FRACTIONS

Jab ek poore unit ko equal parts mein divide karte hain, toh har part ko fractional unit ya unit fraction kehte hain.

• Fractional Unit: Ek unit ka ek part jab us unit ko equal parts mein divide kiya gaya ho. Jaise, agar ek roti ko 2 equal parts mein divide kiya, toh har part \(\frac{1}{2}\) hai, jo ek fractional unit hai.
• Example:
• Ek apple ko 4 equal parts mein kaata, toh har part \(\frac{1}{4}\) hai.
• Ek pizza ko 8 equal slices mein kaata, toh har slice \(\frac{1}{8}\) hai.

Key Idea: Fractions tab aate hain jab hum kisi cheez ko barabar hisson mein baantte hain. Har hissa ek fraction hota hai.

• Numerator: Fraction ka upar wala number. Yeh batata hai ki kitne parts liye gaye hain.
• Denominator: Fraction ka neeche wala number. Yeh batata hai ki poore unit ko kitne equal parts mein divide kiya gaya hai.

Example: \(\frac{3}{5}\) mein:

• 3 hai Numerator (kitne parts liye)
• 5 hai Denominator (total equal parts)

Important: Denominator kabhi bhi zero nahi ho sakta, kyunki hum kisi cheez ko zero equal parts mein divide nahi kar sakte.

Fractions sirf ek unit ke parts hi nahi hote, balki woh ek collection ke parts bhi ho sakte hain.

• Example: Ek box mein 10 chocolates hain. Agar aapne 3 chocolates kha li, toh aapne \(\frac{3}{10}\) chocolates kha li. Yahan, 10 chocolates ka collection ek 'whole' hai.

Visualizing Fractions:

• Shape-based: Ek circle ko 4 equal parts mein divide kiya, 1 part shade kiya, toh woh \(\frac{1}{4}\) hai.
• Collection-based: 5 balls mein se 2 red hain, toh red balls ka fraction \(\frac{2}{5}\) hai.

Different ways to divide a whole:

• Ek chikki ko 6 equal parts mein kaatne ke kayi tareeke ho sakte hain (vertically, horizontally, diagonally), lekin har piece ka value \(\frac{1}{6}\) hi rahega. Shape matter nahi karta, equal division matter karta hai.

Types of Fractions (Basic):

• Proper Fraction: Numerator denominator se chhota hota hai. Value 1 se kam hoti hai. Example: \(\frac{1}{2}, \frac{3}{4}, \frac{5}{8}\)
• Improper Fraction: Numerator denominator se bada ya uske equal hota hai. Value 1 ya 1 se zyada hoti hai. Example: \(\frac{5}{3}, \frac{7}{4}, \frac{6}{6}\)
• Unit Fraction: Numerator 1 hota hai. Example: \(\frac{1}{2}, \frac{1}{5}, \frac{1}{10}\)

Fractions ko hum lengths, weights, capacities, aur time jaise quantities ko measure karne ke liye use karte hain.

• Example (Length): Agar ek paper strip 1 unit long hai. Agar hum usko 3 equal parts mein fold karein, toh har part \(\frac{1}{3}\) unit long hoga.

• Addition of Fractional Units:
• \(\frac{1}{4} + \frac{1}{4} = \frac{2}{4}\)
• \(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3}{5}\)
• Jab denominators same hote hain, toh hum sirf numerators ko add karte hain.

• Repeated Addition as Multiplication:
• \(5 \text{ times } \frac{1}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{5}{4}\)
• Iska matlab \(5 \times \frac{1}{4} = \frac{5}{4}\) bhi hota hai.

Practical Application:

• Recipe mein ingredients ki quantity (\(\frac{1}{2}\) cup sugar).
• Kapde ki length (\(\frac{3}{4}\) meter cloth).
• Time duration (\(\frac{1}{4}\) hour = 15 minutes).

Fractions ko number line par represent karna bahut important skill hai. Har fraction ka number line par ek unique point hota hai.

Steps to mark fractions on a number line:

1. Ek number line draw karo.
2. Whole numbers (0, 1, 2, 3...) ko mark karo.
3. Jis fraction ko mark karna hai, uske denominator ko dekho.
4. Har whole number ke beech ke space ko (jaise 0 aur 1 ke beech) denominator ke equal parts mein divide karo.
5. Numerator ke according point mark karo.

• Example: \(\frac{1}{2}\) ko mark karna hai.
• 0 aur 1 ke beech ke space ko 2 equal parts mein divide karo.
• Pehle part ke end par \(\frac{1}{2}\) mark karo.

