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1. Operations with Fractions
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Grade 6

1.01 Review: Simplify fractions and mixed numbers

Lesson
Practice

Improper fractions and mixed numbers

Fractions like \dfrac{4}{5} make up less than one whole. We know this because the numerator is less than the denominator. We call these proper fractions.

What about a fraction like \dfrac{8}{5}? Notice that the numerator is greater than the denominator. This means that the fraction is greater than a whole. We call these improper fractions.

This image shows 2 horizontal rectangles that contain 5 squares each. Ask your teacher for more information.

Each rectangle in this image has been split into five equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator.

Since the number of shaded parts is more than the number of parts in one whole, a complete rectangle and three more parts have been shaded.

We can write this number as 1\dfrac{3}{5}, which we call one and three fifths.

Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number line.

012

For \dfrac{8}{5} or 1\dfrac{3}{5}, each whole number on the number line is split into 5 equal parts.

For the improper fraction, we would count 8 tick marks from 0, and place the point there.

For the mixed number, we would count 3 tick marks from 1.

Examples

Example 1

What number is plotted on the number line? Give your answer as a mixed number and as an improper fraction.

34
Worked Solution
Create a strategy

For the fraction part, count the number of equal spaces between the two whole numbers, then count how many spaces after the previous whole number the point is located.

Apply the idea

The point is located between 3 and 4. It is greater than 3 but less than 4. So the whole number part is 3.

There are 10 equal spaces between 3 and 4, so each space represents \dfrac{1}{10}. The point is 9 spaces to the right of 3, so the fraction is \dfrac{9}{10}.

The number plotted on the number line is 3\dfrac{9}{10}.

This can also be written as \dfrac{39}{10}.

Example 2

Rewrite 4\dfrac{1}{2} as an improper fraction.

Worked Solution
Create a strategy

To find the improper fraction, multiply the whole number by the denominator, then add the numerator.

Apply the idea
\displaystyle 4\frac{1}{2}\displaystyle =\displaystyle \frac{4\cdot 2 + 1}{2}Multiply the denominator and whole number
\displaystyle =\displaystyle \frac{8+1}{2}Evaluate the multiplication
\displaystyle =\displaystyle \frac{9}{2}Evaluate the addition
Reflect and check
The image shows 5 rows and 2 columns of squares. 9 squares are shaded with blue and 1 square is shaded yellow.

We could also draw an array for 4\dfrac{1}{2} where each part represents \dfrac{1}{2}.

We can see that there are 9 shaded parts.

So the improper fraction is \dfrac{9}{2}.

Example 3

Rewrite \dfrac{11}{3} as a mixed number.

Worked Solution
Create a strategy

Divide the numerator by the denominator. The remainder will be the numerator of the mixed fraction.

Apply the idea

11 divided by 3 is 3 remainder 2.

So, \dfrac{11}{3} is made up of 3 wholes and 2 out of 3 remaining.\dfrac{11}{3}=3\dfrac{2}{3}

Idea summary

Fractions where the numerator is greater than the denominator are called improper fractions.

Improper fractions can be rewritten as mixed numbers.

Equivalent fractions

Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below.

A hexagon divided into 6 equal parts with 3 shaded.

We can see that 3 of the 6 parts have been shaded in, so the area represents \dfrac{3}{6} of the whole shape. But we can also see that \dfrac{1}{2} of the shape has been shaded in.

Since the same area is shaded for both \dfrac{3}{6} and \dfrac{1}{2}, these two fractions must be equal.

These are called equivalent fractions, since they have different numerators and different denominators but are still equal.

Exploration

Explore the following applet to investigate equivalent fractions further.

Loading interactive...
  1. When multiplying the numerator and denominator of the original fraction by 2, how many pieces is the shaded part of the new fraction separated into?
  2. When multiplying the numerator and denominator of the original fraction by 3, how many pieces is the shaded part of the new fraction separated into?
  3. Do you think this is the case for any multiple of the numerator and denominator of any fraction?
  4. Why do you think we have to multiply both the numerator and denominator by the same number?

We can see equivalent fractions in action by representing them visually and seeing how we can create them. Consider this grid:

A rectangle with 1 out of 6 parts shaded. Ask your teacher for more information.

This grid is divided into 6 equal parts as shown by the solid grid lines. 1 out of 6 parts is shaded, so we can represent this area as \dfrac{1}{6} of the grid.

If we separate each of the 6 parts into 3 more equal parts, as shown by the dashed lines, we can create an equivalent fraction.

A rectangle with 3 out of 18 grid squares shaded.

Notice there are now 18 grid squares, and 3 of them are shaded. This means that we can represent the shaded area with the fraction \dfrac{3}{18}.

This shows that \dfrac{1}{6} and \dfrac{3}{18} are equivalent fractions, which we can express using an equal sign:\dfrac{1}{6} = \dfrac{3}{18} By separating each of the original parts into 3 more equal parts, we are multiplying the numerator (the shaded part) and denominator (the entire grid) by 3: \dfrac{1\cdot 3}{6\cdot 3}=\dfrac{3}{18}

We can always find an equivalent fraction by multiplying both the numerator and denominator by the same number.

Examples

Example 4

Rewrite \dfrac{3}{8} with a denominator of 40.

Worked Solution
Create a strategy

Multiply the numerator and denominator by the quotient of 40 divided by 8.

Apply the idea

Dividing 40 by 8 gives 40\div 8=5.

