Pehle Yeh Seekho 2: Fractions, Factors, HCF-LCM, Negative Numbers aur Maths ke Symbols

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🧰 ToolboxIntegers · Rational Numbers · Squares · Probability⏱ har section 2–3 min
Pehle Yeh Seekho 2: Fractions, Factors, HCF-LCM, Negative Numbers aur Maths ke Symbols 🧰

🎬 Yeh Post Kis Liye?

Integers, Number Mastery, Rational Numbers aur Probability ke posts mein kuch chhoti skills pehle se aani chahiye: fraction, decimal, factor, HCF-LCM, negative numbers ka jod-ghatana, letters aur symbols. Agar inmein se kuch nahi aata, koi baat nahi! Yeh post ek toolbox hai. Jo aata hai skip karo, jo nahi aata woh yahan 2–3 minute mein seekho, phir wapas apne post pe jao.

Har section ke neeche “Khud try karo” aur uske answers hain. Un posts ke upar ek 🧰 “Atak gaye?” box hai jo seedha yahan ke sahi section pe laata hai.

IS POST MEIN
Fraction aur “/”Decimal, aur decimal tak bhaagMixed number (1½)Even aur OddFactor, multiple, remainderPrime number aur factor treePower aur square (2⁴, 3²)GCD / HCFLCMNumber line aur negativeNegative + positive jodnaChhote mein se bada ghatana× aur ÷ ke signLetters (x, a, b, p/q)Symbols ≠ < > ≤ | | ⊂ ∈ ℤ ° √Words: angle, plane, strike rate…

1️⃣ Fraction aur “/” ka matlab

3/4 padho “teen bata chaar” ya “4 mein se 3”. Ek roti ke 4 barabar tukde kiye aur 3 liye.

  • Upar wala number (3) = numerator: kitne hisse liye.
  • Neeche wala number (4) = denominator: poori cheez kitne barabar hisson mein bati.
  • “/” line ka matlab “÷” bhi hai: 3/4 = 3 ÷ 4.

Barabar (equivalent) fractions: upar aur neeche dono ko ek hi number se guna ya bhaag karo to value nahi badalti. 1/2 = 2/4 = 50/100. 8/12 → dono ÷ 4 → 2/3.

Simplify karna (sabse chhota roop): upar-neeche ko ek hi number se tab tak bhaag do jab tak koi common factor (1 ke alawa) na bache (common factor kya hai: section 5 dekho). 24/48 → ÷ 2 → 12/24 → ÷ 12 → 1/2. (Sabse jaldi raasta: HCF se ek baar mein bhaag do, section 8.)

Negative fraction: −3/4 ka matlab “3/4 ka ulta”, yaani number line par 0 se left (number line: section 10 dekho). −3/4 = (−3)/4 = 3/(−4), teeno same.

Fractions ka jod (same neeche): 2/7 + 3/7 = 5/7 (sirf upar jodo, neeche wahi). Alag neeche ho to pehle LCM se same banao (section 9).

Aur detail chahiye? Pehle Yeh Seekho (Book 2 Ch 1 toolbox) mein fraction ke 4 aur sections hain.

✍️ Khud try karo:
  1. 2/5 ka matlab kya hai?
  2. 18/24 ko simplify karo.
  3. 1/4 + 2/4 = ?
✅ Answers dekho (pehle khud try karo!)
  1. 5 barabar hisson mein se 2.
  2. Dono ÷ 6 → 3/4.
  3. 3/4 (neeche same, upar 1 + 2 = 3).

— 📌 charumam.com

2️⃣ Decimal, aur decimal answer tak bhaag

Decimal point (.) batata hai ki poore kahan khatam hue. Point ke right: pehla ghar dasvaan (1/10), doosra sauvaan (1/100), teesra hazaarvaan (1/1000).

0.5 = 5/10 = 1/2  |  0.25 = 25/100 = 1/4  |  0.75 = 75/100 = 3/4  |  2.617 = 2617/1000

Decimal → fraction: point ke baad jitne digit, neeche utne zero wala 1 lagao. 0.23: 2 digit → neeche 100 → 23/100. 1.5 = 15/10 = (÷ 5) 3/2. 2.75 = 275/100 = (÷ 25) 11/4.