• Example: \(\frac{3}{4}\) ko mark karna hai.
• 0 aur 1 ke beech ke space ko 4 equal parts mein divide karo.
• Teesre part ke end par \(\frac{3}{4}\) mark karo.

• Improper Fractions on Number Line: Improper fractions 1 ya 1 se bade hote hain. Jaise \(\frac{5}{3}\) ko mark karna hai.
• \(\frac{5}{3}\) ko mixed fraction mein convert karo: \(1 \frac{2}{3}\).
• Yeh fraction 1 aur 2 ke beech mein aayega.
• 1 aur 2 ke beech ke space ko 3 equal parts mein divide karo.
• Dusre part ke end par \(1 \frac{2}{3}\) ya \(\frac{5}{3}\) mark karo.

Important Observation:

• 0 aur 1 ke beech mein infinite fractions hote hain. Aap jitne chaho utne parts mein divide kar sakte ho.

Mixed Fraction ya Mixed Number mein ek whole number part aur ek proper fraction part hota hai.

• Example: \(2 \frac{1}{2}\) (do poore aur aadha).
• 2 hai whole part.
• \(\frac{1}{2}\) hai fractional part.

Conversion: Improper Fraction to Mixed Fraction

1. Numerator ko denominator se divide karo.
2. Quotient (भागफल) whole number part banega.
3. Remainder (शेषफल) naya numerator banega.
4. Denominator wahi rahega.

• Formula: \(\text{Mixed Fraction} = \text{Quotient} \frac{\text{Remainder}}{\text{Divisor}}\)
• Example: \(\frac{7}{3}\)
• 7 ko 3 se divide kiya: \(7 \div 3 = 2\) (Quotient) aur \(1\) (Remainder).
• Toh \(\frac{7}{3} = 2 \frac{1}{3}\).

Conversion: Mixed Fraction to Improper Fraction

1. Whole number part ko denominator se multiply karo.
2. Result mein numerator add karo.
3. Yeh naya numerator banega.
4. Denominator wahi rahega.

• Formula: \(\text{Improper Fraction} = \frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}\)
• Example: \(2 \frac{1}{3}\)
• \((2 \times 3) + 1 = 6 + 1 = 7\).
• Toh \(2 \frac{1}{3} = \frac{7}{3}\).

Why convert? Calculations mein improper fractions zyada convenient hote hain, jabki mixed fractions real-life situations ko describe karne mein better hote hain.

Equivalent Fractions woh fractions hote hain jo same value ya same portion ko represent karte hain, bhale hi unke numerators aur denominators alag-alag hon.

• Example: \(\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}\) sab equivalent fractions hain. Yeh sab aadhe ko represent karte hain.

How to find Equivalent Fractions:

• Multiplication Method: Numerator aur Denominator dono ko same non-zero number se multiply karo.
• Example: \(\frac{2}{3}\) ke equivalent fractions:
• \(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)
• \(\frac{2 \times 3}{3 \times 3} = \frac{6}{9}\)
• \(\frac{2 \times 5}{3 \times 5} = \frac{10}{15}\)

• Division Method: Numerator aur Denominator dono ko unke common factor se divide karo.
• Example: \(\frac{12}{18}\) ke equivalent fractions:
• \(\frac{12 \div 2}{18 \div 2} = \frac{6}{9}\)
• \(\frac{12 \div 3}{18 \div 3} = \frac{4}{6}\)
• \(\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\)

Lowest Terms (Simplest Form):

• Ek fraction lowest terms mein hota hai jab uske numerator aur denominator ka common factor sirf 1 ho. Matlab, unhe aur divide nahi kiya ja sakta.
• How to reduce to lowest terms: Numerator aur denominator ko unke HCF (Highest Common Factor) se divide karo.
• Example: \(\frac{12}{18}\)
• 12 aur 18 ka HCF 6 hai.
• \(\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\). \(\frac{2}{3}\) lowest terms mein hai.

Cross-Multiplication Test for Equivalence:

• Do fractions \(\frac{a}{b}\) aur \(\frac{c}{d}\) equivalent honge agar \(a \times d = b \times c\) ho.
• Example: \(\frac{2}{3}\) aur \(\frac{4}{6}\)
• \(2 \times 6 = 12\)
• \(3 \times 4 = 12\)
• Kyunki \(12 = 12\), toh fractions equivalent hain.

Fractions ko compare karne ka matlab hai yeh pata lagana ki kaunsa fraction bada hai, kaunsa chhota, ya kya woh equal hain.

Case 1: Same Denominators (Like Fractions)

• Agar denominators same hain, toh jis fraction ka numerator bada hota hai, woh fraction bada hota hai.
• Example: \(\frac{5}{7}\) aur \(\frac{3}{7}\). Yahan \(5 > 3\), toh \(\frac{5}{7} > \frac{3}{7}\).

