Basic Multiplication and Division Relationships

Basic Multiplication and Division Relationships

Multiplication counts equal groups. Division can find the size of each group or the number of groups. A fact family connects the same three numbers with multiplication and division.

How to solve these problems

If a × b = c, then c ÷ a = b and c ÷ b = a when the divisors are nonzero. Division by zero is undefined. Keep units attached to rates.

Worked Example 1: 4 groups of 3

4 groups of 3

Solution: 4 × 3 = 12. Repeated addition gives 3 + 3 + 3 + 3 = 12.

Why it works: Equal-sized groups allow multiplication to replace repeated addition.

Worked Example 2: 12 objects shared among 4 people

12 objects shared among 4 people

Solution: 12 ÷ 4 = 3 objects per person. Check: 4 × 3 = 12.

Why it works: The total and number of groups are known; division finds group size.

Worked Example 3: 12 objects packed 3 per box

12 objects packed 3 per box

Solution: 12 ÷ 3 = 4 boxes.

Why it works: The total and group size are known; division finds the number of groups.

Worked Example 4: Write the fact family for 6, 7, 42

Write the fact family for 6, 7, 42

Solution: 6 × 7 = 42; 7 × 6 = 42; 42 ÷ 6 = 7; 42 ÷ 7 = 6.

Why it works: Multiplication is commutative, and division reverses either multiplication fact.

Worked Example 5: 936 flashcards divided into 18 equal sets

936 flashcards divided into 18 equal sets

Solution: 936 ÷ 18 = 52 flashcards per set. Check: 18 × 52 = 936.

Why it works: 18 × 50 = 900 and 18 × 2 = 36; together these account for all 936 cards.

Worked Example 6: 35 ÷ 4 with a remainder

35 ÷ 4 with a remainder

Solution: 4 × 8 = 32; 35 − 32 = 3. Answer: 8 remainder 3, or 8¾.

Why it works: The remainder is smaller than the divisor. Check: 4 × 8 + 3 = 35.

Worked Example 7: 0 ÷ 5

0 ÷ 5

Solution: 0 ÷ 5 = 0. Check: 5 × 0 = 0.

Why it works: Zero items shared among five groups leaves zero in each group.

Worked Example 8: 5 ÷ 0

5 ÷ 0

Solution: There is no answer: 5 ÷ 0 is undefined.

Why it works: No number multiplied by zero can equal 5. Even 0 ÷ 0 is undefined because it has no unique quotient.

Worked Example 9: Six notebooks cost $15. Find the price per notebook.

Six notebooks cost $15. Find the price per notebook.

Solution: $15 ÷ 6 = $2.50 per notebook. Check: 6 × $2.50 = $15.

Why it works: Division creates a unit rate, the amount for one item.

Worked Example 10: A van holds 8 people. How many vans carry 34 people?

A van holds 8 people. How many vans carry 34 people?

Solution: 34 ÷ 8 = 4 remainder 2. Four vans seat only 32 people, so 5 vans are needed.

Why it works: The remainder represents people still needing seats, so the real-world answer rounds up.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.