Yaad rakho: 2 saal, 10% → antar = P/100. 2 saal, 5% → antar = P/400.

✏️ Type 5 — Double/Triple (SI vs CI) SI mein: 5 saal mein double → 10 saal mein triple (byaaj barabar-barabar badhta hai).
CI mein: 5 saal mein double → 10 saal mein 4 guna (double ka double!), 15 saal mein 8 guna. Ye CET ka pasandeeda trap hai — SI aur CI ka rule alag hai!

Trick 1: 2 saal ka CI = successive percentage
Effective rate = 2R + R²/100
10% → 21%, 20% → 44%, 5% → 10.25%, 15% → 32.25%. In char ko ratt lo — CI ke aadhe sawal ek line mein khatam!

Trick 2: SI mein saal-dar-saal barabar
1 saal ka byaaj × saal
"P par 4 saal mein SI 2400" → 1 saal ka 600. Ab kisi bhi saal ka poocho, seedha guna kar do. Formula bhi lagane ki zaroorat nahi.

Trick 3: CI ko step-by-step socho (chhote numbers)
Har saal: pichhla + uska R%
5000 @10%: 5000 → 5500 → 6050. Power lagane se aasan, aur galti ka chance kam. 3 saal tak ye tarika sabse fast hai.

Trick 4: Double/triple ka rule
SI: T = (n−1) × 100/R
Double (n=2) → T = 100/R. Triple (n=3) → T = 200/R. 20% par: double 5 saal, triple 10 saal. Seedha yaad rakho.

Trick 5: CI−SI antar ka fast use
2 saal: P × (R/100)² = "byaaj par byaaj"
10% par antar = P/100, 5% par P/400, 20% par P/25. Antar diya ho to P turant nikal jata hai.

❌ Galti 1: Amount ko Interest samajh lena "₹800, 3 saal mein ₹920 ho gaya" — yahan byaaj ₹120 hai, 920 nahi. Pehle SI = A − P karo, phir formula lagao.

❌ Galti 2: Half-yearly mein sirf time double karna Rate bhi aadha karna hota hai. 10% saalana, half-yearly = 5% × 2 baar. Sirf ek badla to answer galat.

❌ Galti 3: CI ko SI × 2 samajhna 2 saal ka CI @10% = 21% hai, 20% nahi. Wo extra 1% hi "byaaj par byaaj" hai — aur wahi CI−SI ka antar banta hai.

❌ Galti 4: CI mein double-triple ka SI wala rule lagana CI mein 5 saal mein double → 10 saal mein 4 guna (3 guna NAHI). SI mein hi 10 saal par triple hota hai. Dono alag rakho.

❌ Galti 5: Time ko mahine/months mein rakh dena Rate saalana hai to time bhi saal mein chahiye. 6 mahine = 1/2 saal, 9 mahine = 3/4 saal, 8 mahine = 2/3 saal.

SI ka formula?tap 👆

P × R × T / 100

Amount (A) kya hota hai?tap 👆

P + Interest

SI se Rate kaise nikale?tap 👆

R = SI × 100 / (P × T)

CI ka Amount formula?tap 👆

A = P(1 + R/100)ⁿ

CI = ?tap 👆

A − P

2 saal ka effective CI rate?tap 👆

2R + R²/100 (10% → 21%)

20% par 2 saal ka CI rate?tap 👆

44%

Half-yearly mein R aur T?tap 👆

R/2 aur 2n

Quarterly mein R aur T?tap 👆

R/4 aur 4n

2 saal ka CI−SI antar?tap 👆

P (R/100)²

3 saal ka CI−SI antar?tap 👆

P (R/100)² (3 + R/100)

1 saal mein SI aur CI?tap 👆

Barabar (yearly compounding par)

SI mein double hone ka time?tap 👆

T = 100/R (triple = 200/R)

SI: 8 saal mein double → rate?tap 👆

100/8 = 12.5%

CI: 5 saal mein double → 10 saal mein?tap 👆

4 guna (SI hota to 3 guna)

6 mahine = kitna saal?tap 👆

1/2 saal (9 mahine = 3/4)

CI ya SI — kaun bada?tap 👆

CI (1 saal par barabar)

