Dosto, ye chapter fractions (bhinn), decimals (dashamlav) aur surds (root wale numbers) ko cover karta hai โ teeno ek hi cheez ke alag-alag roop hain!
CET Group D mein isse 1โ2 direct questions aate hain (compare karo, simplify karo, root nikalo). Par asli fayda ye hai ki Percentage, Profit-Loss, Time & Work โ har aage wale chapter mein fractions ka istemal hota hai. Yahan strong ho gaye to aage sab aasan.
Sabse zyada kaam ki cheez is chapter mein: fraction compare karne ke shortcuts aur fraction โ decimal table. Ye ratt gaye to calculation ki speed double! ๐ช
1. Fraction ke Types
| Type | Pehchan | Example |
|---|---|---|
| Proper (Uchit) | Upar wala CHHOTA | 3/5, 7/9 |
| Improper (Vishuddh) | Upar wala BADA ya barabar | 7/4, 9/9 |
| Mixed (Mishrit) | Poora number + fraction | 2ยพ = 11/4 |
| Like (Saman) | Denominator SAME | 2/7, 5/7 |
| Unlike (Asaman) | Denominator alag | 2/3, 5/7 |
| Equivalent (Tulya) | Value same, roop alag | 1/2 = 2/4 = 50/100 |
๐ Mixed โ Improper (2 second mein)
2ยพ โ (2 ร 4) + 3 = 11 โ 11/4. Formula: (poora ร niche) + upar, niche wahi rehta hai.
2. Fractions ke Operations (+ โ ร รท)
- Jodna/Ghatana: denominators ka LCM lo (Chapter 2 wala kaam!), phir numerators jodo.
- Multiply: seedha upar ร upar, niche ร niche. LCM ki zaroorat NAHI.
- Divide: doosre fraction ko ulta karo (reciprocal) aur multiply kar do.
- 'Of' ka matlab multiply: "3/4 of 96" = 3/4 ร 96 = 72.
โ๏ธ Solved Example โ 1/2 + 1/3 + 1/6
Step 1: LCM(2, 3, 6) = 6
Step 2: 3/6 + 2/6 + 1/6 = 6/6
Step 3: = 1 โ
Step 2: 3/6 + 2/6 + 1/6 = 6/6
Step 3: = 1 โ
โ๏ธ Solved Example โ 7/8 รท 7/16
Step 1: Doosre ko ulta karo: 7/8 ร 16/7
Step 2: 7 upar-niche cancel, 16/8 = 2 โ 2 โ
Step 2: 7 upar-niche cancel, 16/8 = 2 โ 2 โ
3. Fractions COMPARE karna (exam ka favourite)
| Method | Kab use karo |
|---|---|
| Cross multiplication | Sirf 2 fractions hon: a/b vs c/d โ aรd vs cรb compare karo |
| Decimal banao | 3โ4 options hon (sabse safe tarika) |
| Same numerator trick | Upar same ho to chhota denominator = bada fraction (3/5 > 3/7) |
โ๏ธ Solved Example โ Sabse bada kaunsa: 3/4, 5/7, 7/9, 11/14?
Step 1: Sab ko decimal mein badlo: 3/4 = 0.75, 5/7 = 0.714, 7/9 = 0.778, 11/14 = 0.786
Step 2: Sabse bada = 11/14 โ (Cross-multiply se bhi hota, par 4 options mein decimal fast hai)
Step 2: Sabse bada = 11/14 โ (Cross-multiply se bhi hota, par 4 options mein decimal fast hai)
