Multi-Digit Addition and Subtraction

➕ Multi-Digit Addition & Subtraction

Part 1 of 5 — Place Value & Lining Up the Digits

Topics in This Part

🔑 Key Concept: Every digit in a number has a place — ones, tens, hundreds, thousands. Addition and subtraction only work when you line up digits in the same place and combine them. Get the line-up right and the rest is easy.

What Place Value Means

The value of a digit depends on where it sits in the number. Look at 4,736:

So $4{,}736 = 4{,}000 + 700 + 30 + 6$. Each place is 10 times bigger than the place to its right.

💡 The same digit can mean very different amounts. In 5,208, the 5 is worth $5{,}000$, but in 52 the 5 is only worth $50$. Position is everything.

Concept Check 🎯

Lining Up by Place

When you stack numbers to add or subtract, the ones line up under the ones, tens under tens, and so on. Always work from right to left (ones first).

$\begin{array}{r} 3\,4\,7 \\ +\;2\,8\,5 \\ \hline \end{array}$

The $7$ and the $5$ are both in the ones place, so they get added together. The $4$ and the $8$ are both tens.

⚠️ Watch out! When the numbers have different lengths, still line up the right edge (the ones place). To add $475 + 38$, the $8$ goes under the $5$, not under the $7$:

$\begin{array}{r} 4\,7\,5 \\ +\;\;\;3\,8 \\ \hline \end{array}$

Line Them Up 🔽

You are stacking numbers to add. Choose which digits share each place.

Reading Big Numbers

A comma separates groups of three digits and makes big numbers easy to read. Read $4{,}736$ as "four thousand, seven hundred thirty-six."

🔑 A zero is a placeholder — it holds a place open so the other digits stay in their correct spots. In $6{,}080$, the zero in the hundreds place tells us there are no hundreds.

Name the Value 🧮

For each number, enter the value of the bold digit (not just the digit itself).

1) In $4{,}736$, the value of the $3$ is $\,?$ (it is in the tens place) 2) In $5{,}208$, the value of the $5$ is $\,?$ (it is in the thousands place) 3) In $4{,}736$, the value of the $6$ is $\,?$ (it is in the ones place)

➕ Multi-Digit Addition & Subtraction

Part 2 of 5 — Addition with Regrouping (Carrying)

🔑 The Idea: When a column adds up to $10$ or more, you can't fit two digits in one place. You regroup (carry) the extra ten into the next column to the left. Same trick works for hundreds and thousands.

The Steps for Adding

To add two numbers:

1. Line them up by place (ones under ones).
2. Start at the ones column and add.
3. If a column total is $10$ or more, write the ones digit and carry the ten to the next column.
4. Move left and repeat until every column is done.

Worked Example: $347 + 285$

$\begin{array}{r} \overset{1}{3}\,\overset{1}{4}\,7 \\ +\;2\,8\,5 \\ \hline 6\,3\,2 \end{array}$

• Ones: $7 + 5 = 12$. Write $2$, carry $1$.
• Tens: $4 + 8 + 1 = 13$. Write $3$, carry $1$.
• Hundreds: $3 + 2 + 1 = 6$. Write $6$.

So $347 + 285 = 632$. ✓

Concept Check 🎯

Carrying More Than Once

Sometimes you carry in several columns. Just take it one column at a time.

Worked Example: $1{,}568 + 2{,}745$

$\begin{array}{r} 1\,5\,6\,8 \\ +\;2\,7\,4\,5 \\ \hline 4\,3\,1\,3 \end{array}$

• Ones: $8 + 5 = 13$. Write $3$, carry $1$.
• Tens: $6 + 4 + 1 = 11$. Write $1$, carry $1$.
• Hundreds: $5 + 7 + 1 = 13$. Write $3$, carry $1$.
• Thousands: $1 + 2 + 1 = 4$. Write $4$.

So $1{,}568 + 2{,}745 = 4{,}313$. ✓

💡 Carrying a $1$ never changes — you always carry just one ten (or one hundred, etc.), because a single column can total at most $9 + 9 + 1 = 19$.

Add It Up 🧮

Add each pair. Work right to left and carry when a column is $10$ or more.

1) $475 + 38 = \,?$ 2) $456 + 789 = \,?$ 3) $2{,}649 + 3{,}875 = \,?$

Trace the Carry 🔽

You are adding $3{,}826 + 4{,}597$. Choose what happens at each column.

➕ Multi-Digit Addition & Subtraction

Part 3 of 5 — Subtraction with Regrouping (Borrowing)

🔑 The Idea: When the top digit in a column is smaller than the bottom digit, you can't subtract yet. You borrow $1$ from the next place to the left — that turns it into $10$ extra in the current column. This is regrouping in reverse.

