Working with Fractions

Chapter Overview

Fractions are used in distances, recipes, time, money and measurement. In this chapter we extend fraction operations to multiplication and division.

1. Multiplying a Whole Number by a Fraction

Multiplication can represent repeated addition.

For example:

2. Fraction of a Quantity

The word of usually suggests multiplication.

For example:

3. Multiplying Two Fractions

4. Cancellation

Before multiplying, common factors in numerators and denominators can be cancelled.

5. What Happens to the Size of a Product?

If a positive number is multiplied by a fraction between 0 and 1, the product becomes smaller.

For example:

If multiplied by a number greater than 1, the product becomes larger.

6. Reciprocal

A fraction multiplied by its reciprocal equals 1:

The reciprocal of a whole number $5$ is:

7. Division of Fractions

To divide by a fraction:

1. keep the dividend;
2. take the reciprocal of the divisor;
3. multiply.

8. Dividing by a Whole Number

A whole number can be written as a fraction with denominator 1.

For example:

9. Size of a Quotient

If a positive number is divided by a fraction less than 1, the result becomes larger.

For example:

If it is divided by a number greater than 1, the result becomes smaller:

10. Fraction Word Problems

Key Relations

11. Understanding Multiplication of Fractions

Multiplying by a fraction can be understood as taking a part of a part.

For example:

means taking $\frac{2}{3}$ of $\frac{3}{5}$. Multiply numerators and denominators:

12. Mixed Numbers in Multiplication and Division

Before multiplying or dividing mixed numbers, convert them to improper fractions.

For example:

and:

Then use ordinary fraction multiplication or division.

13. Why Division Uses the Reciprocal

Division asks how many groups of one quantity fit into another. For fractions, multiplying by the reciprocal gives the same result.

The second fraction is inverted because we are replacing division by multiplication with its multiplicative inverse.

14. Predicting the Size of the Answer

Number sense can tell us whether an answer is reasonable.

• Multiplying a positive number by a fraction less than 1 makes it smaller.
• Multiplying by a fraction greater than 1 makes it larger.
• Dividing by a fraction less than 1 makes a positive number larger.
• Dividing by a fraction greater than 1 makes a positive number smaller.

15. Choosing the Operation in Word Problems

Words alone do not determine the operation. Think about the relationship.

• “of” often suggests multiplication: $\frac{3}{4}$ of 20.
• “shared equally” often suggests division.
• “how many pieces of size...” is a division situation.
• Finding one fractional part of another can involve multiplication.

16. Worked Fraction Notes

Fraction of a Quantity

To find a fraction of a quantity, multiply the quantity by the fraction.

For example, $\frac{3}{8}$ of 64 is:

A helpful strategy is to divide by the denominator first when possible, then multiply by the numerator.

Fraction × Fraction

The rule:

comes from taking one fractional part of another. Always simplify the final fraction, and use cancellation before multiplication when possible.

Whole Numbers as Fractions

Any whole number can be written with denominator 1:

This makes multiplication and division rules consistent.

Reciprocal

The reciprocal of a non-zero fraction $\frac{a}{b}$ is $\frac{b}{a}$. Their product is 1:

Zero has no reciprocal because no number multiplied by 0 gives 1.

Division as “How Many?”

Consider:

This asks: how many halves fit into 3 wholes? Each whole contains two halves, so the answer is 6.

This interpretation explains why dividing by a fraction smaller than 1 can give a larger result.

Mixed-number Problems

Convert mixed numbers before operating:

For example:

Reasonableness Checks

Before accepting a fraction answer, ask:

• Is the result expected to be less than or greater than the starting number?
• Is the fraction already in simplest form?
• Does the unit make sense?
• If division was used, can multiplication check the answer?

17. More Fraction Applications

Fraction Unit Check

When a quantity such as length, mass or time is involved, carry the unit through the reasoning. A numerical answer without a unit can be incomplete.

Common Simplification Habit

Always inspect numerator-denominator common factors before multiplication. This can turn a difficult product into simple mental arithmetic.

Important Terms

Common Mistakes

Exam Focus

Quick Revision

Practice Questions

Additional Practice

Chapter Summary