4. Fraction โ Decimal Table (RATT LO!)
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| 1/2 | 0.5 | 1/8 | 0.125 |
| 1/3 | 0.333โฆ | 3/8 | 0.375 |
| 1/4 | 0.25 | 5/8 | 0.625 |
| 1/5 | 0.2 | 7/8 | 0.875 |
| 1/6 | 0.1667 | 1/9 | 0.111โฆ |
| 1/7 | 0.1428 | 1/10 | 0.1 |
| 2/3 | 0.667 | 3/4 | 0.75 |
5. Recurring Decimal โ Fraction
| Decimal | Rule | Fraction |
|---|---|---|
| 0.666โฆ (0.6ฬ) | Repeat digit / 9 | 6/9 = 2/3 |
| 0.333โฆ | 3/9 | 1/3 |
| 0.454545โฆ | 2 digit repeat โ /99 | 45/99 = 5/11 |
| 0.375 (terminating) | 375/1000 simplify | 3/8 |
๐ Terminating ya non-terminating?
Fraction ko simplest form mein lao. Agar denominator mein sirf 2 aur 5 ke factors hain to decimal khatam ho jayega (terminating), warna repeat hoga. Jaise 3/8 (8 = 2ยณ) โ 0.375 โ, par 1/6 (6 = 2ร3) โ 0.1666โฆ
6. Surds (Root wale numbers) โ Basics
- Surd = wo root jiska exact answer poora number nahi aata: โ2, โ3, โ5 โ ye irrational hain. (โ4 = 2 surd NAHI hai.)
- Yaad rakho: โ2 = 1.414 ยท โ3 = 1.732 ยท โ5 = 2.236 ยท โ7 = 2.646
| Rule | Example |
|---|---|
| โa ร โb = โ(ab) | โ2 ร โ8 = โ16 = 4 |
| โa รท โb = โ(a/b) | โ50 รท โ2 = โ25 = 5 |
| โ(a/b) = โa / โb | โ(144/9) = 12/3 = 4 |
| (โa + โb)(โa โ โb) = a โ b | (โ5+โ3)(โ5โโ3) = 5โ3 = 2 |
| โa ร โa = a | โ7 ร โ7 = 7 |
โ๏ธ Solved Example โ โ72 simplify karo
Step 1: 72 ko aise todo ki ek perfect square nikle: 72 = 36 ร 2
Step 2: โ72 = โ36 ร โ2 = 6โ2 โ (aise hi โ50 = โ25รโ2 = 5โ2)
Step 2: โ72 = โ36 ร โ2 = 6โ2 โ (aise hi โ50 = โ25รโ2 = 5โ2)
โ๏ธ Solved Example โ Rationalisation: 1/โ3
Step 1: Niche wala root hatana hai โ upar-niche dono ko โ3 se multiply karo
Step 2: (1 ร โ3) / (โ3 ร โ3) = โ3/3 โ (โ3 ร โ3 = 3, root gayab!)
Step 2: (1 ร โ3) / (โ3 ร โ3) = โ3/3 โ (โ3 ร โ3 = 3, root gayab!)
7. Squares & Roots yaad rakho (calculation half ho jati hai)
| n | nยฒ | n | nยฒ | n | nยณ |
|---|---|---|---|---|---|
| 11 | 121 | 16 | 256 | 2 | 8 |
| 12 | 144 | 17 | 289 | 3 | 27 |
| 13 | 169 | 18 | 324 | 4 | 64 |
| 14 | 196 | 19 | 361 | 5 | 125 |
| 15 | 225 | 25 | 625 | 6 | 216 |
Trick 1: Cross-multiplication se 2 fractions compare
a/b vs c/d โ aรd vs cรb
5/7 vs 7/9 โ 5ร9 = 45, 7ร7 = 49. 49 bada hai to 7/9 bada. Bas tirchha (cross) multiply karo โ 5 second!
a/b vs c/d โ aรd vs cรb
5/7 vs 7/9 โ 5ร9 = 45, 7ร7 = 49. 49 bada hai to 7/9 bada. Bas tirchha (cross) multiply karo โ 5 second!
Trick 2: Same numerator = ULTA rule
Upar same โ chhota niche = bada fraction
3/5, 3/7, 3/11 mein sabse bada = 3/5. Socho: ek roti 5 logon mein bantegi to hissa bada, 11 mein chhota!
Upar same โ chhota niche = bada fraction
3/5, 3/7, 3/11 mein sabse bada = 3/5. Socho: ek roti 5 logon mein bantegi to hissa bada, 11 mein chhota!
Trick 3: Root simplify โ perfect square dhundo
โ72 = โ(36ร2) = 6โ2
Number ko "perfect square ร baaki" mein todo. Perfect squares yaad rakho: 4, 9, 16, 25, 36, 49, 64, 81, 100.