Fraction → decimal: upar ko neeche se bhaag do. 3/4 = 3 ÷ 4: 3 mein 4 nahi jaata, to “3.0” socho: 30 ÷ 4 = 7, baaki 2 (baaki = bhaag ke baad bacha hua; section 5 dekho) → 20 ÷ 4 = 5. Answer 0.75.

Terminating decimal: jo decimal khatam ho jaaye (terminate = khatam hona), jaise 0.75, 0.5, 2.617.

Kabhi bhaag khatam nahi hota (non-terminating): 1/3 = 0.333… (“…” matlab 3 aage chalta hi rahega). 5/3 = 1.666… ≈ 1.67 (lagbhag, exact nahi). Yahan ek hi digit baar-baar repeat hota hai, isliye isse non-terminating repeating kehte hain. √2 = 1.41421356… na khatam hota, na repeat hota: non-terminating non-repeating.

Decimal × poora number (jaise 60 × 0.75)

Point bhool ke guna karo: 60 × 75 = 4500. 0.75 mein point ke baad 2 digit hain, to answer mein right se 2 digit pe point lagao: 45.00 = 45. Aur ek: 0.4 × 130 → 4 × 130 = 520 → 1 digit → 52.0 = 52.

Decimal answer tak bhaag (jaise 45 ÷ 60)

45 ÷ 60: 45 mein 60 nahi jaata → answer 0.something. 450 ÷ 60 = 7, baaki 30 → 300 ÷ 60 = 5. Answer 0.75. Check: 60 × 0.75 = 45 ✅
117 ÷ 130: 1170 ÷ 130 = 9, baaki 0 → 0.9. Check: 130 × 0.9 = 117 ✅

Decimal se bhaag (jaise 130 ÷ 1.3)

Dono numbers ko 10 se guna karo, jab tak bhaajak (÷ ke baad wala) poora number na ban jaaye. Jawab nahi badalta, kyunki dono barabar bade hue. 130 ÷ 1.3 = 1300 ÷ 13 = 100.

✍️ Khud try karo:
  1. 0.4 ko fraction mein likho.
  2. 1/4 ko decimal mein likho.
  3. 36 ÷ 48 = ?
  4. 24 ÷ 1.2 = ?
✅ Answers dekho (pehle khud try karo!)
  1. 4/10 = 2/5.
  2. 1 ÷ 4 = 0.25.
  3. 360 ÷ 48 = 7, baaki 24 → 240 ÷ 48 = 5 → 0.75.
  4. 240 ÷ 12 = 20.

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3️⃣ Mixed number: 1½ kya hai?

1½ (ya “1 1/2”) = 1 + 1/2: ek poori roti aur aadhi roti. Isse mixed number kehte hain.

Improper fraction = jiska upar wala number neeche wale se bada ya barabar ho, jaise 3/2, 7/3.

Mixed → improper: 1½ = 2/2 + 1/2 = 3/2.   2⅓ = 3/3 + 3/3 + 1/3 = 7/3. Shortcut: (poora × neeche + upar)/neeche → (2 × 3 + 1)/3 = 7/3.
Improper → mixed: 11/4: 11 ÷ 4 = 2, baaki 3 → 2¾.
✍️ Khud try karo:
  1. 2½ ko improper fraction banao.
  2. 7/2 ko mixed number banao.
✅ Answers dekho (pehle khud try karo!)
  1. (2 × 2 + 1)/2 = 5/2.
  2. 7 ÷ 2 = 3, baaki 1 → 3½.

— 📌 charumam.com

4️⃣ Even aur Odd

Even (sam) number: jo 2 se poora bat jaaye, koi nahi bache. Inka aakhri digit 0, 2, 4, 6 ya 8 hota hai: 4, 18, 30, 126.

Odd (visham) number: 2 se baanto to 1 bach jaaye. Aakhri digit 1, 3, 5, 7 ya 9: 7, 15, 99.

Pairs mein socho: 6 chappal = 3 jode, sab ke jode bane → even. 7 chappal = 3 jode + 1 akeli → odd. 0 even hai. Poori kahani: MN-07 Even aur Odd.

