027. Multiplying Fractions, Mixed Numerals and Whole Numbers

Learning Intentions

• To understand that multiplying fractions is easier if you first cancel common factors from numerators and denominators in each fraction
• To know that a whole number can be written as a fraction with a denominator of $1$
• multiply fractions, mixed numerals and/or whole numbers, giving an answer in simplest form

Pre-requisite Summary

• Understand that a fraction can represent division and part of a whole
• Know the meaning of numerator and denominator
• Be able to find factors of whole numbers
• Understand equivalent fractions and simplest form
• Know that a whole number such as $4$ can be written as ${\displaystyle \frac{4}{1}}$
• Be able to convert mixed numerals to improper fractions
• Be able to simplify fractions by dividing numerator and denominator by a common factor
• Recall multiplication facts for small whole numbers

Worked Examples

Worked Example 1

a) Write each whole number as a fraction with denominator $1$: $3,7,12$
b) Explain why this does not change the value of the number.

Worked Example 2

Multiply and simplify:

a) ${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{5}{8}}$
b) ${\displaystyle \frac{4}{9}}\times {\displaystyle \frac{3}{10}}$
c) State which common factors can be cancelled first.

Worked Example 3

Multiply by first cancelling common factors:

a) ${\displaystyle \frac{6}{7}}\times {\displaystyle \frac{14}{15}}$
b) ${\displaystyle \frac{12}{25}}\times {\displaystyle \frac{10}{18}}$

Worked Example 4

Multiply a fraction and a whole number:

a) ${\displaystyle \frac{3}{4}}\times 8$
b) $5\times {\displaystyle \frac{2}{9}}$
c) Write the whole number as a fraction first.

Worked Example 5

Convert mixed numerals to improper fractions, then multiply:

a) $1{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{2}{3}}$
b) $2{\displaystyle \frac{1}{4}}\times 3$

Worked Example 6

Multiply mixed numerals and give the answer in simplest form:

a) $1{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{1}{4}}$
b) $3{\displaystyle \frac{1}{5}}\times 1{\displaystyle \frac{1}{2}}$

Problems

Problem 1

a) Write each whole number as a fraction with denominator $1$: $4,9,15$
b) Explain why this does not change the value of the number.

Problem 2

Multiply and simplify:

a) ${\displaystyle \frac{3}{5}}\times {\displaystyle \frac{4}{9}}$
b) ${\displaystyle \frac{7}{12}}\times {\displaystyle \frac{2}{3}}$
c) State which common factors can be cancelled first.

Problem 3

Multiply by first cancelling common factors:

a) ${\displaystyle \frac{8}{9}}\times {\displaystyle \frac{3}{16}}$
b) ${\displaystyle \frac{15}{28}}\times {\displaystyle \frac{14}{25}}$

Problem 4

Multiply a fraction and a whole number:

a) ${\displaystyle \frac{5}{6}}\times 12$
b) $7\times {\displaystyle \frac{3}{14}}$
c) Write the whole number as a fraction first.

Problem 5

Convert mixed numerals to improper fractions, then multiply:

a) $1{\displaystyle \frac{3}{4}}\times {\displaystyle \frac{4}{7}}$
b) $2{\displaystyle \frac{2}{3}}\times 6$

Problem 6

Multiply mixed numerals and give the answer in simplest form:

a) $1{\displaystyle \frac{1}{2}}\times 2{\displaystyle \frac{2}{3}}$
b) $2{\displaystyle \frac{3}{4}}\times 1{\displaystyle \frac{1}{3}}$

Exercises

Understanding and Fluency

1. Write each whole number as a fraction with denominator $1$:
   a) $2$
   b) $6$
   c) $11$

2. Multiply and simplify:
   a) ${\displaystyle \frac{1}{2}}\times {\displaystyle \frac{3}{5}}$
   b) ${\displaystyle \frac{2}{7}}\times {\displaystyle \frac{7}{9}}$
   c) ${\displaystyle \frac{4}{11}}\times {\displaystyle \frac{3}{8}}$

