Addition and subtraction — carrying, borrowing, word problems

WHY place value forces carrying and borrowing

WHAT is place value? A number like $472$ literally means $472 = 4\times 100 + 7\times 10 + 2\times 1.$ Each column is a power of ten: $10^0=1$ (ones), $10^1=10$ (tens), $10^2=100$ (hundreds).

WHY only digits 0–9? Base ten means we have exactly ten symbols. Once a column reaches $10$ of something, that "ten" is by definition one unit of the next column up. So $10$ ones $=1$ ten, $10$ tens $=1$ hundred. This trade is the whole engine.

HOW carrying works (derived): Suppose in the ones column we compute $a+b$ . Write it using the division algorithm (dividing by the base $10$ ): $a+b = 10q + r,\qquad 0\le r \le 9,\ q\in\{0,1\}.$ Here $r$ is the digit we write and $q$ is the carry into the tens column. Because single digits satisfy $a,b\le 9$ (plus a carry of at most $1$ ), the largest sum is $9+9+1=19$ , so $q$ is always $0$ or $1$ .

HOW borrowing works (derived): To compute $t - u$ in a column where the top digit $t < u$ , we borrow $1$ from the next column, which is worth $10$ here. So we replace $t$ with $t+10$ (now $\ge u$ ) and reduce the next top digit by $1$ : $t - u = (t+10) - u \;-\; 10,$ and that $-10$ is absorbed as $-1$ in the next column. This is exact — we added and subtracted the same $10$ .

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

Worked examples

Worked example Word problem — "how many more?"

A library had $1\,204$ books. It gave away $587$ and later bought $349$ . How many books now? Translate: "gave away" → subtract; "bought" → add. $1204 - 587 + 349.$ Step 1 ( $1204-587$ ): ones $4-7$ → borrow, $14-7=7$ ; tens $0-1(\text{borrow})-8$ → borrow chain, $=1$ ; hundreds $=6$ ; but recompute cleanly: $1204-587=617$ . Why order? Left to right is fine because $+/-$ associate; do the subtraction first as written. Step 2 ( $617+349$ ): ones $7+9=16$ → $6$ carry $1$ ; tens $1+4+1=6$ ; hundreds $6+3=9$ . $=966$ . Answer: $966$ books. Check: $966+587-349 = 1204$ . ✓

Common mistake Steel-manning the classic errors

1. "Forgetting the carry." Why it feels right: you already "finished" the column so you move on. The fix: the carry is a loan to the next column; write it small above and add it before you touch that column. A carry left behind means your answer is $10\times$ too small in that place.

2. "Subtracting the smaller from the larger in each column, regardless of position" (e.g. $605-278$ : doing $8-5=3$ in the ones because "8 is bigger"). Why it feels right: subtraction of single numbers is "big minus small," and reversing avoids negatives. The fix: the minuend digit must stay on top; if it's smaller, you must borrow. Reversing changes the problem to $|top-bottom|$ per column, which is not the true difference.

3. Borrowing across zeros wrongly. Why it feels right: "there's nothing in the $0$ to borrow." The fix: the $0$ borrows from its neighbour first, becoming $10$ , lends $1$ (becoming $9$ ), and passes the chain along. Every $0$ in the middle turns into a $9$ .

4. Misplaced digits / not aligning place values. Why it feels right: you write numbers as you read them. The fix: right-align on the ones column; add trailing spaces, not zeros arbitrarily.

Recall Feynman: explain it to a 12-year-old

Imagine money in envelopes: one envelope for pennies (ones), one for dimes (tens), one for dollars (hundreds). Each envelope can hold at most 9 coins of its kind — because $10$ pennies is exactly $1$ dime, so you'd swap them. Adding: dump both piles together; whenever a penny envelope has $10$ or more, trade $10$ pennies for $1$ dime and drop that dime in the next envelope — that's carrying. Subtracting: if you need to pay more pennies than you have, you walk to the dime envelope, break one dime into $10$ pennies (that's borrowing), and now you have enough. Nothing appears or vanishes — you're just repackaging the same money between envelopes.

