Addition and subtraction — carrying, borrowing, word problems

1.1.3 · Maths › Arithmetic & Number Systems

WHY place value carrying aur borrowing ko force karta hai

WHAT hai place value? Ek number jaise $472$ ka matlab literally hai $472 = 4 \times 100 + 7 \times 10 + 2 \times 1.$ Har column ek power of ten hai: $1 0^{0} = 1$ (ones), $1 0^{1} = 10$ (tens), $1 0^{2} = 100$ (hundreds).

WHY sirf digits 0–9? Base ten ka matlab hai hamare paas exactly ten symbols hain. Jaise hi kisi column mein kisi cheez ke $10$ ho jaate hain, woh "ten" by definition agli column ka ek unit ban jaata hai. Toh $10$ ones $= 1$ ten, $10$ tens $= 1$ hundred. Yahi trade poora engine hai.

HOW carrying kaam karti hai (derived): Maano ones column mein hum $a + b$ compute karte hain. Ise division algorithm se likhte hain (base $10$ se divide karke): $a + b = 10 q + r, 0 \leq r \leq 9, q \in {0, 1} .$ Yahaan $r$ woh digit hai jo hum likhte hain aur $q$ woh carry hai jo tens column mein jaati hai. Kyunki single digits mein $a, b \leq 9$ hoti hain (plus zyada se zyada $1$ ki carry), sabse bada sum $9 + 9 + 1 = 19$ hota hai, toh $q$ hamesha $0$ ya $1$ hi hoga.

HOW borrowing kaam karti hai (derived): Kisi column mein $t - u$ compute karne ke liye jab top digit $t < u$ ho, hum agli column se $1$ borrow karte hain, jo yahaan $10$ ke barabar hai. Toh $t$ ko $t + 10$ se replace karte hain (ab $\geq u$ ) aur agli top digit ko $1$ se kam karte hain: $t - u = (t + 10) - u - 10,$ aur woh $- 10$ agli column mein $- 1$ ke roop mein absorb ho jaata hai. Yeh bilkul exact hai — humne usi $10$ ko add aur subtract kiya.

Figure — Addition and subtraction — carrying, borrowing, word problems

Worked examples

Common mistake Classic errors ko steel-man karna

1. "Carry bhool jaana." Kyun sahi lagta hai: us column ka kaam "ho gaya" toh aage bad jaate ho. Fix: carry ek loan hai agli column ko; usse upar chhota likho aur us column ko touch karne se pehle add karo. Carry chhoot jaaye toh answer us jagah $10 \times$ chhota ho jaata hai.

2. "Har column mein chhote se bada subtract karna, position se anjaane mein" (jaise $605 - 278$ mein: ones mein $8 - 5 = 3$ karna kyunki "8 bada hai"). Kyun sahi lagta hai: single numbers ka subtraction "bada minus chhota" hota hai, aur ulta karne se negatives se bacha jaata hai. Fix: minuend digit upar hi rehna chahiye; agar woh chhota hai, toh zaroor borrow karo. Ulta karna problem ko $∣ t o p - b o tt o m ∣$ per column kar deta hai, jo sahi difference nahi hai.

3. Zeros ke paar galat borrowing. Kyun sahi lagta hai: " $0$ mein borrow karne ke liye kuch nahi hai." Fix: $0$ pehle apne paas-wale se borrow karta hai, $10$ ban jaata hai, $1$ deta hai ( $9$ ho jaata hai), aur chain aage pass karta hai. Beech ke har $0$ ka $9$ ho jaata hai.

4. Digits galat jagah / place values align na karna. Kyun sahi lagta hai: numbers jaise padhte ho waise likhte ho. Fix: ones column pe right-align karo; trailing spaces daalo, randomly zeros nahi.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho envelopes mein paise hain: ek envelope pennies (ones) ke liye, ek dimes (tens) ke liye, ek dollars (hundreds) ke liye. Har envelope mein zyada se zyada 9 coins aa sakti hain apni tarah ki — kyunki $10$ pennies exactly $1$ dime hoti hai, toh tum unhe exchange kar loge. Adding: dono dher milao; jab bhi penny envelope mein $10$ ya zyada ho, $10$ pennies ko $1$ dime se badlo aur woh dime agli envelope mein daalo — yahi carrying hai. Subtracting: agar tumhe jitni pennies deni hain utni tumhare paas nahi hain, toh dime envelope pe jaao, ek dime ko $10$ pennies mein toddo (yahi borrowing hai), aur ab tumhare paas kaafi hai. Kuch aata ya jaata nahi — bas wohi paisa envelopes ke beech repackage ho raha hai.

Flashcards

472

number expanded place-value form mein kya hoga?

4 \times 100 + 7 \times 10 + 2 \times 1

Base ten mein har column mein sirf digits 0–9 kyun hoti hain?

Kyunki kisi bhi unit ke ten hone par woh agli place ki ek unit ban jaati hai, toh upar repackage ho jaata hai.

Addition mein "carry" define karo.

Woh

1

jo agli-badi column mein jaata hai jab column ka sum

\geq 10

ho; equals

⌊(a_{i} + b_{i} + c_{i}) /10 ⌋

Subtraction mein "borrow" define karo.

Agli-badi column se

1

(jo

10

ke barabar hai) lena jab top digit subtract karne ke liye bahut chhoti ho.

Addition column mein likhi jaane wali digit ka formula kya hai?

s_{i} = (a_{i} + b_{i} + c_{i}) mod 10

Base-10 addition mein sabse bada possible carry kya hai, aur kyun?

1

, kyunki

9 + 9 + 1 = 19 < 20

, toh

⌊ 19/10 ⌋ = 1

605 - 278

mein beech ka

0

borrow ke baad kya banta hai?

9

(woh borrow karta hai dene ke liye, chain banata hua).

Subtraction answer check kaise karte hain?

Difference ko subtrahend mein add karo; minuend wapas milna chahiye:

(a - b) + b = a

Subtraction ke liye word-problem keywords kya hain?

"gave away", "left", "how many more/fewer", "decrease", "spent".

Addition ke liye word-problem keywords kya hain?

"total", "altogether", "in all", "bought more", "combined", "increase".

Minuend digit upar kyun rehna chahiye, chahe woh chhoti hi kyun na ho?

Kyunki subtraction

∣ t o p - b o tt o m ∣

nahi hai; chhoti top digit ka matlab hai tumhara karz hai aur sahi value rakhne ke liye borrow karna zaroori hai.

Connections

Place Value and Base-Ten System — woh foundation jis par carrying/borrowing tika hua hai.
Multiplication as Repeated Addition — carries multiplication algorithm mein phir dikhte hain.
Division Algorithm — $a = 10 q + r$ bilkul carry decomposition hai.
Negative Numbers and Integers — "zero se paar borrowing" yahan generalize hoti hai.
Estimation and Rounding — compute karne se pehle sums/differences ka sanity-check.
Word Problem Translation — English phrases ko $+$ aur $-$ mein convert karna.