1.5 Divide Whole Numbers

Division of Whole Numbers

Division is the process of splitting a number into equal parts or groups. It's one of the four basic operations, and you'll use it constantly in fractions, ratios, and algebra, so getting comfortable with it now pays off.

Division Notation

There are two common ways to write division:

• Division symbol $\div$: Write the dividend (the number being divided) on the left and the divisor (the number you're dividing by) on the right. Example: $12 \div 3$
• Fraction bar: Place the dividend on top and the divisor on the bottom. Example: $\frac{12}{3}$

Both of these mean the same thing: "split 12 into 3 equal parts." The answer you get is called the quotient.

Visual Models for Division

Arrays arrange objects in equal rows and columns. If you have 12 objects and arrange them into 3 equal rows, you get 4 objects per row. The total (12) is the dividend, the number of rows (3) is the divisor, and the number per row (4) is the quotient.

Number lines work by making equal jumps from zero up to the dividend.

• Start at 0 and make jumps of size 3 (the divisor)
• Count how many jumps it takes to land on 12 (the dividend)
• It takes 4 jumps, so $12 \div 3 = 4$

Dividing Whole Numbers

For simple problems, you can divide in your head: $12 \div 3 = 4$.

When division doesn't come out evenly, you get a remainder. For example, $14 \div 3 = 4$ remainder $2$, because 3 goes into 14 four times (that's 12) with 2 left over.

Checking your answer: Multiply the quotient by the divisor, then add the remainder. The result should equal the dividend.

$4 \times 3 + 2 = 14$ ✓

For larger numbers, use long division. Here's the process for $156 \div 12$:

1. Ask how many times 12 goes into the first part of the dividend. 12 goes into 15 one time.

2. Write 1 above, then multiply: $1 \times 12 = 12$. Subtract: $15 - 12 = 3$.

3. Bring down the next digit (6) to get 36.

4. Ask how many times 12 goes into 36. That's 3 times exactly.

5. Write 3 above, multiply: $3 \times 12 = 36$. Subtract: $36 - 36 = 0$.

6. No remainder, so $156 \div 12 = 13$.

Turning Word Problems into Division

When you read a word problem, look for these clues:

• Dividend: the total amount (e.g., 15 candies)
• Divisor: the number of equal groups or the size of each group (e.g., 3 friends)
• Quotient: what you're solving for (e.g., candies per friend)

Write the division expression and solve: $15 \div 3 = 5$ candies per friend.

Real-World Applications

Division shows up in two main types of problems:

• Equal sharing: You know the total and the number of groups, and you need the size of each group. 20 stickers shared among 4 friends: $20 \div 4 = 5$ stickers each.
• Finding the number of groups: You know the total and the group size, and you need how many groups. 50 seats with 2 seats per row: $50 \div 2 = 25$ rows.

Properties of Division

• Zero property: Dividing by zero is undefined. You simply cannot do it. (But zero divided by any nonzero number equals zero: $0 \div 5 = 0$.)
• Identity property: Any number divided by 1 equals itself. $a \div 1 = a$
• Inverse of multiplication: Division "undoes" multiplication. If $4 \times 3 = 12$, then $12 \div 3 = 4$.
• Not commutative: Order matters. $12 \div 3 = 4$, but $3 \div 12$ does not equal 4.