# Improper Fractions and Mixed Numbers

## Learning text on the topicImproper Fractions and Mixed Numbers

This text is all about improper fractions and mixed numbers, key concepts in Math that you'll find incredibly useful as you move through Middle School. Additionally, understanding these will help you tackle everyday problems more efficiently—like splitting that pizza with your friends or baking cookies using a recipe. Let's get started!

## What are Improper Fractions?

An improper fraction is a type of fraction where the numerator (the top number) is equal to or larger than the denominator (the bottom number). This means you have a fraction that is equal to or greater than 1.

An improper fraction looks like $\frac{5}{3}$ or $\frac{5}{5}$ where the top number is equal to or bigger than the bottom number. It represents more than one whole.

Imagine having two pizzas that have been cut into 4 slices each, and you have 5 of those slices. This situation is perfect for using an improper fraction to describe what you have.

If you have 9 slices of a cake that's cut into 8 slices, what improper fraction represents this?
If you have 15 pieces of chocolate from a bar divided into 12 pieces, what improper fraction represents this?

## What are Mixed Numbers?

A mixed number is a way of expressing a number that's made up of a whole part and a fractional part. It's like having some whole apples and some parts of an apple.

A mixed number is written with a whole number alongside a fraction, such as $2 \frac{1}{2}$, which means 2 whole items plus half of another one.

What mixed number represents 4 whole sandwiches and a quarter of another one?
If you have 3 whole watermelons and 2 slices out of a watermelon cut into 10 slices, what mixed number represents this?

Understanding the concept of Fractions Greater than 1 on a Number Line is extremely helpful when trying to visualise mixed numbers and improper fractions.

## Converting Mixed Numbers to Improper Fractions – Method

When it comes to working with fractions in math problems, it's often easier to deal with them when they're all written in the same form. This is why we learn how to convert a mixed number to an improper fraction.

Here's the step-by-step method for $3 \frac{2}{8}$:

Step What to do? Example
1. Multiply the whole number by the denominator of the fraction. 3 × 8 = 24
2. Add the numerator to the product from step 1. 24 + 2 = 26
3. Write the result over the original denominator. $\frac{26}{8}$

You can also combine these steps into one continuous expression to convert numbers from mixed numbers to improper fractions. The example from the table can also be presented this way:

Can you convert the mixed number $4 \frac{3}{5}$ into an improper fraction?
Convert the mixed number $5 \frac{1}{4}$ into an improper fraction.
Turn $2 \frac{7}{10}$ into an improper fraction.

## Converting Improper Fractions to Mixed Numbers – Method

Sometimes it's more intuitive to work with improper fractions when they are in the form of mixed numbers, especially when you're dealing with amounts or measurements. This is why learning to convert an improper fraction to a mixed number is a valuable skill.

Here's the step-by-step method for $\frac{13}{6}$:

Step What to do? Example
1. Divide the numerator by the denominator to find the whole number part of the mixed number. 13 ÷6 = 2 remainder 1
2. The remainder becomes the new numerator of the fractional part. remainder 1
3. Write the whole number alongside the new fraction with the original denominator. $2 \frac{1}{6}$

Can you convert the improper fraction $\frac{23}{4}$ into a mixed number?
Convert the improper fraction $\frac{29}{6}$ into a mixed number.
Turn $\frac{45}{8}$ into a mixed number.

## Adding and Subtracting Mixed Numbers

When you add or subtract mixed numbers, treat the whole numbers and the fractions separately, ensuring that the fractions have the same denominator.

For example, adding $1 \frac{3}{8}$ and $2 \frac{2}{8}$ together, we combine the whole numbers (1 + 2) and the fractions ($\frac{3}{8} + \frac{2}{8}$) to get $3 \frac{5}{8}$.

What is $2 \frac{1}{4}$ plus $1 \frac{3}{4}$?
If you start with $5 \frac{1}{2}$ bananas and eat $2 \frac{3}{4}$ bananas, how many do you have left?

## Multiplying and Dividing Mixed Numbers

For multiplication and division of mixed numbers, it's best to convert them to improper fractions first. Multiply or divide as usual, and if dividing, remember to use the reciprocal of the divisor.

If you have $2 \frac{1}{2}$ boxes of cookies and triple them, how many boxes do you have?

### Simplifying Fractions

After any operation with mixed numbers or improper fractions, it's important to simplify your answer. This means expressing the fraction in its simplest form by reducing it to the lowest terms.

You may consider converting a mixed number to an improper fraction as it makes solving the problem easier or helps in understanding the result better. However, this conversion is not always necessary and depends on the context or preference in representation.

Convert the mixed number $3 \frac{3}{4}$ to an improper fraction and simplify if possible.
Convert the mixed number $5 \frac{2}{6}$ to an improper fraction and simplify.
Convert the mixed number $7 \frac{5}{8}$ to an improper fraction and simplify if possible.
Convert the mixed number $8 \frac{4}{9}$ to an improper fraction and simplify.
Convert the mixed number $6 \frac{7}{10}$ to an improper fraction and simplify if possible.

## Improper Fractions and Mixed Numbers – Summary

Key Learnings from this Text:

• A mixed number combines a whole number with a fraction.
• Converting between mixed numbers and improper fractions allows for easier calculations.
• To convert Mixed Numbers to Improper Fractions, you need to multiply the whole number by the denominator and add the numerator. This becomes your new numerator, keep the denominator the same.
• Fraction operations with mixed numbers involve ensuring the same denominator for addition/subtraction and converting to improper fractions for multiplication/division.
• Simplifying fractions makes mixed numbers easier to understand and work with.

Ready to test your skills further? Check out other fun math challenges and resources available on our website!

