# Part 1: Fractions | Beginner’s Guide to Year 7 Maths

In this article, we will show you how to solve a variety of questions dealing with fractions, give you worked examples and provide some checkpoint questions at the end!

Are your fraction skills a little rusty? Don’t fear! In this article, we will guide you through everything you need to know about Year 7 fractions.

## Syllabus Outcomes

This article deals with the following NESA Syllabus Outcomes:

 NESA Syllabus Outcomes Syllabus Outcomes Explanation Communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols This means that you will be able to identify fractions in problem questions and solve them. Applies appropriate mathematical techniques to solve problems This means that you will be able to simplify, add/subtract, multiply/divide and order fractions. Recognises and explains mathematical relationships using reasoning This means that you will be able to identify and express equivalent fractions. Operates with fractions, decimals and percentages This means you will solve problems that deal with fractions, decimals and percentages.

## Outline

Fractions are often difficult to grasp initially because of the multiple values involved, what they mean, and their relationship to each other.

They may be hard to understand due to the complexity of the operations.

Eg. When to add/subtract certain numbers versus when to multiply numbers.

However, it is an extremely important fundamental topic that is heavily applied in all areas of maths.

So, make sure you understand how to work with fractions!

## Assumed Knowledge

Students should be familiar with elementary BODMAS operations (how to add, subtract, divide and multiply in the correct order) and simple equations.

Students should know how to find the LCM (lowest common multiple) and HCF (highest common factor) of a group of numbers.

## What are fractions?

Generally, we refer to fractions as part of a whole.

For example:

• $$\frac{1}{2}$$ (half) a pizza
• $$\frac{3}{4}$$ (three-quarters) of an hour.

$$3\frac{2}{3}$$ means there are $$3$$ whole objects, as well as $$\frac{2}{3}$$ of an object.

Example:

You can picture $$2\frac{2}{3}$$ like so:

Fractions can also be used to describe the division of numbers into equal parts; $$\frac{2}{3}$$ means dividing $$2$$ into $$3$$ equal parts.

Fractions are written as one number divided by another.

• The top number is called the numerator, and the bottom number is called the denominator.
• The bar in between them is called the vinculum (you don’t need to remember this), which is another way of writing $$\div$$ (you need to remember this!).
• This then means that the numerator is divided by the denominator. Fractions are just another way to express division!

Eg. $$\frac{2}{3}$$ is just another way to write $$2 \div 3$$.

## Types of fractions

### Proper fractions

In proper fractions, the numerator is less than the denominator

For example, $$\frac{3}{5}$$

### Improper fractions

The numerator is greater than the denominator.

For example, $$\frac{7}{2}$$

### Mixed numbers

A combination of a whole number and a fraction.

For example,  $$3\frac{2}{3}$$

## Expressing mixed numbers as improper fractions and vice versa

A mixed fraction can also be expressed as an improper fraction, and an improper fraction can be expressed as a mixed fraction.

### Expressing mixed fractions as improper fractions

To do this, we:

1. Multiply the whole number by the denominator of the fraction
2. Add this number to the numerator
3. Write the new number over the original denominator

For example

\begin{align*}
\color{blue}{3}\frac{\color{red}{2}}{\color{blue}{5}} =\color{blue}{3} + \frac{\color{red}{2}}{\color{blue}{5}} = \frac{\color{blue}{3×5}+\color{red}{2}}{5} =\frac{17}{5}
\end{align*}

\begin{align*}
2\frac{7}{8} = 2 + \frac{7}{8} = \frac{2×8+7}{8} =\frac{25}{7}\\
\end{align*}

### Expressing improper fractions as mixed numbers

To do this, we:

1. Divide the numerator by the denominator
2. Find the remainder
3. The remainder becomes the new numerator (with the denominator remaining the same), and the number of times the denominator divides into the numerator becomes the whole number.

Example:

We can reverse the process of going from mixed numbers to improper fractions as follows:

\begin{align*}
\frac{25}{9} = \frac{2×9+7}{9} = 2\frac{7}{9}
\end{align*}

But this is a lot of work!

Instead we do the following:

Think: What is $$25 \div 9$$?

The answer is $$2$$ remainder $$7$$.

Then write: $$2 \frac{7}{9}$$

Think: What is $$17 \div 9$$?

The answer is $$8$$ remainder $$1$$.

Then write: $$8 \frac{1}{2}$$.

Note: that there is a negative in this question.