The Steps for Subtracting

To subtract:

1. Line up by place (ones under ones).
2. Start at the ones column.
3. If the top digit is smaller than the bottom, borrow $1$ from the next column left: cross it out, make it one smaller, and add $10$ to the current column.
4. Subtract each column, moving left.

Worked Example: $652 - 318$

$\begin{array}{r} 6\,\overset{4}{5}\,\overset{12}{\cancel{2}} \\ -\;3\,1\,8 \\ \hline 3\,3\,4 \end{array}$

• Ones: $2 - 8$ won't work. Borrow: the $5$ tens becomes $4$, and the ones become $12$. Now $12 - 8 = 4$.
• Tens: $4 - 1 = 3$.
• Hundreds: $6 - 3 = 3$.

So $652 - 318 = 334$. ✓

Concept Check 🎯

Borrowing Across Zeros

The trickiest case is subtracting from a number with zeros, like $700 - 264$. There's nothing to borrow from next door, so you keep borrowing one place further left until you reach a nonzero digit.

Worked Example: $700 - 264$

$\begin{array}{r} \overset{6}{\cancel{7}}\,\overset{9}{\cancel{0}}\,\overset{10}{\cancel{0}} \\ -\;2\,6\,4 \\ \hline 4\,3\,6 \end{array}$

• Ones: $0 - 4$ won't work, but the tens are $0$ too. Borrow from the hundreds: $7$ becomes $6$, the tens become $10$, then lend $1$ ten to the ones — tens become $9$, ones become $10$.
• Ones: $10 - 4 = 6$.
• Tens: $9 - 6 = 3$.
• Hundreds: $6 - 2 = 4$.

So $700 - 264 = 436$. ✓

⚠️ Common mistake: writing $0 - 4 = 4$. You can't skip the borrow! Always regroup before you subtract.

Subtract It 🧮

Subtract each pair. Borrow whenever the top digit is too small.

1) $800 - 356 = \,?$ 2) $623 - 457 = \,?$ 3) $5{,}003 - 2{,}748 = \,?$

Check by Adding 🔽

You can always check a subtraction by adding the answer back. If $a - b = c$, then $c + b$ should equal $a$.

➕ Multi-Digit Addition & Subtraction

Part 4 of 5 — Estimation & Word Problems

🔑 Why estimate? Rounding first gives you a quick "about how much" answer. If your exact answer is far from your estimate, you probably made a regrouping mistake. Estimation is your built-in error checker.

Rounding to Estimate

To round to a place, look at the digit just to the right:

• If it's $5$ or more, round up.
• If it's $4$ or less, round down (keep the digit the same).

Estimating a Sum: $487 + 312$

Round each to the nearest hundred: $500 + 300 = 800$. The exact answer is $799$ — very close, so we trust it. ✓

💡 An estimate is never the exact answer, but it should be near it. Use estimation before you compute (to predict) and after (to check).

Concept Check 🎯

Word Problem Key Words

Word problems tell you whether to add or subtract. Watch for these clues:

Worked Example

A library had $1{,}245$ books. They bought $876$ more. How many books in all?

"In all" means add: $1{,}245 + 876 = 2{,}121$ books.

Worked Example

A stadium holds $2{,}500$ people. If $1{,}750$ seats are filled, how many are empty?

"How many empty" (left over) means subtract: $2{,}500 - 1{,}750 = 750$ empty seats.

⚠️ Key words are clues, not rules — always picture the situation. "How many more" usually means subtract (a difference), even though it contains the word "more."

Solve the Problem 🧮

Read each problem, decide add or subtract, then compute.

1) A school sold $1{,}378$ tickets on Friday and $2{,}456$ on Saturday. How many tickets in all? $\,?$ 2) A truck carried $5{,}280$ pounds of cargo. It dropped off $3{,}675$ pounds. How many pounds are left? $\,?$

Add or Subtract? 🔽

Decide the operation for each problem, then pick the answer.

➕ Multi-Digit Addition & Subtraction

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) line up digits by place, (2) add with carrying, (3) subtract with borrowing (even across zeros), and (4) estimate and solve word problems. Let's put it all together.

Quick Reference

⚠️ The two most common mistakes: forgetting to carry the extra ten in addition, and forgetting to borrow (writing things like $0 - 4 = 4$) in subtraction. Slow down on every column.

Mixed Practice 🧮

Add or subtract as needed.

1) $3{,}456 + 2{,}789 = \,?$ 2) $7{,}000 - 4{,}286 = \,?$ 3) $5{,}604 - 2{,}837 = \,?$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.