3. Multiply by cancelling common factors first:
   a) ${\displaystyle \frac{6}{10}}\times {\displaystyle \frac{5}{9}}$
   b) ${\displaystyle \frac{8}{15}}\times {\displaystyle \frac{3}{4}}$
   c) ${\displaystyle \frac{14}{25}}\times {\displaystyle \frac{5}{21}}$

4. Multiply a fraction and a whole number:
   a) ${\displaystyle \frac{3}{8}}\times 16$
   b) $9\times {\displaystyle \frac{2}{3}}$
   c) ${\displaystyle \frac{5}{12}}\times 24$

5. Convert mixed numerals to improper fractions, then multiply:
   a) $1{\displaystyle \frac{1}{3}}\times {\displaystyle \frac{3}{5}}$
   b) $2{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{4}{9}}$
   c) $3{\displaystyle \frac{1}{4}}\times 2$

6. Multiply mixed numerals and simplify:
   a) $1{\displaystyle \frac{1}{2}}\times 1{\displaystyle \frac{1}{3}}$
   b) $2{\displaystyle \frac{1}{4}}\times 1{\displaystyle \frac{1}{2}}$
   c) $3{\displaystyle \frac{2}{5}}\times 2{\displaystyle \frac{1}{3}}$

7. Mixed practice:
   a) ${\displaystyle \frac{7}{8}}\times 4$
   b) $2\times {\displaystyle \frac{5}{12}}$
   c) ${\displaystyle \frac{9}{10}}\times {\displaystyle \frac{5}{6}}$

8. Mixed practice:
   a) $1{\displaystyle \frac{2}{5}}\times {\displaystyle \frac{10}{21}}$
   b) $2{\displaystyle \frac{3}{4}}\times 8$
   c) $1{\displaystyle \frac{1}{6}}\times 2{\displaystyle \frac{2}{5}}$

Reasoning

9. Explain why writing $5$ as ${\displaystyle \frac{5}{1}}$ is valid.

10. A student says ${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{4}{5}}={\displaystyle \frac{8}{15}}$ but does not simplify first. Explain why cancelling first can make the calculation easier.

11. A student cancels the $2$ in ${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{5}{8}}$ with the $3$ in the same fraction and gets ${\displaystyle \frac{1}{1}}\times {\displaystyle \frac{5}{8}}$. Explain the mistake.

12. Explain why mixed numerals are usually converted to improper fractions before multiplying.

Problem-solving

13. A recipe needs ${\displaystyle \frac{3}{4}}$ cup of milk for one batch. How much milk is needed for $6$ batches?

14. A rope that is $2{\displaystyle \frac{1}{2}}$ m long is cut into pieces each of length ${\displaystyle \frac{2}{3}}$ of a metre for a craft display. Find the product $2{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{2}{3}}$.

15. A garden bed is $1{\displaystyle \frac{1}{2}}$ m wide and $2{\displaystyle \frac{1}{3}}$ m long. Find the product of these measures.

16. A student completes ${\displaystyle \frac{3}{5}}$ of a worksheet, then completes ${\displaystyle \frac{1}{2}}$ of that amount again for revision. Find ${\displaystyle \frac{3}{5}}\times {\displaystyle \frac{1}{2}}$.

17. A machine makes $2{\displaystyle \frac{1}{4}}$ trays each hour for $4$ hours. Find the total number of trays made.

18. A ribbon of length $3{\displaystyle \frac{1}{3}}$ m is used to make decorations, each using ${\displaystyle \frac{3}{4}}$ of a metre in a design calculation. Find the product $3{\displaystyle \frac{1}{3}}\times {\displaystyle \frac{3}{4}}$.

Potential Misunderstandings

• Students may try to add numerators and denominators instead of multiplying
• Students may not recognise that a whole number can be written as a fraction with denominator $1$
• Students may cancel numbers that are being added rather than factors being multiplied
• Students may cancel within a single numerator and denominator incorrectly instead of across factors in a product
• Students may forget to convert mixed numerals to improper fractions before multiplying
• Students may multiply correctly but leave the answer unsimplified
• Students may simplify using numbers that are not common factors
• Students may forget that cancelling common factors does not change the value of the product

Next: 028. Dividing Fractions Using Reciprocals