Negative signs ki ginti: “3 minus” = odd number of negatives, “2 minus” = even.

✍️ Khud try karo:
  1. 46 even hai ya odd?
  2. Die (1–6) ke odd numbers kaun se?
✅ Answers dekho (pehle khud try karo!)
  1. Even (aakhri digit 6).
  2. 1, 3, 5.

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5️⃣ Factor, multiple, divisor, remainder

Remainder (baaki): bhaag ke baad jo bachta hai. 14 ÷ 4 = 3, baaki 2 (kyunki 4 × 3 = 12, 14 − 12 = 2).

Factor / divisor: jo number kisi number ko poora-poora baant de (baaki 0). 12 ke factors: 1, 2, 3, 4, 6, 12. 5, 12 ka factor nahi (baaki 2 bachta hai).

Multiple: kisi number ki table. 4 ke multiples: 4, 8, 12, 16, 20, … Har multiple 4 se poora bat jaata hai.

Common factor: jo dono numbers ka factor ho. 12 aur 18 ke common factors: 1, 2, 3, 6.

✍️ Khud try karo:
  1. 20 ke saare factors likho.
  2. 6 ke pehle 5 multiples?
  3. 17 ÷ 5 mein remainder?
✅ Answers dekho (pehle khud try karo!)
  1. 1, 2, 4, 5, 10, 20.
  2. 6, 12, 18, 24, 30.
  3. 5 × 3 = 15, 17 − 15 = 2.

— 📌 charumam.com

6️⃣ Prime number aur factor tree

Prime number: jiske sirf do factors hon, 1 aur woh khud. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 … (2 akela even prime hai.)

Composite number: jiske 2 se zyada factors hon: 4, 6, 8, 9, 12 … 1 na prime hai na composite (uska ek hi factor hai).

Coprime: do numbers jinka 1 ke alawa koi common factor nahi, jaise 8 aur 15.

Prime factorisation (number ko prime tukdon mein todna)

48 → 2 × 24 → 2 × 2 × 12 → 2 × 2 × 2 × 6 → 2 × 2 × 2 × 2 × 3. Har kadam pe sabse chhota prime (2, phir 3, phir 5 …) se bhaag do, jab tak sab tukde prime na ho jaayein.
12 = 2 × 2 × 3  |  30 = 2 × 3 × 5  |  36 = 2 × 2 × 3 × 3  |  84 = 2 × 2 × 3 × 7
✍️ Khud try karo:
  1. 11 prime hai?
  2. 9 prime hai?
  3. 60 ka prime factorisation?
✅ Answers dekho (pehle khud try karo!)
  1. Haan, factors sirf 1 aur 11.
  2. Nahi, 9 = 3 × 3 (factors 1, 3, 9).
  3. 60 = 2 × 2 × 3 × 5.

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7️⃣ Power aur square (2⁴, 3², n²)

Chhota upar likha number (power) batata hai ki number ko kitni baar khud se guna karna hai.

2⁴ = 2 × 2 × 2 × 2 = 16  |  3² = 3 × 3 = 9  |  2² × 3 = 2 × 2 × 3 = 12  |  10³ = 1000

Square = power 2 = number × wahi number. 5² padho “5 ka square” = 25. Cube = power 3.

Galti mat karo: 3² ≠ 3 × 2. 3² = 3 × 3 = 9.

Negative ka square: (−3)² = (−3) × (−3) = +9 (minus × minus = plus kyun: section 13 dekho). Fraction ka square: (2/3)² = (2 × 2)/(3 × 3) = 4/9.

Ek naya coin = outcomes double: 1 coin → 2, 2 coins → 2 × 2 = 4 (2²), 3 coins → 8 (2³).

✍️ Khud try karo:
  1. 2³ = ?
  2. (−2)² = ?
  3. 2 × 2 × 3 × 3 ko power se likho.
✅ Answers dekho (pehle khud try karo!)
  1. 2 × 2 × 2 = 8.
  2. (−2) × (−2) = 4.
  3. 2² × 3² (= 36).

— 📌 charumam.com

8️⃣ GCD / HCF: sabse bada common factor

GCD (Greatest Common Divisor) aur HCF (Highest Common Factor) ek hi cheez hain: dono numbers ko poora baantne wala sabse bada number.

