3-Digit Addition and Subtraction Study Notes

Understanding 3-digit addition and subtraction helps us work with larger numbers. This lesson will show you how to add and subtract numbers up to 999, using place value and regrouping.

🔢 Place Value Review

Before we add or subtract 3-digit numbers, let's remember place value. Every digit in a number has a value based on its position.

• The digit on the right is in the Ones Place.
• The digit in the middle is in the Tens Place.
• The digit on the left is in the Hundreds Place.

For example, in the number \(345\):

• The \(3\) is in the hundreds place, so it means \(300\).
• The \(4\) is in the tens place, so it means \(40\).
• The \(5\) is in the ones place, so it means \(5\).

So, \(345 = 300 + 40 + 5\).

➕ 3-Digit Addition

When adding 3-digit numbers, always line up the numbers by their place value (hundreds under hundreds, tens under tens, ones under ones). Then, always start adding from the ones place.

Add Without Regrouping (No Carrying Over)

This is when the sum of the digits in any place value column is less than 10.

Example: Add \(234 + 123\)

1. Line up the numbers: \[ \begin{array}{r} 234 \\ + 123 \\ \end{array} \]
2. Add the ones place: \(4 + 3 = 7\). \[ \begin{array}{r} 234 \\ + 123 \\ \text{ \quad \quad } 7 \end{array} \]
3. Add the tens place: \(3 + 2 = 5\). \[ \begin{array}{r} 234 \\ + 123 \\ \text{ \quad } 57 \end{array} \]
4. Add the hundreds place: \(2 + 1 = 3\). \[ \begin{array}{r} 234 \\ + 123 \\ 357 \end{array} \]

So, \(234 + 123 = 357\).

Add With Regrouping (Carrying Over)

Regrouping (or carrying over) happens when the sum of the digits in a place value column is 10 or more. You carry the "extra" tens to the next place value column.

Example: Add \(358 + 264\)

1. Line up the numbers: \[ \begin{array}{r} 358 \\ + 264 \\ \end{array} \]
2. Add the ones place: \(8 + 4 = 12\).

   Since \(12\) is \(1\) ten and \(2\) ones, write down the \(2\) in the ones place and carry over the \(1\) (ten) to the tens place.

   \[ \begin{array}{r} \text{ } \quad ^1 \\ 358 \\ + 264 \\ \text{ \quad \quad } 2 \end{array} \]
3. Add the tens place: \(5 + 6 + 1\) (carried over) \(= 12\).

   Since \(12\) is \(1\) hundred and \(2\) tens, write down the \(2\) in the tens place and carry over the \(1\) (hundred) to the hundreds place.

   \[ \begin{array}{r} ^1 \quad ^1 \\ 358 \\ + 264 \\ \text{ \quad } 22 \end{array} \]
4. Add the hundreds place: \(3 + 2 + 1\) (carried over) \(= 6\). \[ \begin{array}{r} ^1 \quad ^1 \\ 358 \\ + 264 \\ 622 \end{array} \]

So, \(358 + 264 = 622\).

📌 Key Takeaway for Addition: Always start with the ones place and move to the left. Remember to regroup (carry over) when the sum in a column is 10 or more!

➖ 3-Digit Subtraction

When subtracting 3-digit numbers, always line up the numbers by their place value. Then, always start subtracting from the ones place.

Subtract Without Regrouping (No Borrowing)

This is when the top digit in each place value column is greater than or equal to the bottom digit.

Example: Subtract \(457 - 123\)

1. Line up the numbers: \[ \begin{array}{r} 457 \\ - 123 \\ \end{array} \]
2. Subtract the ones place: \(7 - 3 = 4\). \[ \begin{array}{r} 457 \\ - 123 \\ \text{ \quad \quad } 4 \end{array} \]
3. Subtract the tens place: \(5 - 2 = 3\). \[ \begin{array}{r} 457 \\ - 123 \\ \text{ \quad } 34 \end{array} \]
4. Subtract the hundreds place: \(4 - 1 = 3\). \[ \begin{array}{r} 457 \\ - 123 \\ 334 \end{array} \]

So, \(457 - 123 = 334\).

Subtract With Regrouping (Borrowing)

Regrouping (or borrowing) happens when the top digit in a place value column is smaller than the bottom digit. You need to "borrow" from the next place value column to the left.

Example: Subtract \(542 - 128\)

1. Line up the numbers: \[ \begin{array}{r} 542 \\ - 128 \\ \end{array} \]
2. Subtract the ones place: \(2 - 8\).

   You cannot subtract 8 from 2. So, you need to regroup from the tens place.

   The \(4\) in the tens place becomes \(3\). The \(2\) in the ones place becomes \(12\).

   \[ \begin{array}{r} \text{ } \quad ^3 \quad ^{12} \\ 5 \quad \cancel{4} \quad \cancel{2} \\ - 1 \quad 2 \quad 8 \\ \text{ \quad \quad \quad } 4 \end{array} \]

   Now, \(12 - 8 = 4\).

3. Subtract the tens place: \(3 - 2\).

   Remember, the \(4\) became a \(3\). So, \(3 - 2 = 1\).

   \[ \begin{array}{r} \text{ } \quad ^3 \quad ^{12} \\ 5 \quad \cancel{4} \quad \cancel{2} \\ - 1 \quad 2 \quad 8 \\ \text{ \quad } 1 \quad 4 \end{array} \]
4. Subtract the hundreds place: \(5 - 1 = 4\). \[ \begin{array}{r} \text{ } \quad ^3 \quad ^{12} \\ 5 \quad \cancel{4} \quad \cancel{2} \\ - 1 \quad 2 \quad 8 \\ 4 \quad 1 \quad 4 \end{array} \]

So, \(542 - 128 = 414\).

📌 Key Takeaway for Subtraction: Always start with the ones place and move to the left. Remember to regroup (borrow) when the top digit in a column is smaller than the bottom digit!

💡 Pro Tip: Check Your Work!

You can check your subtraction with addition! If \(A - B = C\), then \(C + B\) should equal \(A\).

For example, we found \(542 - 128 = 414\).

Let's check: \(414 + 128\)

1. Add ones: \(4 + 8 = 12\) (write \(2\), carry \(1\)).
2. Add tens: \(1 + 2 + 1\) (carried) \(= 4\).
3. Add hundreds: \(4 + 1 = 5\).

Result: \(542\). It matches the original number! So our subtraction was correct.

📚 Math Vocabulary

Here are some important words to remember:

Term	Meaning
Sum	The answer to an addition problem.
Addend	A number that is added to another number.
Difference	The answer to a subtraction problem.
Minuend	The number from which another number is subtracted (the top number).
Subtrahend	The number being subtracted (the bottom number).