โ72 = โ(36ร2) = 6โ2
Number ko "perfect square ร baaki" mein todo. Perfect squares yaad rakho: 4, 9, 16, 25, 36, 49, 64, 81, 100.
Trick 4: Denominator se root hatao (rationalise)
1/โa โ โa/a
Upar-niche dono ko usi root se multiply karo. 1/โ5 = โ5/5. Agar niche (โa + โb) ho to sign badal ke (โa โ โb) se multiply karo.
1/โa โ โa/a
Upar-niche dono ko usi root se multiply karo. 1/โ5 = โ5/5. Agar niche (โa + โb) ho to sign badal ke (โa โ โb) se multiply karo.
Trick 5: Decimal multiplication mein point ginti
Points jodo, aakhir mein lagao
0.25 ร 0.4 โ 25 ร 4 = 100, total decimal digits = 2+1 = 3 โ 0.100 = 0.1. Pehle poora multiply, baad mein point!
Points jodo, aakhir mein lagao
0.25 ร 0.4 โ 25 ร 4 = 100, total decimal digits = 2+1 = 3 โ 0.100 = 0.1. Pehle poora multiply, baad mein point!
โ Galti 1: Fractions jodte waqt denominator bhi jod dena
1/2 + 1/3 = 5/6 hai, 2/5 NAHI! Jodne mein pehle LCM se same denominator banao. (Multiply mein zaroor upar-upar, niche-niche hota hai โ dono rules mix mat karo.)
โ Galti 2: โ(a + b) ko โa + โb samajh lena
โ(9 + 16) = โ25 = 5, jabki โ9 + โ16 = 3 + 4 = 7. Root ke andar ka plus/minus tootta nahi โ sirf ร aur รท tootte hain!
โ Galti 3: Bade fraction ko "bade numbers wala" samajhna
11/14 (0.786) > 3/4 (0.75), par 5/7 (0.714) sabse chhota hai โ numbers bade hone se fraction bada nahi hota. Hamesha decimal ya cross-multiply se check karo.
โ Galti 4: Decimal mein point ki jagah galat
0.5 ร 0.4 = 0.2 hai (na ki 2.0 ya 0.02). Rule: dono numbers ke decimal digits gino (1+1 = 2) โ answer mein utne hi digits (20 โ 0.20 = 0.2).
โ Galti 5: Mixed fraction ko galat todna
2ยพ = 11/4 (2ร4 + 3), na ki 9/4 ya 23/4. Aur "of" wale sawal mein 3/4 of 96 = 72 โ 'of' ka matlab multiply hota hai, divide NAHI.
Fractions ko divide kaise karte hain?tap ๐
Doosre ka reciprocal (ulta) lekar multiply
2 fractions compare karne ka fast tarika?tap ๐
Cross multiply: a/b vs c/d โ aรd vs cรb
Upar same ho to bada fraction kaunsa?tap ๐
Jiska denominator chhota (3/5 > 3/7)
2ยพ ko improper banaotap ๐
(2ร4)+3 = 11/4
1/8 = ? (decimal)tap ๐
0.125 (3/8 = 0.375, 5/8 = 0.625)
1/6 aur 1/7 decimal mein?tap ๐
0.1667 aur 0.1428
0.666โฆ = ? fractiontap ๐
6/9 = 2/3 (repeat digit รท 9)
0.454545โฆ = ? fractiontap ๐
45/99 = 5/11 (2 digit repeat โ /99)
Decimal kab terminating hota hai?tap ๐
Denominator mein sirf 2 aur 5 ke factors hon
โa ร โb = ?tap ๐
โ(ab)
(โa + โb)(โa โ โb) = ?tap ๐
a โ b
โ72 simplify karotap ๐
โ(36ร2) = 6โ2
1/โ3 rationalise karotap ๐
โ3/3
โ2, โ3, โ5 ki value?tap ๐
1.414 ยท 1.732 ยท 2.236
โ(9 + 16) = ?tap ๐
5 (โ25) โ 7 NAHI! Plus tootta nahi
'Of' ka matlab?tap ๐
Multiply (3/4 of 96 = 72)
15ยฒ aur 25ยฒ?tap ๐
225 aur 625
โ0.25 = ?tap ๐
0.5 (kyunki 0.5 ร 0.5 = 0.25)
| Cheez | Rule / Value |
|---|---|
| Fractions jodna | Denominators ka LCM lo |
| Fractions multiply | Uparรupar, nicheรniche (LCM nahi) |
| Fractions divide | Reciprocal ร |
| 2 fractions compare | Cross multiply |
| Same numerator | Chhota denominator = bada |
| 1/2, 1/4, 1/8 | 0.5, 0.25, 0.125 |
| 1/3, 1/6, 1/7 | 0.333, 0.1667, 0.1428 |
| 0.6ฬ (0.666โฆ) | 2/3 |
| โa ร โb | โ(ab) |
| โ(a+b) | โa + โb NAHI! |
| โ2 ยท โ3 ยท โ5 | 1.414 ยท 1.732 ยท 2.236 |
| 1/โa rationalise | โa / a |
Practice (exam-style)
Q1. Which of the following fractions is the largest?
