## What you need to know

Things to remember:

• Rational numbers are whole numbers, fractions, and decimals. Being able to convert easily between the 3 will make life easier.

There are three points to this topic we need to remember:

• How we compare numbers (inequalities)
• How to convert fractions to decimals.
• How to convert decimals to fractions.

There are 5 symbols used for comparing numbers:

< less than                             > greater than                                    = equals

$\leq$ less than or equal to       $\geq$ greater than or equal to

So, instead of saying something like “6 is less than 11”, we can write this using our inequalities:

6 < 11

Or, instead of “13 is bigger than 7”, we could write:

13 > 7

Converting fractions into decimals is quite nice, and we have two ways of doing this:

Treat them as divisions.

$$\frac{5}{8}=5\div8=0.625$$

$$\frac{9}{10}=9\div10=0.9$$

Or by remembering the conversions.

Write $\dfrac{3}{5}$ as a decimal.

$$\frac{1}{5}=0.2$$

$$\frac{3}{5}=3\times\frac{1}{5}=3\times0.2=0.6$$

Converting decimals to fractions is a little trickier, but not too bad, and we can do it in three steps.

Write 0.245 as a fraction.

Step 1: Write as a fraction over 1.

$$\frac{0.245}{1}$$

Step 2: Multiply the top and bottom until of the fraction by the power of 10 with as many 0s as there are decimal places.

$$\frac{0.245}{1}=\frac{0.245\times1000}{1\times1000}=\frac{245}{1000}$$

$$\frac{0.245}{1}=\frac{245}{1000}$$

Step 3: Simplify

$$\frac{245}{1000} =\frac{245\div5}{1000\div5}=\frac{49}{200}$$

Now that we’ve recapped what inequalities are and how to convert between fractions and decimals, we can start comparing them. To compare, we need to make them both decimals or both fractions.

Fill in the missing inequality $\dfrac{13}{25}\_\_\_0.65$.

Converting to decimals.

Step 1: Convert so that they are both decimals.

$$\frac{13}{25}=13\div25=0.52$$

Step 2: Compare the two decimals.

$$0.52\_\_\_0.65$$

0.52 is clearly less that 0.65, so our missing inequality is “<”.

$$0.52<0.65$$

Converting to Fractions.

Step 1: Convert so that they are either both fractions.

$$0.65=\frac{0.65}{1}=\frac{0.65\times100}{1\times100}=\frac{65}{100}$$

Step 2: Make the denominators the same.

Hint: Sometimes it can be easier to simplify the fraction down before looking for a common denominator.

We can turn the denominator of $\dfrac{13}{25}$ into 100.

$$\frac{13}{25}=\frac{13\times4}{25\times4}=\frac{52}{100}$$

Step 3: Compare the fractions.

$$\frac{52}{100}\_\_\_\frac{65}{100}$$

$\dfrac{52}{100}$ is less than $\dfrac{65}{100}$, so our missing inequality is “<”. This is the same as when we converted to decimals, so it doesn’t matter!

Note: Converting to fractions does have an extra step of making the denominators the same, so this method can be more time consuming that converting to decimals.

## Example Questions

#### Question 1: Fill in the missing inequality $\dfrac{13}{40}\_\_\_0.33$.

Converting to decimals.

Step 1: Convert so that they are both decimals.

$$\frac{13}{40}=13\div40=0.325$$

Step 2: Compare the two decimals.

$$0.325\_\_\_0.33$$

We know that 0.325 is less than 0.33, so the missing inequality it “<”.

$$0.325<0.33$$

Converting to Fractions.

Step 1: Convert so that they are either both fractions.

$$0.33=\frac{0.33}{1}=\frac{0.33\times100}{1\times100}=\frac{33}{100}$$

Step 2: Make the denominators the same.

Hint: Sometimes it can be easier to simplify the fraction down before looking for a common denominator.

We can turn both denominators into 200.

$$\frac{13}{40}=\frac{13\times5}{40\times5}=\frac{65}{200}$$

$$\frac{33}{100}=\frac{33\times2}{100\times2}=\frac{66}{200}$$

Step 3: Compare the fractions.

$$\frac{65}{200}\_\_\_\frac{66}{200}$$

Again, we can see that the missing inequality is “<”

$$\frac{65}{200}<\frac{66}{200}$$

#### Question 2: Fill in the missing inequality $\dfrac{7}{15}\_\_\_0.24$.

Converting to decimals.

Step 1: Convert so that they are both decimals.

$$\frac{7}{15}=7\div15=0.\overline{46}$$

Step 2: Compare the two decimals.

$$0.\overline{46}\_\_\_0.24$$

Although we have a recurring decimal, looking at the values of the tenths, we can see that the missing inequality is “>”.

$$0.\overline{46}>0.24$$

Converting to Fractions.

Step 1: Convert so that they are either both fractions.

$$0.24=\frac{0.24}{1}=\frac{0.24\times100}{1\times100}=\frac{24}{100}$$

Step 2: Make the denominators the same.

Hint: Sometimes it can be easier to simplify the fraction down before looking for a common denominator.

We can turn both denominators into 300.

$$\frac{7}{15}=\frac{7\times20}{15\times20}=\frac{140}{300}$$

$$\frac{24}{100}=\frac{24\times3}{100\times3}=\frac{72}{100}$$

Step 3: Compare the fractions.

$$\frac{140}{300}\_\_\_\frac{72}{300}$$

Again, we can see that the missing inequality is “>”

$$\frac{140}{300}>\frac{72}{300}$$