CI @10%, 2 saal, A = 4840 → P?tap 👆

4840 × 100/121 = ₹4000

Cheez	Formula / Value
SI	PRT/100
P · R · T (SI se)	SI×100/RT · SI×100/PT · SI×100/PR
CI Amount	P(1 + R/100)ⁿ
2 saal effective CI rate	2R + R²/100
5% · 10% · 20% (2 saal CI)	10.25% · 21% · 44%
Half-yearly · Quarterly	R/2, 2n · R/4, 4n
CI−SI (2 saal)	P(R/100)²
CI−SI (3 saal)	P(R/100)²(3+R/100)
1 saal mein SI vs CI	Barabar
SI double / triple	T = 100/R / 200/R
CI double in n saal → 2n saal	4 guna
6 / 8 / 9 mahine	1/2 · 2/3 · 3/4 saal

🎯 Score: 0 / 0 Total questions: 14

Practice (exam-style)

Q1. The simple interest on ₹5000 at 8% per annum for 3 years is:

💡 PRT/100 = 5000×8×3/100 = 1200.

Practice (exam-style)

Q2. What amount will ₹2000 become in 4 years at 5% per annum simple interest?

💡 SI = 400 → Amount = 2000 + 400 = 2400.

Practice (exam-style)

Q3. A sum gives ₹720 as simple interest in 3 years at 6% per annum. The sum is:

💡 P = SI×100/(R×T) = 72000/18 = 4000.

Practice (exam-style)

Q4. In what time will ₹800 amount to ₹920 at 5% per annum simple interest?

💡 SI = 120 → T = 120×100/(800×5) = 3 saal.

Practice (exam-style)

Q5. ₹1200 amounts to ₹1560 in 3 years at simple interest. The rate of interest is:

💡 SI = 360 → R = 360×100/(1200×3) = 10%.

Practice (exam-style)

Q6. The compound interest on ₹5000 at 10% per annum for 2 years is:

💡 5000 → 5500 → 6050 → CI = 1050 (ya 21% of 5000).

Practice (exam-style)

Q7. The difference between compound interest and simple interest on ₹8000 at 5% per annum for 2 years is:

💡 P(R/100)² = 8000 × (0.05)² = 20.

Practice (exam-style)

Q8. A sum of money doubles itself in 8 years at simple interest. The rate of interest is:

💡 Double → T = 100/R → R = 100/8 = 12.5%.

Practice (exam-style)

Q9. At 10% per annum compound interest, a sum amounts to ₹4840 in 2 years. The sum is:

💡 P × 1.21 = 4840 → P = 4840 × 100/121 = 4000.

Practice (exam-style)

Q10. Find the compound interest on ₹8000 for 1 year at 10% per annum, compounded half-yearly.

💡 5% × 2 baar: 8000 → 8400 → 8820 → CI = 820.

Practice (exam-style)

Q11. A sum doubles itself in 5 years at simple interest. In how many years will it become three times?

💡 R = 20% → triple ke liye SI = 2P → T = 200/20 = 10 saal.

Practice (exam-style)

Q12. The difference between CI and SI on a sum for 2 years at 10% per annum is ₹50. The sum is:

💡 P(10/100)² = 50 → P/100 = 50 → P = 5000.

Practice (exam-style)

Q13. What sum will give ₹2400 as simple interest in 4 years at 10% per annum?

💡 P = 2400×100/(10×4) = 6000.

Practice (exam-style)

Q14. The compound interest on ₹10,000 at 20% per annum for 2 years is:

💡 2 saal ka effective rate = 44% → 10000 ka 44% = 4400.

SI = PRT/100 — har saal barabar byaaj, isliye seedha. Amount diya ho to pehle SI = A − P nikalo, tab formula. CI: A = P(1+R/100)ⁿ, aur 2 saal ka shortcut 2R + R²/100 (10% → 21%, 20% → 44%) — ye Chapter 4 wala successive formula hi hai. Half-yearly mein rate aadha, time double (quarterly mein R/4, 4n). CI − SI (2 saal) = P(R/100)² — yahi "byaaj par byaaj" hai. Double-triple mein SI ka rule T = 100/R aur 200/R, par CI mein 5 saal me double → 10 saal me 4 guna (3 guna nahi!). 🚀