Tareeka 1: factors ki list

12: 1, 2, 3, 4, 6, 12  |  30: 1, 2, 3, 5, 6, 10, 15, 30 → common: 1, 2, 3, 6 → HCF = 6

Tareeka 2: prime tukde (bade numbers ke liye)

48 = 2 × 2 × 2 × 2 × 3  |  84 = 2 × 2 × 3 × 7 → jo tukde dono mein hain unhe chuno (2, 2, 3) aur guna karo: 2 × 2 × 3 = 12. To HCF(48, 84) = 12.

Kaam kya hai? Fraction simplify: 48/84 → dono ÷ 12 → 4/7, ek hi kadam mein.

Tareeka 3 (Euclid, shortcut): bade ko chhote se bhaag do, phir chhote ko baaki se, jab tak baaki 0 na ho. Aakhri bhaajak = HCF. 84 ÷ 48 = 1 baaki 36; 48 ÷ 36 = 1 baaki 12; 36 ÷ 12 = 3 baaki 0 → 12.

✍️ Khud try karo:
  1. HCF(18, 24)?
  2. HCF(36, 24) prime tukdon se?
  3. 36/24 simplify karo.
✅ Answers dekho (pehle khud try karo!)
  1. 18: 1,2,3,6,9,18; 24: 1,2,3,4,6,8,12,24 → 6.
  2. 36 = 2×2×3×3, 24 = 2×2×2×3 → common 2×2×3 = 12.
  3. Dono ÷ 12 → 3/2.

— 📌 charumam.com

9️⃣ LCM: sabse chhota common multiple

LCM (Least Common Multiple): sabse chhota number jo dono ki table mein aaye. Alag-alag denominator wale fractions ko jodne/compare karne ke liye chahiye.

Tareeka 1: tables likho

4: 4, 8, 12, 16 …  |  6: 6, 12, 18 … → pehla common = 12. LCM(4, 6) = 12.

Tareeka 2: prime tukde

4 = 2 × 2, 6 = 2 × 3 → har prime ko jitni zyada baar kisi ek mein aaya, utni baar lo: 2 do baar (4 mein), 3 ek baar → 2 × 2 × 3 = 12.

LCM se fractions same denominator mein

1/4 + 1/6: LCM = 12. 1/4 = (1 × 3)/(4 × 3) = 3/12; 1/6 = (1 × 2)/(6 × 2) = 2/12. Jod = 5/12.
✍️ Khud try karo:
  1. LCM(3, 4)?
  2. LCM(6, 8)?
  3. 1/2 + 1/3 = ?
✅ Answers dekho (pehle khud try karo!)
  1. 3: 3,6,9,12; 4: 4,8,12 → 12.
  2. 6: 6,12,18,24; 8: 8,16,24 → 24.
  3. LCM 6: 3/6 + 2/6 = 5/6.

— 📌 charumam.com

🔟 Number line aur negative numbers

Ek seedhi line par 0 beech mein. Right = bada, left = chhota.

−5−4−3−2−1012345
  • Negative numbers (−1, −2, −3 …) 0 se left: 0 se chhote. Udhaar, 0 se neeche temperature, basement floor.
  • −3 bada hai −5 se, kyunki number line par −3, −5 ke right mein hai (0 ke zyada paas). Minus hata ke “5 bada” dekhna galti hai.
  • Integers (ℤ) = … −3, −2, −1, 0, 1, 2, 3 … (poore numbers, plus aur minus dono).
  • Natural numbers = 1, 2, 3 … (ginti wale). Whole numbers = 0, 1, 2, 3 …
✍️ Khud try karo:
  1. −7 aur −2 mein bada kaun?
  2. 0 aur −4 mein chhota kaun?
✅ Answers dekho (pehle khud try karo!)
  1. −2 (number line par right mein).
  2. −4.

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1️⃣1️⃣ Negative + positive jodna

Number line walk: 0 se shuru. Plus = right chalo, minus = left chalo. Jahan ruko, wahi answer.

