Subtracting fractions may seem intimidating, but it’s actually quite simple.
When you subtract fractions, you’re essentially finding the difference between two values that are represented as fractions.
To do this, you subtract the numerators once the denominators are the same.
The numerator is the top number in a fraction, and the denominator is the bottom number.
\frac{numerator}{denominator}
To subtract fractions, you must have a common denominator . If the denominators are not the same, you’ll need to find a common denominator before you can subtract the fractions. Once you have a common denominator, you can subtract the numerators and write the answer over the common denominator.
Understanding Fractions
Fractions are a way of representing parts of a whole. They are used to describe quantities that are not whole numbers. A fraction consists of two parts: the numerator and the denominator. The numerator represents the number of parts that are being considered, while the denominator represents the total number of parts that make up the whole.
Types of Fractions
There are three main types of fractions: proper fractions, improper fractions, and mixed numbers.
A proper fraction is a fraction where the numerator is smaller than the denominator. For example, \frac{1}{2} is a proper fraction because the numerator is smaller than the denominator.
An improper fraction is a fraction where the numerator is greater than or equal to the denominator.
For example, \frac{5}{3} is an improper fraction because the numerator is greater than the denominator.
A mixed number is a combination of a whole number and a proper fraction.
For example, 3\frac{1}{2} is a mixed number because it represents 3 whole units and \frac{1}{2} of another unit.
Fractions can also be classified as equivalent fractions, like fractions, and unlike fractions. Equivalent fractions are fractions that represent the same quantity, but are written in different forms. Like fractions have the same denominator, while unlike fractions have different denominators.
Understanding the different types of fractions is important when working with fractions, as it helps you to identify and compare fractions more easily.
You can read more about fractions here.
Subtracting Fractions with the Same Denominator
When subtracting fractions with the same denominator, the process is quite simple. You only need to subtract the numerators, as the denominator remains the same.
To subtract two fractions with the same denominator, you need to find the difference between their numerators.
For example, suppose you want to subtract \frac{3}{5} from \frac{2}{5} , as the denominators are the same, you can focus on the numerators by subtracting 3 from 2 , which gives you 1 . So the answer is \frac{1}{5} .
\frac{3}{5} - \frac{2}{5} = \frac{3 - 2}{5} = \frac{1}{5}
In summary, when subtracting fractions with the same denominator, you only need to subtract the numerators, and the denominator remains the same.
Subtracting Fractions with Different Denominators
When subtracting fractions with different denominators, you need to find a common denominator. This can be done by identifying the least common multiple of the two denominators. After finding the common denominator, you can subtract the numerators as you would with any two fractions.
Finding the Common Denominator
To find the common denominator of two fractions, you need to identify the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators can divide into evenly.
For example, let’s say you want to subtract \frac{3}{8} from \frac{1}{12} . The denominators are 8 and 12 .
To find the LCM, list the multiples of each denominator until you find a common multiple.
\{ 8, 16, 24, 32, 40, 48, ... \} \\ \{12, 24, 36, 48, ... \}
The smallest number that appears in both lists is 24 , so the LCM of 8 and 12 is 24 .
\{ 8, 16, \fcolorbox{red}{yellow}{24}, 32, 40, \fcolorbox{red}{lightyellow}{48}, ... \} \\ \{12, \fcolorbox{red}{yellow}{24}, 36, \fcolorbox{red}{lightyellow}{48}, ... \}
Converting Fractions to Equivalent Fractions
Once you have found the common denominator, you need to convert each fraction to an equivalent fraction with the common denominator. To do this, you need to multiply the numerator and denominator of each fraction by the same number so that the denominator becomes the LCM.
For example, to convert \frac{3}{8} to an equivalent fraction with a denominator of 24 , you need to multiply both the numerator and denominator by 3 :
\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}
To convert \frac{1}{12} to an equivalent fraction with a denominator of 24 , you need to multiply both the numerator and denominator by 2 :
\frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24}
Subtracting the Fractions
Once both fractions have the same denominator, you can subtract the numerators as you would with any two fractions. Simply subtract the numerators and write the result over the common denominator.
\frac{9}{24} - \frac{10}{24} = \frac{9 - 2}{24} = \frac{7}{24}
Remember to simplify the resulting fraction if possible .