Keep the negative symbol where it is! The conversion still follows the same process.

## Equivalent fractions

Equivalent fractions are fractions that have the same mathematical value but have different numerators and denominators.

Although they may look different from each other, they are mathematically the same.

Eg. $$\frac{1}{2}$$, $$\frac{2}{4}$$, and $$\frac{3}{6}$$ are all the same.

To change one fraction to another equivalent fraction, we multiply (or divide) the numerator and denominator by the same number.

For example:

We can find an equivalent for $$\frac{1}{2}$$ by multiplying both the numerator and denominator by $$3$$.

$$\rightarrow$$ This gives us $$\frac{3}{6}$$.

We can find an equivalent for$$\frac{10}{15}$$by dividing both number and denominator by $$5$$.

$$\rightarrow$$ This gives us $$\frac{2}{3}$$.

Examples:

1. What are some equivalent fractions for$$\frac{3}{4}$$ and $$\frac{16}{28}$$?

\begin{align*}
\frac{3}{4} \rightarrow \frac{6}{8}, \frac{9}{12}, \frac{30}{40}
\end{align*}

\begin{align*}
\frac{16}{20} \rightarrow \frac{8}{14}, \frac{4}{7}, \frac{32}{56}
\end{align*}

## Simplifying fractions

Simplifying a fraction means to rewrite the fraction as an equivalent fraction, so that the numerator and denominator are as small as possible.

Like equivalent fractions, you can simplify a fraction if its numerator and the denominator have a common factor.

We can divide both numerator and denominator by this number to create a simplified fraction that is equivalent to the original fraction.

You keep simplifying a fraction until the numerator and denominator don’t have a common factor anymore – this is its simplest form.

Examples:

1. Simplify $$\frac{14}{22}$$

Both $$14$$ and $$22$$ are divisible by $$2$$ , so we can divide both top and bottom:

\begin{align*}
\frac{14}{22} = \frac{7}{11}
\end{align*}

$$7$$ and $$11$$ don’t have any common factors. So, this is its simplest form.

What to ask yourself:

1. Do both the numerator and denominator have a common factor?
1. Yes – Divide both numerator and denominator by this number
2. No – This is the fraction’s simplest form.
2. Repeat step 1 until there are no numbers which will divide exactly into the numerator and denominator.

We usually look for the highest common factor when simplifying fractions. Don’t worry if you can’t identify it at first, you can always continue simplifying the fraction.

2. Simplify $$\frac{56}{64}$$

This looks like a hard fraction to simplify, but we can start off with an easy factor: $$2$$.

Dividing both numerator and denominator by $$2$$:

\begin{align*}
\frac{56}{64} = \frac{28}{32}
\end{align*}

Now it’s a little easier to identify common factors.

$$28$$ and $$32$$ are both divisible by $$4$$, so:

\begin{align*}
\frac{28}{32} = \frac{7}{8}
\end{align*}

$$7$$ and $$8$$ don’t have a common factor.

So, this $$\frac{7}{8}$$ is the simplest form!

## Ordering fractions

### Comparing the size of fractions

In order to compare the size of two fractions, the first step is to choose a new denominator for both fractions.

The new denominator should be a number which both denominators divide into exactly – preferably the Lowest Common Multiple (LCM) of the two numbers.

Say we want to compare $$\frac{3}{4}$$ and $$\frac{5}{7}$$.

The denominators are $$4$$ and $$7$$.

We can choose $$28$$ as the new denominator since it is the smallest number that both $$4$$ and $$7$$ are factors of.

Then we change both fractions into an equivalent fraction with $$28$$ as the denominator.
\begin{align*}
\frac{3}{4} = \frac{21}{28}
\end{align*}

(by multiplying both numerator and denominator by $$7$$ to create an equivalent fraction)

\begin{align*}
\frac{5}{7} = \frac{20}{28}
\end{align*}

(by multiplying both numerator and denominator by $$4$$ to create an equivalent fraction)

We now compare the numerators ONLY $$-21$$ is larger than $$20$$.

So we know that $$\frac{21}{28} > \frac{5}{7}$$.

However, we want to write it in terms of the original question.

Hence, $$\frac{3}{4} > \frac{5}{7}$$.

## Addition and subtraction of fractions

Fractions can only be added or subtracted if they have the same denominator.

### With same denominator

If the fractions in the question have the same denominator already, we simply add or subtract the numerators without changing the denominator.

You should then simplify the answer (mixed fraction if applicable).