(−3) + 5: 3 left (−3 pe), phir 5 right → 2.
−4−3−2−10123

Rule (bina line ke):

  • Same sign: size jodo, wahi sign. (−5) + (−2) = −7.
  • Alag sign: bade size mein se chhota size ghatao, aur bade size wale ka sign lagao. (−9) + 10: 10 − 9 = 1, bada 10 (plus) → +1. (−7) + 3: 7 − 3 = 4, bada 7 (minus) → −4.

Paise se socho: ₹3 ka udhaar (−3) aur ₹5 mile (+5) → udhaar chuka ke ₹2 bache (+2).

✍️ Khud try karo:
  1. (−6) + 2 = ?
  2. (−4) + (−3) = ?
  3. 8 + (−11) = ?
✅ Answers dekho (pehle khud try karo!)
  1. 6 − 2 = 4, bada 6 minus → −4.
  2. Same sign: −7.
  3. 11 − 8 = 3, bada 11 minus → −3.

— 📌 charumam.com

1️⃣2️⃣ Chhote mein se bada ghatana (3 − 5)

3 − 5 ka matlab: 3 se 5 kadam left chalo → 3, 2, 1, 0, −1, −2. To 3 − 5 = −2. (Jab bada number ghatate ho, answer negative aata hai.)

−3−2−101234
  • 9 − 10 = −1  |  2 − 7 = −5  |  (−5) − 3 = −8 (aur 3 left)
  • Minus ek negative = plus: −2 − (−5) = −2 + 5 = 3. Udhaar ka ₹5 “hata diya” to tum ₹5 amir ho gaye.
  • Ghatana = ulta jodna: 7 − 3 = 7 + (−3); 7 − (−3) = 7 + 3 = 10.
✍️ Khud try karo:
  1. 4 − 9 = ?
  2. (−2) − 6 = ?
  3. 5 − (−4) = ?
✅ Answers dekho (pehle khud try karo!)
  1. −5.
  2. −8.
  3. 5 + 4 = 9.

— 📌 charumam.com

1️⃣3️⃣ × aur ÷ ke sign: (−) × (−) = (+) kyun?

Pattern dekho, har baar pehla number 1 ghatao aur × (−2) karo:

3 × (−2) = −6  →  2 × (−2) = −4  →  1 × (−2) = −2  →  0 × (−2) = 0  →  (−1) × (−2) = +2  →  (−2) × (−2) = +4

Har kadam pe answer 2 badhta hai (−6, −4, −2, 0 …). Pattern ko chalne do to 0 ke baad +2, +4 hi aayega. Isliye negative × negative = positive.

SignsAnswer ka signExample
same (+,+) ya (−,−)+(−4) × (−3) = 12; (−12) ÷ (−3) = 4
alag (+,−) ya (−,+)−(−15) ÷ 5 = −3; 18 ÷ (−6) = −3

Bahut saare numbers: minus gino. Odd (1, 3, 5) → answer −. Even (0, 2, 4) → answer +.

Bhaag ka matlab: 84 ÷ 4 = “84 ko 4 barabar hisson mein baanto” = 21 (check: 4 × 21 = 84).

✍️ Khud try karo:
  1. (−5) × 4 = ?
  2. (−36) ÷ (−9) = ?
  3. (−1) × (−2) × (−3) = ?
✅ Answers dekho (pehle khud try karo!)
  1. −20.
  2. 4.
  3. 3 minus (odd) → −6.

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1️⃣4️⃣ Letters: x, a, b, n, p/q

Maths mein letter = kisi number ka naam (dabba), jab number abhi pata nahi ya “koi bhi number” chal sakta hai.

  • x + 3 = 7: x ek anjaan number; yahan x = 4.
  • 2x ka matlab 2 × x. Letters saath-saath likhe hon to guna: ab = a × b, ps = p × s, 2bd = 2 × b × d.
  • a − b = a + (−b): “koi bhi do number” ke liye rule. Example: a = 7, b = 3 → 7 − 3 = 7 + (−3).
  • p/q: “koi integer p, upar; koi integer q, neeche (q zero nahi)”. Jaise 3/4 mein p = 3, q = 4.
  • LHS / RHS: “=” ke left wala hissa (Left Hand Side) aur right wala (Right Hand Side).
  • a₁, a₂: chhote neeche likhe number sirf naam hain (pehla a, doosra a), guna nahi.

