1.1.5
Division ka jawab hota hai "kitne equal groups ban sakte hain, aur kitna bacha?"
Agar tumhare paas 17 5 3 quotient )
aur 2 remainder ). Division basically divisor ki repeated subtraction hai jab tak aur subtract na ho sake — long division usi repeated subtraction ka ek fast, organised tarika hai jo digit-by-digit kaam karta hai.
Long division, division algorithm ko ek baar mein ek place-value column par apply karna hai, sabse bade place se shuru karke. Kyunki 728 7 hundreds + 2 tens + 8 ones
728 ÷ 6 Step 1 — hundreds divide karo. 7 ÷ 6 = 1 1 1 × 6 = 6 7 − 6 = 1 Kyun? Hum poochh rahe hain "7 hundreds mein kitne 6 1
Step 2 — tens bring down karo. Bacha hua 1 = 10 2 → 12 12 ÷ 6 = 2 2 2 × 6 = 12 → 0 Kyun? Bacha hua hundred, dah tens ban jaata hai — yahi "bring down" ka trick hai jo place value express karta hai.
Step 3 — ones bring down karo. 8 8 ÷ 6 = 1 1 1 × 6 = 6 → 2 Kyun? Ones column par same logic; 2 6
Answer: quotient = 121 = 2 Check (hamesha karo!): 6 × 121 + 2 = 726 + 2 = 728
945 ÷ 7
9 ÷ 7 = 1 2 Kyun: nine mein ek seven fit hota hai, do bacha.Bring down 4 → 24 24 ÷ 7 = 3 3 × 7 = 21 3 Kyun: 7 × 3 = 21 ≤ 24 < 28
Bring down 5 → 35 35 ÷ 7 = 5 0
Quotient = 135 = 0 Check: 7 × 135 = 945 0 7 945 divides karta hai.
Worked example Divisor pehle digit se bada:308 ÷ 4
3 ÷ 4 = 0 0 30 30 ÷ 4 = 7 7 × 4 = 28 2 Kyun: 4 × 7 = 28 ≤ 30 < 32 Bring down 8 → 28 28 ÷ 4 = 7 0
Quotient = 77 0 Check: 4 × 77 = 308
Recall Compute karne se pehle predict karoPehle 623 ÷ 8 8 × 80 = 640 8 × 70 = 560 70 62 ÷ 8 = 7 6 3 → 63 63 ÷ 8 = 7 7 Quotient 77 7 Tumhara forecast (70
Common mistake "Remainder divisor ke barabar ho sakta hai."Kyun sahi lagta hai: agar r = d Fix yeh hai: ek poore group ko quotient mein count karna zaroori hai . Agar r ≥ d 1 d 0 ≤ r < d 17 ÷ 5 = 2 7 3 2
Common mistake Placeholder zero bhool jaana.Kyun sahi lagta hai: 308 ÷ 4 7 7 7 Fix yeh hai: jab bhi current running number divisor se chhota ho ek digit bring down karne ke baad , quotient mein 0 6048 ÷ 6 1008 18
Common mistake Dividend aur divisor aapas mein badal dena.Kyun sahi lagta hai: "5 17 17 5 Fix yeh hai: dividend = bracket ke andar wali badi poori amount ; divisor = bahar wala group size. 17 ÷ 5 17 5
Mnemonic Long-division steps ka order"Does McDonald's Sell Burgers?" → D ivide, M ultiply, S ubtract, B ring down. Repeat karo.
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho 728 6 100 ke bade bags baanto: har box mein 1 6 1 deal nahi ho sake woh remainder hain. Har box ko jo milta hai woh quotient hai, jo pile se shuru kiya woh dividend hai, aur boxes ki sankhya divisor hai.
a ÷ b = q r a b q r
Division Algorithm equation batao. Dividend = × + 0 ≤ r <
Remainder divisor se chhota kyun hona chahiye? Agar r ≥
728 ÷ 6 Quotient 121 2 6 × 121 + 2 = 728
Long division ke 4 repeating steps? Divide, Multiply, Subtract, Bring down (DMSB).
Koi bhi long-division answer check kaise karte hain? divisor× +
Quotient mein 0 Jab ek digit bring down karne ke baad bhi running number divisor se chhota ho.
Remainder = 0 Divisor, dividend ko exactly divide karta hai (dividend, divisor ka multiple hai).
Digit-sum rule 3 Kyunki 10 ≡ 1 ( mod 3 )
Division = share into equal groups
Repeated subtraction of divisor
Quotient - number of groups
Division Algorithm a = dq + r
0 le remainder less than divisor
Long division, column by column
Place value, bring down trick
Check by multiplying back
