Multiplication Basics

Multiplication as Repeated Addition

See why 5 × 3 equals 3 + 3 + 3 + 3 + 3. Counting five groups of 3 shows that multiplication is just a faster way to add the same number over and over.

Example

Show 5 x 3 by adding five groups of 3. Repeated addition is exact for whole numbers only; once fractions and decimals appear in later chapters, multiplication is extended beyond it.

highlighted = computed this step

Step 1 — Set up

Start with equal groups.

5×3=□5\times 3=\square
Start with 5 equal groups of 3.group 13group 23group 33group 43group 53Each group contributes 3; add one group at a time.3691215totals

Step 2 — Group 1

Group 1: add 3, total 3.

3×1=33\times 1=\hl{3}
Add group 1: total 3.group 13group 23group 33group 43group 533 = 33691215totals

Step 3 — Group 2

Group 2: add 3, total 6.

3×2=63\times 2=\hl{6}
Add group 2: total 6.group 13group 23group 33group 43group 533 + 3 = 63691215totals

Step 4 — Group 3

Group 3: add 3, total 9.

3×3=93\times 3=\hl{9}
Add group 3: total 9.group 13group 23group 33group 43group 533 + 3 + 3 = 93691215totals

Step 5 — Group 4

Group 4: add 3, total 12.

3×4=123\times 4=\hl{12}
Add group 4: total 12.group 13group 23group 33group 43group 533 + 3 + 3 + 3 = 123691215totals

Step 6 — Group 5

Group 5: add 3, total 15.

3×5=153\times 5=\hl{15}
Add group 5: total 15.group 13group 23group 33group 43group 533 + 3 + 3 + 3 + 3 = 153691215totals

Step 7 — Result

So 5 groups of 3 make 15.

5×3=155\times 3=15
Read the product: 5 groups of 3 make 15.group 13group 23group 33group 43group 533 + 3 + 3 + 3 + 3 = 153691215totals
repeated-addition Multiplying a × b means adding b to itself a times. Each group adds one more copy of b, and the total grows by b each step.