1/15
00:00

S.3 Relationship between multiplication and division

Loading questions...

The connection between multiplication and division

Multiplication and division are inverse operations. This means they undo each other, just like addition and subtraction. If you know a multiplication fact, you automatically know two related division facts.

Example:
  • If 6 rows of 4 columns equals 24 (6 × 4 = 24), then:
  • 24 divided into 6 equal groups gives 4 in each group (24 ÷ 6 = 4).
  • 24 divided into 4 equal groups gives 6 in each group (24 ÷ 4 = 6).
Note

Think of a rectangle. The total number of squares (product) can be divided by the number of rows to find the columns, or by the number of columns to find the rows.

Understanding fact families

A fact family is a group of four related math facts created from the same three numbers. For multiplication and division, these three numbers are the two factors and their product.

Example with 5, 7, and 35:
  • 5 × 7 = 35
  • 7 × 5 = 35
  • 35 ÷ 5 = 7
  • 35 ÷ 7 = 5
Note

Every fact family shows the same relationship in four different ways. Knowing one fact helps you know all four.

Using arrays to see the relationship

An array is an arrangement of objects in rows and columns. It is a powerful tool to visualize how multiplication and division are connected.

Array example (3 rows, 6 columns):
  • Multiplication: 3 rows × 6 columns = 18 total objects.
  • Division (by rows): 18 objects ÷ 3 rows = 6 objects in each column.
  • Division (by columns): 18 objects ÷ 6 columns = 3 objects in each row.
Note

The array doesn't change. Whether you count the total (multiplication) or split the total into equal groups (division), you are describing the same arrangement.

Solving for unknowns using the relationship

If a number in a multiplication or division equation is missing, you can use the inverse operation to find it. This is often called solving for an unknown.

Examples:
  • Find the missing factor: 7 × ? = 42. Think: 42 ÷ 7 = 6. So, ? = 6.
  • Find the missing dividend: ? ÷ 4 = 5. Think: 5 × 4 = 20. So, ? = 20.
  • Find the missing divisor: 28 ÷ ? = 7. Think: 28 ÷ 7 = 4. So, ? = 4.
Note

Ask yourself: "What operation would undo this problem?" Use the fact family to help you rearrange the numbers and find the answer.

Key patterns to remember

Recognizing patterns can make working with multiplication and division facts up to 7×7 easier and faster.

Important patterns:
  • Any number multiplied by 1 equals itself (4 × 1 = 4). Dividing a number by 1 also equals itself (4 ÷ 1 = 4).
  • Any number multiplied by 0 equals 0 (7 × 0 = 0). 0 divided by any number (except 0) is 0 (0 ÷ 5 = 0).
  • Dividing a number by itself always equals 1 (6 ÷ 6 = 1).
Note

These special rules are consistent for all numbers. Memorizing them helps you solve problems quickly and check your work for reasonableness.