Calculate inverse sums
Scenario
Aap pizza ke alag-alag fractions add kar rahe ho, jaise `1/3` aur `1/6`.
Lekin jab denominator alag ho, toh fractions add kaise karte hain?
Mental model
Fractions add karne ke liye sabka base ek jaisa banana padta hai.
Inverse sums mein (jaise `1/R = 1/R1 + 1/R2`), hum fractions ko directly nahi jod sakte. Pehle LCM (Least Common Multiple) le kar common denominator banate hain, fractions add karte hain, aur aakhir mein result ka ulta (reciprocal) karte hain.
Explanation
Physics mein parallel circuits solve karte waqt, `1/R = 1/R1 + 1/R2` jaisi equations banti hain. Isko solve karne ke liye fraction addition ke mathematical rules zaroori hain.
Sabse pehle, denominators `R1` aur `R2` ka LCM nikalte hain. LCM woh sabse chhota number hai jo dono denominators ki table mein aata hai. Jab common base mil jaye, tab hum numerators ko us hisaab se adjust karte hain aur add kar dete hain.
Ye step poora hone par humein `1/R` ki value milti hai. Lekin humein total resistance `R` chahiye hota hai. Isliye final fraction ko ulta (reciprocal) karna sabse critical step hai.
Bina reciprocal kiye answer chhod dena sabse common galti hoti hai. Hamesha dhyan rakho, formula `1/R` deta hai, aapko ise palat kar `R` nikalna hai.
Worked example
Problem: Calculate `R` if `1/R = 1/3 + 1/6`.
- Pehle `3` aur `6` ka LCM find karte hain.
- Fractions ko common base par convert karte hain.
- Dono ko add karte hain.
- `R` nikalne ke liye fraction ko flip karte hain.
Answer: 2
Problem: Find `x` if `1/x = 1/10 + 1/40`.
- Denominators `10` aur `40` ka LCM lete hain.
- Base same banakar fractions add karte hain.
- Fraction simplify karte hain.
- Final answer ke liye flip karte hain.
Answer: 8
Key points
- Common Denominator: Use LCM to create a common base before adding fractions.
- Final Reciprocal: Always flip the resulting fraction to find `R` instead of `1/R`.
Common mistakes
- Forgetting the final flip: `1/R = 1/2` nikalte hain aur `0.5` likh kar answer chhod dete hain. Isko flip karna zaruri hai taaki `R = 2` aa jaye.
- Adding numerators and denominators directly: Fractions add karte time `1/3 + 1/6` ko `(1+1)/(3+6) = 2/9` likh dete hain jo mathematically bilkul galat hai.
Recall questions
- What is the LCM of 2 and 4?
- Why do we need inverse sums in physics?
- What is the sum of 1/2 and 1/2?
Questions & answers
Evaluate 1/R = 1/20 + 1/20
R = 10
Approach: Base is already common. 1/R = 2/20 = 1/10. Flip to get R = 10.
Solve for f: 1/f = 1/3 + 1/6
f = 2
Approach: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2. Therefore, f = 2.