Multiplying a Mixed Number and a Whole Number

Learn to multiply mixed numbers by whole numbers using the convert-multiply-convert method

1. The Method: Convert → Multiply → Convert

Three Steps

Convert the mixed number to an improper fraction
Multiply the improper fraction by the whole number
Convert the result back to a mixed number

Example: $1\frac{1}{2} \times 3$

Visual: 3 groups of $1\frac{1}{2}$

3 wholes + 3 halves = 3 + $1\frac{1}{2}$ = $4\frac{1}{2}$

Step 1: Convert $1\frac{1}{2}$ to improper fraction

$1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$

Step 2: Multiply by 3

$\frac{3}{2} \times 3 = \frac{3 \times 3}{2} = \frac{9}{2}$

Step 3: Convert back to mixed number

$\frac{9}{2} = 4\frac{1}{2}$ (9 ÷ 2 = 4 remainder 1)

Answer: $4\frac{1}{2}$

Reminder: Converting Mixed to Improper

$3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$

Multiply whole by denominator, add numerator, keep same denominator.

2. More Examples

Example: $6 \times 1\frac{2}{9}$

Step 1: $1\frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{11}{9}$

Step 2: $6 \times \frac{11}{9} = \frac{66}{9}$

Step 3: $\frac{66}{9} = 7\frac{3}{9} = 7\frac{1}{3}$ (simplified)

Answer: $7\frac{1}{3}$

Example with Cancellation: $4 \times 2\frac{1}{6}$

Step 1: $2\frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{13}{6}$

Step 2: 4 and 6 have a common factor 2. Divide both by 2, then multiply.

$\displaystyle \overset{2}{\cancel{4}\vphantom{\Big|}} \times \frac{13}{\underset{3}{\cancel{6}\vphantom{\big|}}} = \frac{2 \times 13}{3} = \frac{26}{3}$

Step 3: $\frac{26}{3} = 8\frac{2}{3}$ (26 ÷ 3 = 8 remainder 2)

Answer: $8\frac{2}{3}$

3. Word Problems

Example: Rope Problem

The length of a piece of rope is $2\frac{1}{5}$ m. What is the total length of 4 such pieces of rope?

Solution: 4 × $2\frac{1}{5}$

= 4 × $\frac{11}{5}$

= $\frac{44}{5}$

= $8\frac{4}{5}$ m

The total length of 4 pieces of rope is $8\frac{4}{5}$ m.

Example: Sharing Pies

Mrs Li has 3 children. She gives each child $1\frac{1}{2}$ pies. How many pies does she give them altogether?

Solution: 3 × $1\frac{1}{2}$

= 3 × $\frac{3}{2}$

= $\frac{9}{2}$

= $4\frac{1}{2}$ pies

She gives them $4\frac{1}{2}$ pies altogether.

4. Practice Problems

Practice 1: Calculate $5 \times 2\frac{3}{4}$

ShowHide solution

$2\frac{3}{4} = \frac{11}{4}$

$5 \times \frac{11}{4} = \frac{55}{4} = 13\frac{3}{4}$

Answer: $13\frac{3}{4}$

Practice 2: Calculate $8 \times 3\frac{1}{6}$

ShowHide solution

$3\frac{1}{6} = \frac{19}{6}$

$8 \times \frac{19}{6} = \frac{152}{6} = \frac{76}{3} = 25\frac{1}{3}$

Answer: $25\frac{1}{3}$

Practice 3: A container holds $1\frac{3}{8}$ litres of juice. How much juice is in 6 such containers?

ShowHide solution

$1\frac{3}{8} = \frac{11}{8}$

$6 \times \frac{11}{8} = \frac{66}{8} = \frac{33}{4} = 8\frac{1}{4}$ litres

There are $8\frac{1}{4}$ litres of juice in 6 containers.

Practice 4: Each bag of rice has a mass of $2\frac{2}{3}$ kg. What is the total mass of 9 bags?

ShowHide solution

$2\frac{2}{3} = \frac{8}{3}$

$9 \times \frac{8}{3} = \frac{72}{3} = 24$ kg

The total mass of 9 bags is 24 kg.

Key Takeaways

Three-step method: Convert → Multiply → Convert
Mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator. For example, $3\frac{2}{5} = \frac{17}{5}$
Multiply fraction by whole: multiply the numerator by the whole number, keep the denominator. For example, $\frac{17}{5} \times 2 = \frac{17 \times 2}{5} = \frac{34}{5}$
Improper to mixed: Divide numerator by denominator
Use cancellation when possible to simplify calculations
Always simplify final answers
Include units in word problem answers

Multiplying a Mixed Number and a Whole Number

1. The Method: Convert → Multiply → Convert

Three Steps

Example: 112×31\frac{1}{2} \times 3121​×3

Reminder: Converting Mixed to Improper

2. More Examples

Example: 6×1296 \times 1\frac{2}{9}6×192​

Example with Cancellation: 4×2164 \times 2\frac{1}{6}4×261​

3. Word Problems

Example: Rope Problem

Example: Sharing Pies

4. Practice Problems

Practice 1: Calculate 5×2345 \times 2\frac{3}{4}5×243​

Practice 2: Calculate 8×3168 \times 3\frac{1}{6}8×361​

Practice 3: A container holds 1381\frac{3}{8}183​ litres of juice. How much juice is in 6 such containers?

Practice 4: Each bag of rice has a mass of 2232\frac{2}{3}232​ kg. What is the total mass of 9 bags?

Key Takeaways

Example: $1\frac{1}{2} \times 3$

Example: $6 \times 1\frac{2}{9}$

Example with Cancellation: $4 \times 2\frac{1}{6}$

Practice 1: Calculate $5 \times 2\frac{3}{4}$

Practice 2: Calculate $8 \times 3\frac{1}{6}$

Practice 3: A container holds $1\frac{3}{8}$ litres of juice. How much juice is in 6 such containers?

Practice 4: Each bag of rice has a mass of $2\frac{2}{3}$ kg. What is the total mass of 9 bags?