Case 2: Same Numerators

• Agar numerators same hain, toh jis fraction ka denominator chhota hota hai, woh fraction bada hota hai.
• Example: \(\frac{3}{5}\) aur \(\frac{3}{8}\). Yahan \(5 < 8\), toh \(\frac{3}{5} > \frac{3}{8}\). (Socho, 3 roti ko 5 logon mein baantna vs 8 logon mein baantna).

Case 3: Different Numerators and Denominators (Unlike Fractions)

• Yeh sabse common case hai. Isme hum fractions ko equivalent fractions mein convert karte hain jinka denominator same ho (LCM method).

Steps:

1. Dono denominators ka LCM (Least Common Multiple) find karo.
2. Har fraction ko uske equivalent fraction mein convert karo jiska denominator LCM ke equal ho.
3. Ab, Case 1 ki tarah, numerators ko compare karo.

• Example: \(\frac{2}{3}\) aur \(\frac{3}{4}\)

1. 3 aur 4 ka LCM = 12.
2. \(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)
3. \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
4. Ab \(\frac{8}{12}\) aur \(\frac{9}{12}\) ko compare karo. Kyunki \(8 < 9\), toh \(\frac{8}{12} < \frac{9}{12}\).
5. Iska matlab \(\frac{2}{3} < \frac{3}{4}\).

Cross-Multiplication Method (Quick Check):

• \(\frac{a}{b}\) aur \(\frac{c}{d}\) ko compare karne ke liye, \(a \times d\) aur \(b \times c\) ko compare karo.
• Agar \(a \times d > b \times c\), toh \(\frac{a}{b} > \frac{c}{d}\).
• Agar \(a \times d < b \times c\), toh \(\frac{a}{b} < \frac{c}{d}\).
• Agar \(a \times d = b \times c\), toh \(\frac{a}{b} = \frac{c}{d}\).

Ascending/Descending Order:

• Jab kayi fractions ko order karna ho, toh sabko same denominator wale equivalent fractions mein convert karke numerators ko compare karte hain.

Fractions ko add ya subtract karne ke liye, unke denominators same hone chahiye. Agar same nahi hain, toh unhe equivalent fractions mein convert karna padta hai.

Case 1: Same Denominators (Like Fractions)

• Addition: Numerators ko add karo aur denominator wahi rakho.
• Formula: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
• Example: \(\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}\)

• Subtraction: Numerators ko subtract karo aur denominator wahi rakho.
• Formula: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
• Example: \(\frac{5}{9} - \frac{2}{9} = \frac{5-2}{9} = \frac{3}{9}\)

Case 2: Different Denominators (Unlike Fractions) - Brahmagupta's Method

• Steps:

1. Dono denominators ka LCM find karo.
2. Har fraction ko uske equivalent fraction mein convert karo jiska denominator LCM ke equal ho.
3. Ab, Case 1 ki tarah, numerators ko add ya subtract karo.
4. Final answer ko lowest terms mein reduce karo, agar possible ho.

• Example (Addition): \(\frac{1}{2} + \frac{1}{3}\)

1. 2 aur 3 ka LCM = 6.
2. \(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\)
3. \(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)
4. \(\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}\)

• Example (Subtraction): \(\frac{3}{4} - \frac{1}{6}\)

1. 4 aur 6 ka LCM = 12.
2. \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
3. \(\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)
4. \(\frac{9}{12} - \frac{2}{12} = \frac{9-2}{12} = \frac{7}{12}\)

Adding/Subtracting Mixed Fractions:

• Method 1 (Convert to Improper): Mixed fractions ko pehle improper fractions mein convert karo, phir add/subtract karo. Final answer ko wapas mixed fraction mein convert kar sakte ho.
• Method 2 (Separate Whole and Fractional Parts): Whole parts ko alag se add/subtract karo, aur fractional parts ko alag se add/subtract karo. Agar fractional part improper aaye, toh use mixed mein convert karke whole part mein add karo.

• Example (Mixed Addition): \(2 \frac{1}{2} + 1 \frac{1}{3}\)
• Method 1: \(\frac{5}{2} + \frac{4}{3} = \frac{15}{6} + \frac{8}{6} = \frac{23}{6} = 3 \frac{5}{6}\)
• Method 2: \((2+1) + (\frac{1}{2} + \frac{1}{3}) = 3 + (\frac{3}{6} + \frac{2}{6}) = 3 + \frac{5}{6} = 3 \frac{5}{6}\)

Word Problems: Fractions ke addition/subtraction par based word problems bahut common hain. Carefully read karo ki kya add karna hai aur kya subtract.