Sometimes multiple fractions can represent the same amount.
Consider the shaded area of the hexagon below.
These are called equivalent fractions, since they have different numerators and denominators but are still equal.
We can see equivalent fractions in action by representing them visually and seeing how we can convert from one to the other.
Consider the grid below.
But what about equivalent fractions?
Notice that the placement of shaded squares is the same on each row. So, by removing some of the grid lines, we get:
This shows us that \dfrac{1}{16} and \dfrac{3}{18} are equivalent fractions, which we can express using the equivalence:\dfrac{1}{6} = \dfrac{3}{18}
Explore the following applet to investigate equivalent fractions further.
For each fraction, we can find an equivalent fraction by multiplying both the numerator and denominator by the same number.
We can increase the denominator by multiplying it by any whole number. However, if we do this we also have to multiply the numerator by the same number.
Looking at \dfrac{1}{6} we can change 6 to 18 by multiplying it by 3. So we get:
\displaystyle \dfrac{1}{6} | \displaystyle = | \displaystyle \dfrac{1\times3}{6\times3} | Multiply the numerator and denominator by 3 |
\displaystyle = | \displaystyle \dfrac{3}{18} | Evaluate the multiplications |
Which gives us the same equivalent fractions as before.
We can also decrease the denominator by cancelling common factors in both the numerator and denominator. Since we know that\dfrac{3}{18} = \dfrac{1\times3}{6\times3}
And from cancelling out the 3s we get:\dfrac{3}{18} = \dfrac{1}{6}
We can write each step of the process like this. Notice that we could also follow the process in reverse to increase the denominator.
Select the two fractions which are equivalent to \dfrac{8}{12}.
The same amount of a whole can be written as many different fractions.
When two fractions represent the same amount of a whole they are equivalent fractions.
We can convert a fraction into an equivalent fraction by multiplying the numerator and denominator by the same number, or cancelling a common factor from the numerator and denominator.
When we cancel common factors from the numerator and denominator of a fraction, the numbers become smaller. We call this simplifying the fraction.
In the grid below, \dfrac{20}{32} parts have been shaded. Use the grid to simplify \dfrac{20}{32}.
When we find an equivalent fraction by cancelling common factors from the numerator and denominator we are simplifying the fraction.
When the fraction has no common denominators between the numerator and denominator (other than 1) it is a fully simplified fraction.
Consequently, two fractions are equivalent when they can be simplified to the same fraction.
In order to compare fractions with different denominators we can find equivalent fractions with the same denominator and compare the equivalent fractions.
For example, which is larger, \dfrac{5}{12} or \dfrac{3}{8}?
We can try comparing these fractions visually:
Because the size of a twelfth is different from the size of an eighth, we have to find equivalent fractions with the same denominator in order to compare these two fractions.
But what denominator should we try to get?
When comparing fractions we want to make their new denominators equal to the lowest common multiple of their current denominators.
Since 24 is the lowest common multiple of 12 and 8, we want to find the equivalent fractions of \dfrac{5}{12} and \dfrac{3}{8} that have a denominator of 24.
We can do this by multiplying the numerator and denominator of \dfrac{5}{12} by a factor of 2, and the numerator and denominator of \dfrac{3}{8} by a factor of 3,
\text{Multiply the numerator and denominator by }2 | \text{Multiply the numerator and denominator by }3 |
---|---|
\dfrac{5}{12}\to\dfrac{5\times2}{12\times2}\to\dfrac{10}{24} | \dfrac{3}{8}\to\dfrac{3\times3}{8\times3}\to\dfrac{9}{24} |
Since \dfrac{10}{24} is larger than \dfrac{9}{24} we can see that \dfrac{5}{12} is larger than \dfrac{3}{8}.
We could also add some gridlines to the fraction grids so that they both have 24 equal parts in order to compare them.
We can see that after adding the gridlines, the grid for \dfrac{5}{12} now has \dfrac{10}{24} shaded parts and the grid for \dfrac{3}{8} now has \dfrac{9}{24} shaded parts. This tells us that \dfrac{5}{12} is greater than \dfrac{3}{8}.
Arrange the following fractions from biggest to smallest. \dfrac{2}{3},\, \dfrac{3}{7},\, \dfrac{4}{5}
To compare fractions with different denominators:
Find the lowest common denominator
Find the equivalent fraction with this denominator
Compare the numerators of the equivalent fractions