Flashcards

What does the number

472

equal in expanded place-value form?

4\times100+7\times10+2\times1

Why can each column hold only digits 0–9 in base ten?

Because ten of any unit equals one unit of the next place, so it gets repackaged upward.

Define "carry" in addition.

The

1

moved to the next-higher column when a column's sum is

\ge 10

; equals

\lfloor(a_i+b_i+c_i)/10\rfloor

Define "borrow" in subtraction.

Taking

1

(worth

10

) from the next-higher column when the top digit is too small to subtract.

Formula for the digit written in an addition column?

s_i=(a_i+b_i+c_i)\bmod 10

What is the largest possible carry in base-10 addition, and why?

1

, because

9+9+1=19<20

, so

\lfloor 19/10\rfloor=1

605-278

, what does the middle

0

become after borrowing?

9

(it borrows to lend, chaining the borrow).

How do you check a subtraction answer?

Add the difference to the subtrahend; you should recover the minuend:

(a-b)+b=a

Word-problem keywords for subtraction?

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

Word-problem keywords for addition?

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

Why must the minuend digit stay on top even if it's smaller?

Because subtraction is not

|top-bottom|

; a smaller top digit means you owe and must borrow to keep the true value.

Connections

Place Value and Base-Ten System — the foundation carrying/borrowing rests on.
Multiplication as Repeated Addition — carries reappear in the multiplication algorithm.
Division Algorithm — $a=10q+r$ is exactly the carry decomposition.
Negative Numbers and Integers — what "borrowing past zero" generalises to.
Estimation and Rounding — sanity-check sums/differences before computing.
Word Problem Translation — converting English phrases into $+$ and $-$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, hamara number system "place value" pe based hai — matlab har digit ki value uske column pe depend karti hai: ones, tens, hundreds. Jaise $472$ ka matlab hai $4\times100 + 7\times10 + 2\times1$ . Har column mein sirf $0$ se $9$ tak hi aa sakta hai, kyunki $10$ ones ka matlab hota hai exactly $1$ ten — toh hum use upar wale column mein "repackage" kar dete hain. Yahi repackaging ka naam hai carrying (addition mein) aur borrowing (subtraction mein).

Carrying tab hoti hai jab kisi column ka sum $10$ ya usse zyada ho jaata hai. Jaise $7+6=13$ — hum $3$ likhte hain aur $1$ ko agle (tens) column mein le jaate hain. Formula simple hai: $13 = 10\times 1 + 3$ , toh carry $=1$ , likhne wala digit $=3$ . Borrowing ulta hai — jab upar wala digit chhota ho (jaise $5-8$ ), toh hum next column se $10$ udhaar le lete hain: $5$ ban jaata hai $15$ , aur $15-8=7$ . Jo column se udhaar liya, uski value $1$ kam ho jaati hai. Zero ke across borrow karte waqt $0$ pehle udhaar leta hai (banke $10$ ), phir aage deta hai, toh $0$ ban jaata hai $9$ .

Word problems mein sabse important hai keywords pehchanna: "total, altogether, in all" matlab jodo (add); "gave away, left, how many more, spent" matlab ghatao (subtract). Problem ko English se maths mein translate karo, phir column method chalao.

Ek zabardast trick hamesha yaad rakho: subtraction ka answer verify karne ke liye usmein subtrahend wapas jodo — agar minuend wapas mil gaya toh answer sahi hai, kyunki $(a-b)+b=a$ . Common galtiyan: carry bhool jaana, aur column mein "bada minus chhota" karna position dekhe bina — dono se answer galat aata hai, toh dhyan rakhna!

Addition and subtraction — carrying, borrowing, word problems

WHY place value forces carrying and borrowing

Worked examples

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Arithmetic & Number Systems

Connections