## Improper Fractions and Mixed Numbers – Frequently Asked Questions

How do you convert an improper fraction to a mixed number?
Why do you need to simplify fractions?
Can you add mixed numbers without converting to improper fractions?
What role does the denominator play in fractions?
How does converting mixed numbers to decimals help?
Are improper fractions and mixed numbers equivalent?
How do you multiply mixed numbers?
Is a fraction calculator always accurate?
When would you use mixed numbers in real life?
How can mixed numbers and improper fractions be used in problem-solving?

## Improper Fractions and Mixed Numbers exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Improper Fractions and Mixed Numbers.
• ### What does each keyword mean?

Hints

A fraction is made up of a numerator and denominator. Which way round do they go?

Proper fractions are those with a quantity less than one whole. Improper fractions have a quantity equal to or greater than one whole.

Solution

Numerator - The top number in a fraction.
Denominator - The bottom number in a fraction.
Improper Fraction - When the numerator is equal to or larger than the denominator.
Proper Fraction - When the numerator is smaller than the denominator.
Mixed Number - A number made up of a whole part and a fractional part.

• ### What is represented by the diagram?

Hints

The denominator is the number of equal parts per cake.

The numerator is the number of pieces of cake highlighted.

There are two possible answers, an improper fraction and a mixed number.

Solution

Each cake has 8 equal parts, this tells us that the denominator is 8.
There are 9 pieces of cake highlighted, this tells us the numerator is 9.
The correct answers are $\mathbf{\frac{9}{8}}$ and $\mathbf{1 \frac{1}{8}}$.

• ### What are the equivalent mixed numbers and improper fractions?

Hints

To convert a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. Write this number over the original denominator.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part of the mixed number and the remainder becomes the new numerator over the original denominator.

For example: $\frac{5}{3}$ = $1 \frac{2}{3}$

Solution

$\mathbf{2 \frac{3}{4}}$ = $\mathbf{\frac{11}{4}}$ because 2 x 4 + 3 = 8 + 3 = 11 giving $\frac{11}{4}$
$\mathbf{3 \frac{1}{2}}$ = $\mathbf{\frac{7}{2}}$ because 3 x 2 + 1 = 6 + 1 = 7 giving $\frac{7}{2}$
$\mathbf{\frac{7}{4}}$ = $\mathbf{1 \frac{3}{4}}$ because 7 $\div$ 4 = 1 remainder 3 giving $1 \frac{3}{4}$
$\mathbf{\frac{18}{5}}$ = $\mathbf{3 \frac{3}{5}}$ because 18 $\div$ 5 = 3 remainder 3 giving $3 \frac{3}{5}$

• ### Adding and subtracting fractions.

Hints

You can add the whole numbers and the fractions separately then put them together to form the answer.

You can convert the mixed numbers to improper fractions before adding or subtracting, then convert the answer back to a mixed number.

If you subtract using the mixed numbers watch out for negative fractions. You must subtract the fractional part from the whole number. If you want to avoid negative fractions you should convert the mixed numbers to improper fractions first.

• For example, using a negative fraction,
${2 \frac{1}{5}} - {1 \frac{4}{5}} = {1 \frac{-3}{5}} = {\frac{2}{5}}$

• Or, changing to improper fractions,
${2 \frac{1}{5}} - {1 \frac{4}{5}} = {\frac{11}{5}} - {\frac{9}{5}} = {\frac{2}{5}}$

Solution

$\mathbf{{2 \frac{2}{9}}} + \mathbf{{3 \frac{5}{9}}} = \mathbf{{5 \frac{7}{9}}}$
because $2 + 3 = 5$ and ${\frac{2}{9}} + {\frac{5}{9}} = {\frac{7}{9}}$
$\mathbf{{6 \frac{3}{5}}} - \mathbf{{2 \frac{2}{5}}} = \mathbf{{\frac{21}{5}}}$
because ${6 \frac{3}{5}} = {\frac{33}{5}}$ and ${2 \frac{2}{5}} = {\frac{12}{5}}$ then ${\frac{33}{5}} - {\frac{12}{5}} = {\frac{21}{5}}$
$\mathbf{{5 \frac{1}{7}}} - \mathbf{{3 \frac{5}{7}}} = \mathbf{{1 \frac{3}{7}}}$
because $5 - 3 = 2$ and ${\frac{1}{7}} - {\frac{5}{7}} = { \frac{-4}{7}}$ then $2 - {\frac{4}{7}} = {1 \frac{3}{7}}$

• ### Can you identify the fraction type?

Hints

In a proper fraction the numerator is smaller than the denominator.

In an improper fraction the numerator is equal to or more than the numerator.

A mixed number has a whole part and a fractional part.

There are three items to be assigned to each group.

Solution

Proper Fractions - $\mathbf{\frac{3}{4}}$, $\mathbf{\frac{1}{4}}$, $\mathbf{\frac{3}{8}}$
Improper Fractions - $\mathbf{\frac{4}{3}}$, $\mathbf{\frac{8}{5}}$, $\mathbf{\frac{10}{5}}$
Mixed Numbers - $\mathbf{2 \frac{1}{2}}$, $\mathbf{3 \frac{3}{4}}$, $\mathbf{1 \frac{1}{5}}$

• ### What is $2\frac{3}{10}$ $\times$ $3$?

Hints

When multiplying mixed numbers remember to convert them to improper fractions first.

When multiplying a fraction by a whole number we only multiply the numerator by the whole number i.e. $\frac{5}{4} \times 2 = \frac{10}{4}$

Solution

$2\frac{3}{10} \times 3 = \frac{23}{10} \times 3 = \frac{69}{10} = 6\frac{9}{10}$

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