๐ก Decimals: 0.75, 0.714, 0.778, 0.786 โ 11/14 sabse bada.
Practice (exam-style)
Q2. Simplify: 0.25 + 0.5 ร 0.4 = ?
๐ก BODMAS: pehle 0.5ร0.4 = 0.2, phir 0.25+0.2 = 0.45.
Practice (exam-style)
Q3. 0.375 expressed as a fraction in lowest terms is:
๐ก 375/1000 โ 3/8 (yaad rakho: 3/8 = 0.375).
Practice (exam-style)
Q4. The recurring decimal 0.666โฆ is equal to:
๐ก Repeat digit รท 9 = 6/9 = 2/3.
Practice (exam-style)
Q5. The value of โ0.25 is:
๐ก 0.5 ร 0.5 = 0.25, isliye โ0.25 = 0.5.
Practice (exam-style)
Q6. โ72 in simplest form is:
๐ก 72 = 36ร2 โ โ36 ร โ2 = 6โ2 (baaki options simplest form nahi).
Practice (exam-style)
Q7. On rationalising the denominator, 1/โ3 becomes:
๐ก Upar-niche โ3 se multiply: (1รโ3)/(โ3รโ3) = โ3/3.
Practice (exam-style)
Q8. The value of (โ5 + โ3)(โ5 โ โ3) is:
๐ก (โa+โb)(โaโโb) = aโb = 5โ3 = 2.
Practice (exam-style)
Q9. Find 2/3 of 3/4 of 96.
๐ก 96 ร 3/4 = 72, phir 72 ร 2/3 = 48.
Practice (exam-style)
Q10. 5/8 written as a decimal is:
๐ก 5 รท 8 = 0.625 (7/8 hota to 0.875).
Practice (exam-style)
Q11. Which fraction lies between 1/3 and 1/2?
๐ก 1/3 = 0.333, 1/2 = 0.5; 5/12 = 0.4167 beech mein hai.
Practice (exam-style)
Q12. Simplify: 7/8 รท 7/16 = ?
๐ก 7/8 ร 16/7 = 16/8 = 2 (divide = reciprocal se multiply).
Practice (exam-style)
Q13. The value of โ(144/9) is:
๐ก โ144/โ9 = 12/3 = 4.
Practice (exam-style)
Q14. 1/2 + 1/3 + 1/6 = ?
๐ก LCM 6 โ 3/6 + 2/6 + 1/6 = 6/6 = 1.
Fractions jodne mein LCM, multiply mein seedha upar-niche, divide mein reciprocal. Compare karne ke 2 hathiyar: cross-multiply (2 fractions) aur decimal (4 options). Fractionโdecimal table (1/8 = 0.125, 1/6 = 0.1667, 1/7 = 0.1428) ratt lo โ Percentage wale chapter mein yahi kaam aayega. Surds mein โaรโb = โ(ab) aur (โa+โb)(โaโโb) = aโb; root simplify karne ke liye andar se perfect square nikalo (โ72 = 6โ2), aur denominator ka root rationalise karke hatao. Sabse badi galti se bacho: โ(a+b) โ โa + โb. ๐