For example

$$\frac{1}{8} + \frac{5}{8} = \frac{6}{8} = \frac{3}{4}$$ (simplified)

$$\frac{5}{9} + \frac{7}{9} = \frac{14}{9} = 1\frac{5}{9}$$ (simplified into mixed fraction)

Examples:

1. What number should replace in the following?

\begin{align*}
\frac{1}{6} + \frac{\alpha}{6} = \frac{4}{6}
\end{align*}

Since both fractions have the same denominator, $$\alpha$$ must add to $$1$$ to give $$4$$.

Hence, $$\alpha$$ is $$3$$ .

### Different denominator

If the two fractions have a different denominator (which is the case most of the time), we need to change them so that they have the same denominator.

Remember, we can add or subtract fractions ONLY when they have the same denominator.

To do this:

1. Find a common denominator (lowest common multiple of the two denominators)
2. Convert each fraction to an equivalent fraction with the new denominator
3. Add/subtract the numerators without changing the denominator.

Examples:

1. What is the common denominator of $$\frac{5}{8}$$,$$\frac{7}{6}$$ and $$\frac{2}{3}$$?

The lowest common multiple of $$8$$, $$6$$ and $$3$$ is $$24$$.

2. Simplify $$\frac{2}{3} + \frac{5}{6}$$

The lowest common multiple of $$3$$ and $$6$$ is $$6$$.

This means we have to change $$\frac{2}{3}$$into an equivalent fraction with $$6$$ as the denominator, in order for us to add the two fractions.

\begin{align*}
\frac{2}{3} + \frac{5}{6} &= \frac{4}{6} + \frac{5}{6} \\
&= \frac{9}{6} \\
&= \frac{3}{2} \\
&= 1 \frac{1}{2}\\
\end{align*}

3. What number $$\alpha$$ should replace in the following?

After changing each fraction to an equivalent with denominator $$15$$, we get:

\begin{align*}
\frac{5}{15} + \frac{3\alpha}{15} = \frac{11}{15}
\end{align*}

Then $$5 + 3\alpha = 11$$

So, $$\alpha = 2$$

### Mixed fractions

When a question involved mixed fractions, it is sometimes easier to add/subtract the whole numbers and then add/subtract the fractional parts.

Example:

1. Simplify $$1 \frac{3}{4} + 2 \frac{1}{3} = 1\frac{1}{2}$$

\begin{align*}
1\frac{3}{4} +2 \frac{1}{3} – 1 \frac{1}{2} &= (1+2-1) + \frac{3}{4} + \frac{1}{3} – \frac{1}{2} \\
&= 2 \frac{9+4-6}{12} \\
&= 2\frac{7}{12}
\end{align*}

The other method is to simply convert the mixed fractions into improper fractions before adding or subtracting.

You may choose to convert the answer back to a mixed number.

Example:

1. Simplify $$1 \frac{3}{4} + 2 \frac{1}{3} = 1\frac{1}{2}$$

\begin{align*}
1 \frac{3}{4} + 2\frac{1}{3} – 1\frac{1}{2} &= \frac{7}{4} + \frac{7}{3} – \frac{3}{2} \\
&= \frac{21 + 28 – 18}{12} \\
&= \frac{31}{12} \\
&= 2 \frac{7}{12}
\end{align*}

## Multiplication and division of fractions

It is IMPORTANT that you rewrite all the fractions as improper fractions before starting operations.

Unlike for addition and subtraction, it doesn’t matter if the denominators are different in multiplication and division.

### Multiplication of fractions

The product of two fractions is found by multiplying the numerators and multiplying the denominators separately.

Example:

$$\frac{3}{4} \times \frac{2}{7} = \frac{3\times 2}{4\times 7} = \frac{6}{28} = \frac{3}{14}$$ (after simplification)

Another trick here is that fractions can be simplified before multiplying… you can ‘cancel’ out numbers using common factors.

This is similar to simplifying a single fraction, but this involves dividing a common factor into the numerator and denominator of different fractions.

In the example above, we see that $$2$$ (the numerator of the 2nd fraction) and $$4$$ (the denominator of the 1st fraction) have a common factor of $$2$$.

Thus, we can divide both numbers by $$2$$ first to convert our equation into a simpler multiplication step:

\begin{align*}
\frac{3}{\color{red}{4}}\times \frac{\color{red}{2}}{7} = \frac{3}{2}\times \frac{1}{7} = \frac{3}{14}
\end{align*}

Remember, you CANNOT do this for addition and subtraction.

