Lagnever AI English

Multiplication ke shortcuts: bade guna jaldi kaise karein

Short mein jawab

5 se guna karna aadha karke 10 se guna karne jaisa hai, kyunki 5, 10 ka aadha hai. Isi tarah × 25 matlab ÷ 4 phir × 100, aur × 125 matlab ÷ 8 phir × 1000. Saath hi, m-digit sankhya guna n-digit sankhya mein hamesha m + n − 1 ya m + n digits hote hain.

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Pehle aadha, phir ek zero

10 se guna karna aasaan hai: bas aakhir mein 0 likh do. 5 se guna karna bhi lagbhag utna hi aasaan hai, kyunki 5, 10 ka aadha hai. Roxie 116 × 5 aise nikaalti hai: 116 ka aadha 58, phir 10 se guna, jawab 580. Ye har sankhya pe chalta hai, kyunki kisi cheez ke paanch samooh uske das samoohon ke theek aadhe hote hain.

× 5, × 25 aur × 125 ke shortcuts: pehle 2, 4 ya 8 se bhaag, phir 10, 100 ya 1000 se guna
Pehle bhaag, phir zero

25 ke liye chauthaai, 125 ke liye aathwaan hissa

Yahi vichaar doosri sankhyaon pe bhi chalta hai. 4 × 25 = 100, isliye 25 se guna karna 4 se bhaag aur 100 se guna karne jaisa hai. Estu 824 × 25 ko 206 × 100 = 20,600 nikaalta hai. 8 × 125 = 1000, isliye 125 se guna karna 8 se bhaag aur 1000 se guna hai, to 72 × 125 = 9 × 1000 = 9,000. Shortcut tab sabse tez hai jab bhaag poora jaaye, par na jaaye tab bhi chalta hai: 30 × 25 = 7.5 × 100 = 750.

Friendly pairs aur regrouping

Lambe guna mein aise friendly pairs dhoondho jinka guna-phal 10, 100 ya 1000 ho, aur pehle unhe guna karo. Factors ka kram badalne se guna-phal nahi badalta, to 2 × 1768 × 50 = 1768 × (2 × 50) = 1768 × 100 = 1,76,800. Aur 125 × 40 × 8 × 25 = (40 × 25) × (125 × 8) = 1000 × 1000 = 10,00,000. Sankhya ko factors mein todna bhi kaam aata hai: 250 × 120 = (25 × 12) × 100 = 300 × 100 = 30,000.

Ulta bhi chal sakte ho. Kaunsi do sankhyaon ka guna 12,00,00,000 hai? Ek jawab 1200 × 1,00,000 hai; doosra 2500 × 48,000, kyunki 25 × 48 = 1200.

Guna-phal mein kitne digits?

Roxie kehti hai ki do 2-digit sankhyaon ka guna hamesha 3-digit ya 4-digit sankhya hota hai. Wo har joda nahi aazmaati. Sabse chhota guna-phal 10 × 10 = 100 hai, jisme 3 digits hain. Har 2-digit sankhya 100 se kam hai, to har guna-phal 100 × 100 = 10,000 se kam hai, yaani zyaada se zyaada 4 digits. Dono chhor check karne se sawaal hal ho jaata hai.

Isi soch se niyam milta hai: m-digit sankhya guna n-digit sankhya mein m + n − 1 ya m + n digits hote hain. To do 3-digit sankhyaon se kabhi 4-digit guna-phal nahi banta (sabse chhota 10,000 hai), par 4-digit guna 2-digit se 5-digit sankhya ban sakti hai, jaise 1000 × 10 = 10,000.

Digits ki table: 2-digit × 2-digit se 3 ya 4 digits, 5-digit × 5-digit se 9 ya 10, waghera
Guna-phal mein kitne digits?

Aage badhaane waale patterns

Kuch guna-phal kamaal ke pattern banaate hain: 11 × 11 = 121, 111 × 111 = 12321 aur 1111 × 1111 = 1234321, to 11111 × 11111 = 123454321. Phir, 3 × 5 = 15, 33 × 35 = 1155 aur 333 × 335 = 111555. Har line ke digits bhi gino: wo upar waale niyam pe chalte hain.