\displaystyle \frac{3}{8}\displaystyle =\displaystyle \frac {3 \cdot 5}{ 8 \cdot 5} Multiply the numerator and denominator by 5
\displaystyle =\displaystyle \frac {15}{ 40} Evaluate

\dfrac {3}{8} = \dfrac {15}{40}

Idea summary

When two fractions represent the same amount of a whole, they are equivalent fractions.

We can create an equivalent fraction by multiplying the numerator and denominator by the same number.

Simplify fractions

We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction.

Previously, we saw that\frac{3}{18} = \frac{1\cdot 3}{6\cdot 3}From here, we can remove the common factor of 3:

\dfrac{3}{18} \to \dfrac{1 \cdot 3}{6 \cdot 3} \to \dfrac{1 \cdot \cancel{3}}{6 \cdot \cancel{3}} \to \dfrac{1}{6}

This shows:\dfrac{3}{18} = \dfrac{1}{6}

The fraction \dfrac{1}{6} has no common factors between the numerator and denominator (other than 1), so it is in simplest form. This also means that 3 was the greatest common factor of 3 and 18.

Examples

Example 5

In the grid shown, \dfrac{20}{32} parts have been shaded. Use the grid to simplify \dfrac{20}{32}.

A grid 4 rows of 8 squares. The first 5 squares in each row are shaded. A total of 20 out of 32 squares are shaded.
Worked Solution
Create a strategy

Find the fraction of columns that is shaded.

Apply the idea

The grid can be divided into 8 identical columns as shown.

A grid 4 rows of 8 squares. The first 5 squares in each row are shaded. Ask your teacher for more information.

5 out of 8 columns are shaded. So,\dfrac{20}{32} = \dfrac{5}{8}

Reflect and check

We also could have simplified the fraction by writing the numerator and denominator as a multiple of a common factor, and then remove the common factor.

The common factor of 20 and 32 is 4 so we can write both as multiples of 4:

\displaystyle \frac{20}{32}\displaystyle =\displaystyle \frac{5\cdot 4}{8 \cdot 4}Write both numbers as multiples of 4
\displaystyle =\displaystyle \frac{5}{8}Remove the common factor of 4

Example 6

Select the two fractions which are equivalent to \dfrac{8}{12}.

A
\dfrac{20}{30}
B
\dfrac{7}{3}
C
\dfrac{2}{6}
D
\dfrac{2}{3}
Worked Solution
Create a strategy

Simplify \dfrac{8}{12} by dividing out the greatest common factor (GCF) of the numerator and denominator. Next simplify each fraction in the answer choices. Any fractions that have the same simplified form as \dfrac{8}{12} are equivalent.

Apply the idea

The numerator, 8, and the denominator, 12 have a greatest common factor of 4. So we can simplify the fraction by dividing by 4 in both the numerator and denominator.

\displaystyle \frac{8}{12}\displaystyle =\displaystyle \frac{2\cdot\cancel{4}}{3\cdot\cancel{4}}Factor out the GCF of 4
\displaystyle =\displaystyle \frac{2}{3}Divide

Now, simplify each of the fractions in the answer choices and compare them to \dfrac{2}{3}:

  • Option A

    \displaystyle \dfrac{20}{30}\displaystyle =\displaystyle \dfrac{2 \cdot \cancel{10}}{3 \cdot \cancel{ 10}}Factor out the GCF of 10
    \displaystyle \dfrac{20}{30}\displaystyle =\displaystyle \dfrac{2}{3}Divide

    The fraction \dfrac{20}{30} is equivalent to \dfrac{8}{12} because they both simplify to \dfrac{2}{3}.

  • Option B

    The fraction \dfrac{7}{3} is already in its simplest form because 7 and 3 have no common factors other than 1. The fraction \dfrac{7}{3} is not equivalent to \dfrac{8}{12} because it is greater than \dfrac{2}{3} and not equal to it.

  • Option C

    \displaystyle \dfrac{2}{6}\displaystyle =\displaystyle \dfrac{1 \cdot \cancel{2}}{3 \cdot \cancel{2}}Factor out the GCF of 2
    \displaystyle \dfrac{2}{6}\displaystyle =\displaystyle \dfrac{1}{3}Divide

    The fraction \dfrac{2}{6} is not equivalent to \dfrac{8}{12} because it is smaller than \dfrac{2}{3} and not equal to it.

  • Option D

    The fraction \dfrac{2}{3} is already in its simplest form and is equivalent to \dfrac{8}{12} which we already found simplifies to \dfrac{2}{3}.

So options A and D are the correct answers.

Reflect and check

Simplifying creates equivalent fractions by making the numerator and denominator smaller. We can also make equivalent fractions by making numerators and denominators larger.

\dfrac{8}{12} is simplified to \dfrac{2}{3}. We can also scale up \dfrac{2}{3} to form another equivalent fraction:

\displaystyle \frac{2}{3}\displaystyle =\displaystyle \frac{2\cdot10}{3\cdot10}Multiply the numerator and denominator by 10
\displaystyle =\displaystyle \frac{20}{30}Evaluate

This confirms our answer, showing that we have correctly identified the equivalent fractions \dfrac{2}{3} and \dfrac{20}{30}.

Idea summary

When we find an equivalent fraction by removing common factors from the numerator and denominator, we are simplifying the fraction.

When the fraction has no common factors between the numerator and denominator (other than 1), it is in simplest form.

To write a fraction in simplest form, divide both the numerator and denominator by their greatest common factor.

Two fractions are equivalent when they can be simplified to the same fraction.

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