The cancellation technique between different fractions ONLY works for MULTIPLICATION (and division), when the numerator and denominator cancel.

You cannot cancel across two numerators.

### Multiplying mixed numbers

To multiply mixed numbers, we have to change them to improper fractions first. You cannot multiply the whole numbers and fractions separately.

Once converted, we can multiply them as we usually do – by multiplying the numerator and denominator separately.

Example:

\begin{align*}
\frac{1}{5} \times 2\frac{1}{7} &= \frac{6}{5} \times \frac{15}{7}\\
&=\frac{90}{35}\\
&=\frac{18}{7}\\
&=2\frac{2}{7}
\end{align*}

Note: In this question, we could also cancel before multiplication!

\begin{align*}
\frac{6}{5} \times \frac{15}{7} &= \frac{6}{1} \times \frac{3}{7}\\
&=\frac{18}{7}
\end{align*}

## Division of fractions

### Reciprocal

The reciprocal of a fraction is essentially the fraction turned upside down.

For example, the reciprocal of $$\frac{5}{12}$$ is $$\frac{12}{5}$$.

Reciprocals are always used in the division of fractions.

### Division

To divide two fractions, we change the question into a multiplication.

We keep one fraction the same, then multiply it by the reciprocal of the other fraction.

Example:

\begin{align*}
\frac{3}{4} \div \color{red}{\frac{6}{7}} &= \frac{3}{4}\times \color{red}{\frac{7}{6}}\\
&=\frac{1}{4}\times \frac{7}{2}\\
&=\frac{7}{8}
\end{align*}

## Applications

### The unitary method

This is a technique for solving problems by first finding the value of ONE unit, then finding the value of $$X$$ units by multiplication.

Example:

1. If $$5$$ pens cost $$10$$ , how much do $$8$$ pens cost?

The cost of $$1$$ pen is $$105 = 2$$.

The cost of $$8$$ pens is $$28 = 16$$ .

This can be solved in one step by multiplying by  $$\frac{8}{5}$$. This fraction multiplication carries out the same division by $$5$$, then multiplication by $$8$$.

## Summary

### 1. Mixed fraction

To express a mixed fraction as an improper fraction, multiply the denominator by the whole number and add the numerator (this is the new numerator).

The denominator stays the same.

### 2. Equivalent fractions

Multiply a number to both numerator and denominator, or divide the numerator and denominator by a common factor.

### 3. Simplifying fractions

The lowest equivalent form that the fraction can have.

### 4. Ordering fractions

Change each fraction to an equivalent fraction with the same denominator, then compare the numerators

### 5. Adding and subtracting fractions

If the denominators are different, find the lowest common multiple of the two denominators.

Then, find equivalents of each fraction with the new denominator.

Add/subtract the numerators without changing the denominator

### 6. Multiplying fractions

If the fraction is mixed, convert it to an improper fraction first.

Multiply the numerators, then multiply the denominators separately.

Your answer is $$\frac{top \times top}{bottom \times bottom}$$.

### 7. Dividing fractions

Multiply one fraction by the reciprocal (flipped) of the other.

## Checkpoint questions and solutions

### Questions

1. Rewrite $$\frac{-17}{2}$$ as a mixed number.

2. Simplify $$\frac{84}{144}$$

3. What are $$\alpha$$ and $$\beta$$ in:

4. Arrange the following group of fractions in ascending order (from smallest to largest)

5. Simplify $$\frac{7}{13} – \Big{(} \frac{2}{13} – \frac{11}{13} \Big{)}$$

6. Simplify $$\frac{5}{8} – \Big{(} \frac{9}{10} – \frac{3}{4} \Big{)}$$

7. Calculate $$\frac{4}{21} \times \frac{5}{8} \times \frac{-13}{15}$$

8. Simplify $$\frac{3}{4} + 1\frac{1}{15} \times 4 \frac{2}{7} – 1\frac{1}{2}$$

9. Evaluate $$3\frac{5}{9} \div 9 \frac{1}{3}$$

### Solutions

1. Rewrite $$\frac{-17}{2}$$ as a mixed number.

Solutions:

\begin{align*}
– \frac{17}{2} =  -\frac{(8 \times 2) + 1}{2} = -8 \frac{1}{2}
\end{align*}

2. Simplify $$\frac{84}{144}$$

Solutions:

\begin{align*}
\frac{84}{144} = \frac{7}{12}
\end{align*}

(both $$84$$ and $$144$$ are divisible by $$12$$)

3. What are $$\alpha$$ and $$\beta$$ in:

\begin{align*}
\frac{5}{7} = \frac{ \alpha}{63} = \frac{-35}{ \beta}
\end{align*}

Solutions:

This question converts $$\frac{5}{7}$$ to equivalent fractions.