Chaar guna ke patterns, har ek ka agla kadam dashed line ke neeche
Guna ke patterns

Guna se khulte tathya

Chapter hisaab ke peechhe tathya chhupaata hai. 1250 × 380 = 4,75,000, utne kirtan jo kahaaniyon ke mutaabik Purandaradasa ne rache. 2100 × 70,000 = 14,70,00,000, kilometre mein Dharti se Sooraj ki lagbhag doori, yaani American system mein 147 million. Aur 10,50,00,000 ÷ 700 = 1,50,000: badi blue whale ka wazan itne kilo se bhi zyaada ho sakta hai.

Guna ya bhaag se khule chhe badi sankhyaon ke tathya, har ek dono system mein padha gaya
Tathya kholo

Ye kyun zaroori hai

Aise shortcuts se badi sankhyaon ka guna dimaag mein ho jaata hai aur calculator ke jawab bhi check ho jaate hain. Jawab padhne ke liye lakh aur crore dekho; unka size jaldi aankne ke liye rounding aur estimation. Kitaab ke har sawaal ke hal Chapter 1 ke NCERT solutions mein hain; poore Chapter 1 ke notes bhi dekho.

Practice karo

  1. 48 × 25 kitna hai?

    1. (a) 120
    2. (b) 1,200
    3. (c) 12,000
    4. (d) 600
    Jawab dekho

    Jawab: (b) 1,200

  2. 4-digit sankhya guna 2-digit sankhya mein kitne digits hote hain?

    1. (a) 4 ya 5
    2. (b) 5 ya 6
    3. (c) 6 ya 7
    4. (d) sirf 6
    Jawab dekho

    Jawab: (b) 5 ya 6

  3. 4 × 37 × 25 ko kaunsa samooh aasaan banaata hai?

    1. (a) (4 × 37) × 25
    2. (b) (4 × 25) × 37
    3. (c) (37 × 25) × 4
    4. (d) 4 + 37 + 25
    Jawab dekho

    Jawab: (b) (4 × 25) × 37

  4. 64 × 125 jaldi nikaalo.

    Jawab dekho

    Jawab: 125 = 1000 ÷ 8, to 64 × 125 = (64 ÷ 8) × 1000 = 8 × 1000 = 8,000.

  5. Pattern 9 × 9 = 81, 99 × 99 = 9801, 999 × 999 = 998001 ko ek kadam aage badhaao.

    Jawab dekho

    Jawab: 9999 × 9999 = 99980001. Har kadam pe shuru mein ek 9 aur aakhri 1 se pehle ek 0 badh jaata hai.

Students ke sawal

5 se guna karna 2 se bhaag aur 10 se guna karne jaisa kyun hai?

Kyunki 5, 10 ka aadha hai. Kisi sankhya ke paanch samooh uske das samoohon ke aadhe hote hain, to n × 5 = n × 10 ÷ 2. Pehle aadha karo ya pehle 10 se guna, jawab wahi aata hai: 116 × 5 = 58 × 10 = 580.

Bina guna kiye kaise pata chale ki guna-phal mein kitne digits honge?

Dono factors ke digits jodo: guna-phal mein utne digits honge ya ek kam. Kyun, ye dono chhor dekhkar samjho. Do 3-digit sankhyaon ke liye 100 × 100 = 10,000 mein 5 digits hain aur 999 × 999 = 9,98,001 mein 6, isliye har aise guna-phal mein 5 ya 6 digits hote hain.

Guna mein friendly pairs kya hote hain?

Aise jode jinka guna-phal gol sankhya ho, jaise 2 aur 50, 4 aur 25, ya 8 aur 125. Lambe guna mein ye milein to pehle inhe jodo: 125 × 40 × 8 × 25 = (40 × 25) × (125 × 8) = 1000 × 1000 = 10,00,000.

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