Atak gaye to letter ki jagah koi chhota number rakh ke dekho, rule samajh aa jaayega.

✍️ Khud try karo:
  1. Agar x = 5, to 3x = ?
  2. a = 2, b = 6 to ab = ?
  3. 5/9 mein p aur q?
✅ Answers dekho (pehle khud try karo!)
  1. 3 × 5 = 15.
  2. 2 × 6 = 12.
  3. p = 5, q = 9.

— 📌 charumam.com

1️⃣5️⃣ Maths ke symbols

SymbolPadhoExample
=barabar hai2 + 3 = 5
≠barabar nahi haiq ≠ 0 (q zero nahi ho sakta); 305 ≠ 35
<se chhota hai (nukila side chhote ki taraf)−2 < 0; 3 < 8
>se bada hai (khula muh bade ki taraf)7 > 0; −3 > −5
≤ / ≥chhota ya barabar / bada ya barabarx ≤ 3 matlab x = 3 ya 3 se chhota; 0 ≤ P ≤ 1
| |absolute value: 0 se doori (sign hatao)|−4| = 4; |3| = 3; |0| = 0
≈lagbhag5/3 ≈ 1.67
±plus ya minusE ± E = E (jodo ya ghatao, even hi aayega)
⇒isliye / matlab2x = 10 ⇒ x = 5
ℤsaare integers ka group… −2, −1, 0, 1, 2 …
∈“ka member hai”p ∈ ℤ: p ek integer hai
⊂“ke andar aata hai” (subset)Natural ⊂ Whole: har natural number whole number bhi hai
°degree (angle ki unit) / °C temperature90°, 180°; −2°C
√square root: kis number ka square yeh hai?√25 = 5 (kyunki 5 × 5 = 25); √2 ≈ 1.414… (kabhi khatam nahi)
π“pai”: circle ka khaas numberπ ≈ 3.14159… (kabhi khatam nahi, repeat nahi)
∞infinity: kabhi khatam na honumbers ∞ tak chalte hain
… aage aise hi chalta rahe0.333…
( )bracket: pehle andar ka kaam2 × (3 + 4) = 2 × 7 = 14
✍️ Khud try karo:
  1. 5 ___ −5 (sahi symbol)
  2. |−9| = ?
  3. “7 ∈ ℤ” ka matlab?
✅ Answers dekho (pehle khud try karo!)
  1. 5 > −5.
  2. 9.
  3. 7 ek integer hai.

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1️⃣6️⃣ Words jo posts mein aate hain

WordMatlab
Angle (kon)Do lines ek point se niklein to unke beech ka “khulapan”. Degree (°) mein naapte hain.
90° (right angle)Notebook ke kone jaisa seedha kona.
180°Seedhi line ka angle: aadha chakkar. Poora chakkar = 360°.
PlaneEk bilkul samtal (flat) sataha jo har taraf chalti rahe, jaise notebook ka page ya table ka top.
AreaKisi jagah ko dhakne mein kitne 1 × 1 ke chhote square lagenge. 1 m × 1 m = 1 square metre (sq m).
Strike rateCricket mein: har 100 ball pe kitne run bante. Formula: (run ÷ ball) × 100. 30 run, 20 ball → 30 ÷ 20 = 1.5 → × 100 = 150.
Average (ausat)Sab jodo, ginti se bhaag do. 4 aur 6 ka average = (4 + 6) ÷ 2 = 5.
Magnitude / sizeNumber ka sign hata ke bacha hissa: −7 ka size 7.
Lakh, crore1 lakh = 1,00,000; 1 crore = 100 lakh = 1,00,00,000.
Even, odd, primeSection 4 aur 6 dekho.
UndefinedJiska koi jawab hi nahi, jaise 5 ÷ 0.
✍️ Khud try karo:
  1. Seedhi line ka angle?
  2. 40 run, 50 ball. Strike rate?
  3. 10 aur 20 ka average?
✅ Answers dekho (pehle khud try karo!)
  1. 180°.
  2. 40 ÷ 50 = 0.8 → × 100 = 80.
  3. (10 + 20) ÷ 2 = 30 ÷ 2 = 15.

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