We multiply $$7$$ in the denominator by $$9$$ to get $$63$$ , so we multiply the numerator by $$7$$ as well.

\begin{align*}
\alpha = 5 \times 7 = 35
\end{align*}

Similarly, for $$-35$$ as the numerator, we have to multiply $$7$$ by $$-7$$ .

\begin{align*}
\beta = 7 \times -7 = 35
\end{align*}

4. Arrange the following group of fractions in ascending order (from smallest to largest)

\begin{align*}
\frac{3}{4}; \frac{13}{24}; \frac{5}{12}; \frac{5}{6}
\end{align*}

Solutions:

To compare these fractions, we much change their denominators.

The lowest common multiple of all the denominators is $$24$$.

Find the equivalent of each fraction with $$24$$ as the denominator, then compare the numerator.

\begin{align*}
\frac{3}{4} = \frac{3\times 6}{4\times 6} &= \frac{18}{24} \\
\frac{5}{12} = \frac{5\times 2}{12\times 2} &= \frac{10}{24} \\
\frac{5}{6} = \frac{5\times 4}{6\times 4} &= \frac{20}{24}
\end{align*}

Now, $$\frac{10}{24} < \frac{13}{24} < \frac{18}{24} < \frac{20}{24}$$.

Using the original fractions, $$\frac{5}{12} < \frac{13}{24} < \frac{3}{4} < \frac{5}{6}$$.

5. Simplify $$\frac{7}{13} – \Big{(} \frac{2}{13} – \frac{11}{13} \Big{)}$$

Solution:

Using BODMAS, we need to compute the inside of the brackets first.

\begin{align*}
\frac{2}{13} – \frac{11}{13} = \frac{-9}{13}
\end{align*}

(Note: be careful when subtracting negative fractions!)

\begin{align*}
\frac{7}{13} – \Big {(} \frac{-9}{13} \Big{)} = \frac{7}{13} + \frac{9}{13} = \frac{16}{13}
\end{align*}

6. Simplify $$\frac{5}{8} – \Big{(} \frac{9}{10} – \frac{3}{4} \Big{)}$$

Solution:

The lowest common multiple of $$8$$, $$10$$ and $$4$$ is $$40$$.

Changing all the fractions to this equivalent denominator gives us:

\begin{align*}
\frac{25}{40} – \Big{(} \frac{36}{40} – \frac{30}{30} \Big{)}\\
&= \frac{25}{40} – \Big{(} \frac{6}{40} \Big{)}\\
&= \frac{19}{40}
\end{align*}

7. Calculate $$\frac{4}{21} \times \frac{5}{8} \times \frac{-13}{15}$$

Solution:

Try to cancel numbers before you start multiplying.

\begin{align*}
\frac{1}{21} \times \frac{1}{2} \times \frac{-13}{3}\\
&=  – \frac{13}{21 \times 2 \times 3}\\
&=  – \frac{13}{126}
\end{align*}

Do a quick check for your solution: based off the original question, should it be negative or positive?

8. Simplify $$\frac{3}{4} + 1\frac{1}{15} \times 4 \frac{2}{7} – 1\frac{1}{2}$$

Solution:

Using BODMAS, we should compute the multiplications first, then addition/subtraction.

\begin{align*}
\frac{3}{4} + 1\frac{1}{15} \times 4 \frac{2}{7} – 1\frac{1}{2}\\
&= \frac{3}{4} + \frac{32}{7} – \frac{3}{2}\\
&= \frac{21}{28} + \frac{128}{28}-\frac{42}{28}\\
&=\frac{107}{28}
\end{align*}

9. Evaluate $$3\frac{5}{9} \div 9 \frac{1}{3}$$

Solution:

In divisions, remember to convert all mixed fractions to improper fractions.

\begin{align*}
3 \frac{5}{9} \div 9 \frac{1}{3} &= \frac{32}{9} \div \frac{28}{3}\\
&=\frac{32}{9} \times \frac{3}{2} \\
&=\frac{8}{3} \times \frac{1}{7} \\
&= \frac{8}{21}
\end{align*}

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