Fractions may be hard to grasp but they’re the foundation for other math skills. Perhaps the most important thing fractions teach us is proper calculation. This skill is essential especially for technical and medical professions.
Check out the different types of fractions below.
Types of Fractions
Decimal fractions consist of both a whole number and a decimal fraction. A decimal fraction is a fraction with a denominator that has been divided into the numerator. Decimal fractions are those that have a denominator in the power of ten and whose numerator is expressed by placing figures to the right of a decimal point.
They also include the type called periodicals, which means there is a sequence of digits behind the period, or the decimal point, that is endlessly repeated. These are examples:
1/3 = 0.33333 …
1/7 = 0.142857142857 …
Equivalent fractions are fractions that have the same value as each other. In other words, 9/12 is equal to ¾. Equivalent fractions may look different, but they each have the same value. They are the same for one main reason – when you multiply or divide both the top and the bottom by the same number, the fraction keeps its value.
These are fractions where the numerator, or top number, is bigger than the denominator, or bottom number. In other words, this type of fraction is always greater than one. Improper fractions can be rearranged as a mixed number. In other words, 1 7/8 can become 15/8.
If a fraction is written after a whole number, this is an example of a mixed fraction.
Also called vulgar fractions or common fractions, these include fractions which are smaller than one but greater than zero.
All fractions have three parts – the numerator (top number), the denominator (bottom number), and a division sign. Simple fractions refer to those fractions that have integers as the numerator and the denominator. They are further broken down into two sub-types, which are as follows:
- Irregular: fractions with a denominator that is smaller than the numerator.
- Regular: fractions with a numerator that is smaller than the denominator.
Unit fractions have the number 1 as the numerator and a whole integer as the denominator. Examples of unit fractions include 1/10, 1/3, 1/25, and 1/100.
How to Work with Fractions
To add fractions, there are three steps that you must take. These steps are as follows:
- Make sure the denominators, or the bottom numbers, are the same.
- Add the numerators, or the top numbers, and put that answer over the denominator.
- If necessary, simplify the fraction.
Dividing fractions can be done if you take the following steps:
- Take the fraction you wish to divide by, then turn it upside down. It is now called a reciprocal fraction.
- Multiply the first fraction by that reciprocal fraction.
- Simply the fraction if needed.
Here are the three steps you need to take in order to multiply fractions:
- Multiply the top numbers or the numerators.
- Multiply the bottom numbers or the denominators.
- If necessary, simply the fraction.
Simplifying (Reducing) Fractions
There will be times when you are unable to work with fractions until you get them into their simplest form. To do this, you must reduce, or simplify, the fractions, which is a lot less complicated than it seems. There are two ways to do this, which are as follows:
- Using whole numbers only, keep dividing both the top and bottom number by 2, 5, 7, etc., until you can no longer divide them.
- Find the greatest common factor first; then, divide both the top and the bottom numbers of the fraction by this number.
Subtracting fractions involve three steps, which are described below:
- Make sure the denominators, or the bottom numbers, are both the same.
- Subtract the numerators, or the top numbers, and place that answer over the denominator.
- Simplify the fraction if needed.
As you can see, with both addition and subtraction of fractions, the denominator has to be the same in both fractions in order to proceed to the second step of the process.
Comparing Fractions with Like Denominators
If two fractions have the same denominator, the fraction that has the largest numerator is the larger fraction. For instance, 5/8 is larger than 3/8 because all pieces are the same, and three pieces are less than five pieces.
Comparing Fractions with Unlike Denominators
If you have two fractions that have the same numerators, the fraction with the smaller denominator is the bigger fraction. For example, 5/8 is bigger than 5/16. This is because each of these fractions says there are five pieces.
However if an item is divided into eight pieces, each piece is going to be bigger than if the item was divided into 16 pieces. Because of this, five bigger pieces are greater than five smaller pieces.
Comparing Unlike Fractions
If you have two fractions with different numerators and denominators, it can be a challenge to find out which of them is bigger, but there is a way to do this. First, you can multiply both the numerator and the denominator of the same fraction by the same numbers, which enables both fractions to have the same denominator.
As an example, if you are comparing 5/12 and 1/3, you should multiply 1/3 by 4/4. If you do this, you end up with 4/12, which you can then easily compare to 5/12.
In some instances, you have to multiply both fractions by different numbers in order to get the same denominator for both fractions. Take a look at this: if you’re comparing ¾ and 2/3, multiply 2/3 by 4/4 to get 8/12; then, you can multiply ¾ by 3/3 so that you get 9/12. This means you can correctly compare the two fractions.
Furthermore, the fraction ¾, which is equal to 9/12, is larger than the fraction ⅔, which is equal to 8/12. The fraction with the bigger numerator is the larger fraction since the denominators are the same.
Comparing Fractions and Decimals
Fractions and decimal numbers can be compared, and one number is always less than, greater than, or equal to the other number. If you’re comparing fractions to decimal numbers, simply convert the fraction to a decimal number by using division, then compare the decimal numbers. If a decimal has a higher number to the left of the decimal point, then it is larger.
If the numbers to the left of the decimal point are equal, but one of the decimals has a higher number in the tenths place, that number is larger. If the numbers in the tenths place are equal, you compare the hundredths, etc., until one decimal is larger or there are no more places to compare.
