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7 The real numbers
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Stage 4

7.07 Fractions and decimals

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Lesson

Introduction

We will now combine everything we have learned about  directed numbers and the order of operations  with our knowledge of fractions and decimals.

Compare and order rational directed numbers

The symbol < represents the phrase is less than. For example, -\dfrac{3}{2} is less than \dfrac{3}{4} can be represented by -\dfrac{3}{2}<\dfrac{3}{4}.

The symbol > represents the phrase is greater than. For example, \dfrac{4}{3} is greater than -\dfrac{2}{3} can be represented by \dfrac{4}{3}>-\dfrac{2}{3}.

On the number line below, each tick is labelled with a multiple of the fraction \dfrac{1}{5}. We can see that the point further to the left is plotted at the fraction -\dfrac{3}{5}, and the point further to the right is plotted at the fraction \dfrac{6}{5}.

-\frac{7}{5}-\frac{6}{5}-1-\frac{4}{5}-\frac{3}{5}-\frac{2}{5}-\frac{1}{5}0\frac{1}{5}\frac{2}{5}\frac{3}{5}\frac{4}{5}1\frac{6}{5}\frac{7}{5}

This means \dfrac{6}{5} is greater than -\dfrac{3}{5}. It is the numbers' positions on the number line that helps us decide which number is greater (not their magnitudes).

Examples

Example 1

Which decimal is greater?

A
-0.74
B
0.8
Worked Solution
Create a strategy

Plot the numbers on the number line.

Recall that the number further to the right is always greater than the number to the left.

Apply the idea

Here are the decimals -0.74 and 0.8 on the number line.

-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.9

We can see from the number line above that 0.8 is to the right of -0.74.

So 0.8 is greater than -0.74. The correct answer is option B.

Idea summary

The symbol < represents the phrase is less than.

The symbol > represents the phrase is greater than.

Multiply and divide rational directed numbers

We follow the exact same rules as before, we just need to take care when dealing with negative numbers.

We can divide fractions with keep, change, flip.

  • Keep the first fraction the same

  • Change division to multiplication

  • Flip the second fraction to the reciprocal

Examples

Example 2

Evaluate: \dfrac{2}{7}\div\dfrac{5}{3}.

Worked Solution
Create a strategy

Use the keep, change, flip process to divide by a fraction. A positive divided by a positive is a positive.

Apply the idea
\displaystyle \dfrac{2}{7}\div\dfrac{5}{3}\displaystyle =\displaystyle \dfrac{2}{7}\times\dfrac{3}{5}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{2\times 3}{7 \times 5}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{6}{35}Evaluate

Example 3

Evaluate: -\dfrac{2}{5}\times\left(-\dfrac{9}{7}\right).

Worked Solution
Create a strategy

Multiply numerators and denominators separately.

The product of two negative integers is a positive integer.

Apply the idea
\displaystyle -\dfrac{2}{5}\times\left(-\dfrac{9}{7}\right)\displaystyle =\displaystyle \dfrac{-2\times-9}{5\times7}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{18}{35}Evaluate
Reflect and check

It is useful to know whether to expect a positive or negative answer before evaluating, as this can help us know if we have made a mistake if we end up with an answer that is not the expected sign.

Idea summary

We can divide fractions with keep, change, flip.

  • Keep the first fraction the same

  • Change division to multiplication

  • Flip the second fraction to the reciprocal

Multiply and divide directed decimals

As with fractions, we follow the same rules as before, taking into account if our numbers are positive and/or negative to decide whether our answer will be positive or negative.

For example, to find -0.5\times (-0.3) we first multiply 5 and 3 to get 15. Since the question had two decimal places we insert the decimal point into our answer so that it also has two decimal places to get 0.15. And since a negative times a negative gives a positive, this is our final answer.

Examples

Example 4

Evaluate the quotient 7.36\div(-0.08).

Worked Solution
Create a strategy

Multiply both numbers by a power of 10 to make them whole numbers then perform the division.

The quotient of a negative and a positive integer is a negative integer.

Apply the idea
\displaystyle 7.36\div(-0.08)\displaystyle =\displaystyle (7.36\times100)\div(-0.08\times100)Multiply both decimals by 100
\displaystyle =\displaystyle 736\div(-8)Evaluate multiplication
\displaystyle =\displaystyle -92Evaluate division
Idea summary

The reciprocal of a number is 1 divided by that number.

  • The reciprocal of a whole number is 1 over that number.

  • The reciprocal of a fraction can be found by swapping the numerator and denominator.

The magnitude of a number is its distance from zero.

  • For example, 6 and -6 have the same magnitude.

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

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