Fractions are parts of a whole, i.e., they represent a section of a collection. The word fraction comes from ‘fractio,’ a Latin word that means ‘to break.’ The Egyptians used fractions to solve mathematical problems, including dividing food and supplies. Ancient Romans wrote fractions as words and not numbers. Indians first wrote fractions as numbers that appeared as one number above another. The Arabs were the first to add the line between the numbers differentiating them as numerators and denominators.
When we divide something whole into different parts, each piece becomes a fraction of that whole.
For example, when you cut a whole watermelon into two parts, both the parts become one-half of the whole watermelon or a fraction of the watermelon. One half of the watermelon is then mathematically represented as 12. When you further cut the two halves of the watermelon into two more parts, the whole watermelon is divided into four parts or fourths of the whole watermelon. Then, one part of the watermelon would be represented as ¼.
Another perfect example that can illustrate this concept is a pizza. Pizzas come as a circle that has been divided or sectioned into 4-6 parts or more, depending on the size. In this case, each slice of pizza represents a fraction of the whole, which is the pizza.
What are the Parts of a Fraction?
The numerator and denominator make up the two parts of a fraction. The horizontal bar which separates the numerator and denominator is called the fractional bar.
- The denominator is the number of sections into which the whole has been divided. Its place is below the fractional bar.
- The numerator indicates the number of sections of the fraction that are represented or selected. Its place is above the fractional bar.
For example, in the fraction 1227, 12 is the numerator, and 27 is the denominator.
Why do we use fractions?
Fractions tell us what portion of a whole you have, need, or want. Fractions are also easier to understand than decimals. They help to visualize the concept or system better.
Types of Fractions:
There are four basic types of fractions. They are:
- Unit Fraction – A fraction with 1 as the numerator. For example, 12, 14
- Proper Fraction – They have a numerator with a value less than the denominator. Example: 49, 910
- Improper Fraction – They have a numerator with a value greater than the denominator. Example: 37, 128
- Mixed Fraction – These consist of a whole number with a proper fraction. Example 5 34, 10 12
How do you simplify fractions?
To simplify a fraction, you can follow any of the following steps according to whichever is more appropriate:
- Find the Highest Common Factor:
To do this, note down the factors for both the numerator and the denominator. For example:
To simplify the fraction 824,
Step 1: Factors of numerator and denominator:
24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 8 = 1, 2, 4, 8
Step 2: Greatest/Highest common factor is the number 8.
Step 3: Divide both numerator and denominator by 8
8 divided by 8 = 1
24 divided by 8 = 3
Step 4: The simplified fraction is 13
- Simplifying Improper Fractions to Mixed Numbers
To simplify an improper fraction to a mixed number, the steps are:
- Divide the numerator by the denominator
- Write down the whole number as the result
- Write the remainder as the numerator of the fraction
- The denominator will stay the same
Addition and Subtraction of Fractions
To add or subtract fractions, let us first understand the types of fractions on this basis:
- Like fractions – these fractions have the same denominator.
Example: 34, 54, 94. These fractions all have the same denominator. Thus they are like fractions.
- Unlike fractions – these are fractions that have different denominators.
Example: 53, 49, 27. These fractions all have different denominators. Thus they are unlike fractions.
- Equivalent fractions – These are fractions that, once simplified, have the same value.
Example: 25 and 410. Once 410 is simplified, it becomes 25. Thus making 25 and 410 equivalent fractions.
Addition of Fractions:
To add fractions, the first step is to identify whether the denominators of the fractions are the same or different, i.e., whether they are like or unlike fractions.
For like fractions, add the numerators and put the answer over the same denominator.
26 + 56 = 2+56 = 76
Then, simplify if possible. In this example, simplification is not possible.
For unlike fractions, first, make the denominators the same number. The numerator and the denominator of fractions should be multiplied with their LCM (least common multiple). To find the LCM, all you have to do is identify the multiples of the denominators and then identify the least common multiple. The least common multiple is the smaller number on the list of multiples.
To add 211 and 34
The Lowest common multiple of 4 and 11 is 44. So, multiply both the numerator and denominator of each fraction with a number that will give 44 as the denominator.
1×411×4 + 3×114×11 = 444 + 3344
Now, add the two fractions as like fractions
444 + 3344 = 4 + 3344 = 3744
Subtraction of Factors:
Now, you might be asking how to subtract fractions with unlike denominators? Like adding fractions, the first step in subtracting fractions is to identify whether the denominators of the fractions are the same or different, i.e., whether they are like or unlike fractions.
For subtracting fractions with like denominators, subtract the numerators and put their difference over the common denominator.
Example: 47 − 37 = 4-37 = 17
For Subtracting fractions with unlike denominators, there are two methods:
To cross-multiply two fractions, first, subtract the second number from the first to get the answer’s numerator.
Example: Find the difference between 67 and 25
To get the numerator, cross multiply the two fractions. Then, subtract their products.
(6 x 5) − (2 x 7) = 30 − 14 = 16
Next, multiple the two denominators together.
7 x 5 = 35
Put the product of the cross-multiplication over the denominator to get your answer.
So your answer becomes 67 − 25 = 1635
- LCM Method:
This method is used for the addition of unlike fractions as well. [Tip: Be careful of the placement of the fractions since any interchanging of the two fractions will have two completely different results.]
To use this method, follow these steps:
- Step 1: Convert them to like fractions. First, find the LCM of the denominators. Follow the same steps taken in the addition of unlike denominators to find the LCM of the denominators.
- Step 2: Then, subtract the numerators.
- Step 3: Simplify the fraction if needed.
Example: Subtract 16 from 12
- First, find the LCM of the denominators.
Multiples of 2 = 2, 4, 6, 8
Multiples of 6 = 6, 12, 18
The least common denominator is 6.
- Second, find the equivalent fraction by multiplying both the numerator and denominator with a number so that the resulting denominator is 6.
1×32×3 = 16
- Rewrite the problem using the fractions
12 – 16 = 36 – 16
- Next, subtract the numerators since the denominators are now the same
36 – 16 = 3 – 16 = 26
- Simplify if possible
26 = 13
- So, the final answer is 12 – 16 = 13
So now you know the steps of adding and subtracting fractions with unlike denominators and like denominators. Fractions are important because we apply them in our everyday lives. While buying food, dividing money, or calculating distance and time, fractions help make it easier to find a solution to your problem. Knowingly or unknowingly, we need fractions in our day-to-day lives. Learning to add and subtract fractions is just a simple introduction to the world of fractions. There are many more methods, applications, and uses of fractions. Once you learn to add and subtract fractions, your foundation is solid for moving on to harder concepts such as multiplication and division.
Worksheet for Subtracting Factors with Unlike Denominators.
- Find the difference between:
- 59 – 26 = __________________
- 916 – 18 = __________________
- 820 – 618 = __________________
- 49 – 25 = __________________
- 314 – 29 = __________________
- Solve these word problems using the LCM method for subtracting fractions with unlike denominators
- Annie and Mark are competing in a running race. Annie has covered 24 of the total distance, and Mark has covered 15 of the total distance. What is the difference in the distance covered by both of them?
- Tom has 57 papayas. If he gives 23 of it to Jerry, what fraction of the papaya is left?
- A recipe needs 34 teaspoons of black pepper and 14 teaspoons of red pepper. How much more black pepper than red pepper does the recipe need?
- Kylie lives 49 miles from the zoo. Sandy lives 24 miles from the zoo. How much closer is Sandy to the zoo than Kylie?