1.3 Subtract Whole Numbers

Subtraction of whole numbers is about finding the difference between two numbers or taking one amount away from another. You use it constantly: calculating change, figuring out how much time is left, or comparing two quantities.

This section covers subtraction notation, visual models, the standard subtraction algorithm (including regrouping), word problems, and how the base-10 place value system makes it all work.

Subtraction of Whole Numbers

Correct subtraction symbols and notation

Every subtraction problem has three parts:

• Minuend: the starting number (the number you subtract from)
• Subtrahend: the number being taken away
• Difference: the result you get after subtracting

In the problem $15 - 6 = 9$, the minuend is 15, the subtrahend is 6, and the difference is 9. The minus sign (−) tells you to subtract.

One thing to watch out for: subtraction is not commutative. That means you can't swap the order and get the same answer. For example, $8 - 3 = 5$, but $3 - 8 = -5$. Order matters every time.

Visual models for subtraction

Two common models can help you picture what subtraction is doing:

Counters or blocks: Start with a group of objects equal to the minuend, then physically remove the number of objects equal to the subtrahend. Whatever remains is the difference. For example, lay out 12 counters, remove 5, and you're left with 7. So $12 - 5 = 7$.

Number line: Start at the minuend on the number line and move to the left by the value of the subtrahend. Where you land is the difference. To solve $10 - 6$, start at 10, hop 6 units to the left, and you land on 4.

Efficient subtraction of whole numbers

Subtracting without regrouping is straightforward. Work from right to left, subtracting each place value column separately:

Subtracting with regrouping (borrowing) is needed when a digit in the minuend is smaller than the digit below it. Here's the step-by-step process:

1. Start at the ones column. If the top digit is smaller than the bottom digit, you need to borrow.
2. Go to the next column to the left and reduce that digit by 1.
3. Add 10 to the current column's top digit.
4. Now subtract in that column.
5. Move left and repeat for each remaining column.

Mental math shortcut: For quick calculations, try rounding. To solve $83 - 47$ in your head, round 47 up to 50, subtract ($83 - 50 = 33$), then add back the 3 you over-subtracted: $33 + 3 = 36$.

Word problems to subtraction expressions

To turn a word problem into a subtraction expression:

1. Find the starting amount or total. That's your minuend.
2. Find the amount being removed or taken away. That's your subtrahend.
3. Write the expression: minuend − subtrahend.
4. Solve to find the difference.

Look for clue words like left, remaining, fewer, difference, how many more, or how much less. These signal that subtraction is the right operation.

Real-world applications of subtraction

• Comparing quantities: Two buildings are 100 meters and 85 meters tall. The difference in height is $100 - 85 = 15$ meters.
• Finding what remains: A class has 25 students, and 3 are absent. That means $25 - 3 = 22$ students are present.
• Measuring change over time: A town's population drops from 10,000 to 9,500. The decrease is $10{,}000 - 9{,}500 = 500$ people.
• Finding distance between two points: Cities sit at mile markers 120 and 95. The distance between them is $120 - 95 = 25$ miles.

Understanding the base-10 system and its role in subtraction

The base-10 system is what makes the standard subtraction algorithm work. Each digit's value depends on its place value: ones, tens, hundreds, thousands, and so on. Each place is worth 10 times the place to its right.

This matters for subtraction because regrouping depends on that ×10 relationship. When you "borrow" from the tens column, you're really converting 1 ten into 10 ones. When you borrow from the hundreds column, you're converting 1 hundred into 10 tens. Without understanding place value, regrouping is just a memorized trick. With it, you can see why the borrowing process actually works.

One more useful fact: subtraction and addition are inverse operations. You can always check a subtraction answer by adding. If $52 - 17 = 35$, then $35 + 17$ should equal $52$. This is a reliable way to catch mistakes.