Subtracting Mixed Fractions
When subtracting mixed fractions, you need to follow a few steps. There are two approaches when subtracting fractions with the most common being the conversion of all the mixed numbers into improper fractions. The second approach separates the fraction component from the mixed numbers and applies the normal subtraction of fractions as performed above and handles the mixed numbers separately.
Converting Mixed Fractions to Improper Fractions
First, you need to convert the mixed fractions to improper fractions. Then, you can subtract the fractions and convert the result back to a mixed fraction.
To convert a mixed fraction to an improper fraction, you need to multiply the whole number by the denominator of the fraction and then add the numerator. The result will be the numerator of the improper fraction, and the denominator will stay the same.
For example, if you want to convert 3\frac{1}{2} to an improper fraction, you would multiply 3 by 2 and add 1 to get 7. The denominator would stay 2, so the improper fraction would be \frac{7}{2} .
Once you have converted the mixed fractions to improper fractions, you can subtract the fractions. To do this, you need to find a common denominator for the fractions. You can do this by multiplying the denominators together.
For example, if you want to subtract 2\frac{1}{3} from 3\frac{2}{5} , you would convert both mixed fractions to improper fractions. 2\frac{1}{3} becomes \frac{7}{3} , and 3\frac{2}{5} becomes \frac{17}{5} .
2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \\ \ \\ 3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}
Now that you have two fractions you can apply the same technique as above by finding the lowest common denominator and obtaining the equivalent fractions for both.
The LCM of 3 and 5 is 15 , therefore, to obtain the same denominator for both fractions you would need to multiply by a common factor:
\frac{7}{3} = \frac{7 \times 5}{3 \times 5} = \frac{35}{15}
\frac{17}{5} = \frac{17 \times 3}{5 \times 3} = \frac{51}{15}
Finally, you can subtract the fractions by subtracting the numerators and keeping the denominator the same.
\frac{35}{15} - \frac{51}{15} = \frac{35 - 51}{15} = \frac{-16}{15} = -1\frac{1}{15}
Separate Out Mixed Numbers And Fractions
Another way to perform the same process is to separate out the mixed numbers from the fractions.
Using the same example above, suppose:
2\frac{1}{3} - 3\frac{2}{5}
Separate the fractions from their mixed number, ensuring you keep the same sign as at the front of the mixed number . This would look as follows with the above subtraction problem:
2 + \frac{1}{3} - 3 - \frac{2}{5}
Notice each fraction needs to have the same sign as that in front of the mixed number. Therefore, as the 2\frac{1}{3} is positive, separating these two numbers means they are broken up as 2 + \frac{1}{3} .
Whereas the second fraction -3\frac{2}{5} needs to be broken up into -3 - \frac{2}{5} .
Focus on operating on the fractions first:
\frac{1}{3} - \frac{2}{5} = \frac{1 \times 5}{3 \times 5} - \frac{2 \times 3}{5 \times 3} = \frac{5}{15} - \frac{6}{15} = \frac{5 - 6}{15} = \frac{-1}{15}
The same process with operating on fractions applies and has been shown above giving you the result of \frac{-1}{15} .
The last aspect is to perform the operations on the mixed numbers:
2 - 3 - \frac{1}{15} = -1 - \frac{1}{15} = -1\frac{1}{15}
As both the resulting mixed number of 2 - 3 = -1 has the same sign as the fraction they can easily be merged together to form the same result.
This method may be easier to perform as it doesn’t require converting the mixed number into improper fractions. However, this method does mean you need to be aware of what you are doing when separating out a fraction from it’s mixed number.
Remember: whatever the sign is in front of the mixed number is the same that needs to be placed in front of the fraction.
Subtracting Fractions Summary
Subtracting fractions involves finding the difference between two fractions. Fractions are parts of a whole, and they consist of a numerator and a denominator. The numerator represents the number of parts being considered, while the denominator represents the total number of parts in the whole.
To subtract fractions, you need to make sure that the denominators are the same. If the denominators are different, you need to find a common denominator before you can subtract the fractions. Once you have a common denominator, you can subtract the numerators and simplify the resulting fraction if necessary.
When subtracting mixed fractions, you need to convert them to improper fractions first. To do this, you need to multiply the whole number by the denominator and add the numerator. The resulting fraction will have the same denominator as the original mixed fraction.
Subtracting fractions can be a bit tricky, but with practice, it becomes easier. Remember to find a common denominator, subtract the numerators, and simplify the resulting fraction if necessary.