In the previous lesson we reviewed fractions, equivalent fractions and conversions to decimal and percent. In this lesson, we will focus on operations with fractions.
When the denominators are the same, you can add/subtract the numerators and leave the denominators the same:
\( \cfrac{4}{15} + \cfrac{5}{15} = \cfrac{4+5}{15} = \cfrac{9}{15}\)
\( \cfrac{12}{17} - \cfrac{5}{17} = \cfrac{12-5}{17} = \cfrac{7}{17}\)
However, when the denominators are different we have to change the fractions to make the demoninators the same. We do this by using equivalent fractions. You cannot the numerators when the demoninators are different because the "size" of each peice is different.
Let's say you have one pizza and cut it into \(2\). Each slice is \(\cfrac{1}{2} \). Take another pizza and cut it into \(4\). Each slice is \(\cfrac{1}{4}\) which is smaller than the other slice from the other pizza. The slices are different, so when you add them together it is not super obvious what fraction you have.
\(\cfrac{1}{2} + \cfrac{1}{4} = ?\)
To add these fractions together, let's find an equivalent fraction for \(\cfrac{1}{2}\) that has a denominator of \(4\):
\(\cfrac{1*?}{2*?} = \cfrac{?}{4} = ?\)
We will need to multiply the denominator by \(2\) in order to get a \(4\). That means, multiply the numerator by \(2\) as well:
\(\cfrac{1*2}{2*2} = \cfrac{2}{4} = ?\)
This means the fractions \( \cfrac{1}{2}\) and \( \cfrac{2}{4}\) are equivalent. Now, let's rewrite the original addition problem and add now that the demoninators are the same:
\(= \cfrac{1}{2} + \cfrac{1}{4}\)
\(= \cfrac{2}{4} + \cfrac{1}{4}\)
\(= \cfrac{2+1}{4}\)
\(= \cfrac{3}{4}\)
Typically, you want to find the lowest common denominator (LCD) between the two fractions and make both fractions over the LCD. To find the LCD, you want to find the smallest common multiple of the denominators. To find the multiples, just count by that number:
\(2: 2, 4, 6, 8, 10\)
\(4: 4, 8, 12, 16, 20\)
The LCD here is \(\boldsymbol{4}\) because that is the smallest common multiple.
These fractions do not have the same denominator so we cannot just subtract the numerators. Let's find the LCD. First, list the multiples of each number:
\(8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80\)
\(7: 7, 14, 21, 28, 35, 42, 49, 56, 83, 70\)
The lowest common multiple is \(56\) so we will find equivalent fractions with denominators of \(56\).
We will need to work with each fraction separately since they have different denominators and we want to write an equivalent fraction with a denominator of \(56\).
For the first fraction we want to find a number such that \(8x = 56\).
\(\cfrac{7*?}{8*?} = \cfrac{?}{56}\)
We need to multiply by \(7\) on both the top and bottom:
\(\cfrac{7*7}{8*7} = \cfrac{49}{56}\)
For the second fraction we want to find a number such that \(7x = 56\).
\( \cfrac{6*?}{7*?} = \cfrac{?}{56} \)
We need to multiply by \(8\) on both the top and bottom:
\(\cfrac{6*8}{7*8} = \cfrac{48}{56}\)
Now we can rewrite the question with the new equivalent fractions:
\(=\cfrac{7}{8} - \cfrac{6}{7}\)
\(=\cfrac{49}{56} - \cfrac{48}{56}\)
The fractions have the same denominators so just subtract the numerators:
\(= \cfrac{1}{56}\)
Therefore, we can determine that the final result is \(\boldsymbol{\cfrac{1}{56}}\).
To multiply fractions, simply multiply across: multiply the numerators and then multiply the denominators. Don't forget your rules for integer multiplication!
Evaluate \(\cfrac{3}{7} * \cfrac{-2}{3}\).
First, we can multiply the respective numerators and denominators across both fractions:
\(= \cfrac{3*(-2)}{7*3}\)
\(= \cfrac{-6}{21}\)
Next, we can reduce the final answer to its lowest terms. Here, we can divide both top and bottom by \(3\):
\(= \cfrac{-6/3}{21/3}\)
\(= \cfrac{-2}{7}\)
Therefore, we can determine that the final result is \(\boldsymbol{\cfrac{-2}{7}}\).
To divide fractions, we are going to multiply by the reciprocal. A reciprocal is a fraction where you switch the top and bottom. Once you change the division question to multiplication by the reciprocal, use the rules to multiply fractions.
Evaluate \(\cfrac{3}{7} \div \cfrac{-2}{3}\).
First, we can change division to multiplication by finding the reciprocal of the second fraction:
\(\cfrac{3}{7} \div \cfrac{-2}{3}\)
\(= \cfrac{3}{7} * \cfrac{3}{-2} \).
Next, we can multiply the respective numerators and denominators across both fractions:
\(= \cfrac{3*3}{7*(-2)}\)
\(= \cfrac{9}{-14}\)
\(= \cfrac{-9}{14}\)
In this example, you cannot reduce the fraction any further. However, we typically put the negative sign on top.
Therefore, we can determine that the final result is \(\boldsymbol{\cfrac{-9}{14}}\).
First, we can change the division sign to multiplication by finding the reciprocal of the third fraction:
\(= \cfrac{2}{5} * \cfrac{-1}{6} * \cfrac{10}{4}\)
Next, we multiply the numerators across and denominators across. With three fractions, we can go all the way across:
\(= \cfrac{2*(-1)*10}{5*6*4} \)
\(= \cfrac{-20}{120}\)
Finally, we want to reduce the fraction to its lowest terms:
\(= \cfrac{-20/20}{120/20}\)
\(= \cfrac{-1}{6}\)
Therefore, we can determine that expression evaluated is \(\boldsymbol{\cfrac{-1}{6}}\).