Sometimes you can estimate the decimal from a fraction. If the estimated decimal is obviously much smaller or larger than the decimal it is being compared to, you do not have to convert the fraction to a decimal.
Things You Need to Know about Fractions
Any Number Can be Written in a Fraction
Try it by taking any whole number and writing it in fraction form as the numerator. For example, 4 is always equal to 4/1.
At Any Time, You Can Multiply by Any Form of One
The number 1 is always called the multiplicative identity. This is because it can be multiplied by any number and that number will remain the same. This is an important aspect of fractions because oftentimes, the appearance of a fraction needs to be altered without changing its actual value.
For example, the number 1 can change the fraction 1/3 into the equivalent fraction 3/9 simply by multiplying it by 3/3, which represents the number 1. So,
1/3 x 3/3 = 3/9
Always Change Mixed Numbers First
A mixed number is a combination of a whole number and a fraction, and when working with fractions, you need to change that mixed number before you do anything else. If you change the mixed number 2 2/3 into 8/3, you can work with it much more easily and in any capacity (multiplication, division, etc.).
When You Compare Fractions, Use the Cross Point
For example, if we start with the fractions (remember, never used mixed numbers) 5/12 and 6/13, and we wish to show which one is bigger, we can take the following steps to find out:
- Multiply a diagonal (the denominator of the first fraction by the numerator of the second one), then write the answer above the numerator. Therefore, 12 x 6 = 72, so write the number 72 in the numerator section of the fraction 6/13.
- Multiply the other diagonal and write that answer above the numerator of the fraction 5/12. So you would have the number 65 (13 x 5) above the numerator in the fraction 5/12.
- Compare the two final numbers (the products). If you compare 72 and 65, you’ll see that 72 is the largest number, which means 6/13 is bigger than 5/12.
You Can Use Multiplication to Divide Fractions
Let’s say you have a complicated fraction, for example, 8/9 divided by 2/3. First, we have to keep the following two facts in mind:
- You can multiply by any form of one (i.e., anything that is over itself).
- If you multiply by the reciprocal of 3/2 (i.e., 2/3), it results in 1 after cancelling.
This is the best way to figure out the answer to this problem:
- First, write the equation like this: 8/9 divided by 2/3, multiplied by 3/2 divided by 3/2. This is because you want to multiply by the reciprocal over itself.
- Cancel out anything that divides to 1 in the bottom fraction, which should always result in the number 1. So, now you have 8/9 multiplied by 3/2 on top of the fraction, and 2/3 multiplied by 3/2 on the bottom of the fraction. After you figure this out, you end up with 8/9 multiplied by 3/2 (since the number at the bottom is going to be 1).
- Cancel in order to reduce the fractions. So, you will multiply 4/3 by 1/1, which will equal 4/3.
So, in this problem, your answer is 4/3, or 3 1/3.
Fractions Always Equal the Number of Parts Divided by the Total Parts
So, fractions can be 4/6, 2/3, 7/8, 10/12, etc.
Always Multiply Fractions Straight Across
So, 3/5 multiplied by 7/8 is going to total 21/40, since 3 x 7 (the two numerators) is equal to 21, and 5 x 8 (the two denominators) is equal to 40.
Always Add and Subtract Equal Sized Parts
If you are going to add or subtract fractions, the denominators have to be the same. Therefore, if you subtract 7/21 from 15/21, you get the answer 8/21, since 15-7 (the numerators) is equal to 8, and you keep the denominator at 21.
Converting a Multiple Number Requires Using Addition
Let’s say you want to convert the number 2 4/5 into an improper fraction. The following are the steps you’d take:
- Rewrite 2 as 2/1, then add 2/1 to 4/5.
- Next, multiply 2/1 by 5/5 (i.e. 1) to get 10/5, so that each fraction has the same denominator.
- Add 10/5 to 4/5, getting the result 14/5. As you can see, 14/5 converts back to 2 4/5, but you have now converted the number.
You Can Cancel Anything That Divides to One
Fractions are easy to cancel out, which makes them a lot easier to work within certain situations. For example, in the fraction 8/10, both 8 and 10 can be rewritten using the factor of two. Therefore, you can start by writing it like this: 8/10 = 2 x 4 / 2 x 5. Therefore, 8/10 = 4/5 after you cancel out the 2’s. In other words, 4/5 is the reduced fraction of 8/10.
Fun and Interesting Facts about Fractions
Fractions Are Very Old
Fractions date way back, thousands of years. In the year 2000 BC, Egyptians used fractions to calculate taxes. Various parcels of land were divided into sections, and each of the sections was taxed separately. In 1550 AD, the British adopted this idea and paid for the profits of voyages using fractions.
You Can Explain the Infinite Chocolate Trick with Fractions
In this trick, you divide an entire bar of chocolate into two trapezoids, one rectangle, and a small square. In this instance, an extra piece of chocolate is obtained when you rearrange the divided sections, but how?
Here is the answer: when rearranging the divided sections, the extra piece is equal to the tiny piece of chocolate that is lost while you are aligning the sides of the trapezoid.
Pi Has No Exact Fractional Representative
Pi is an irrational number with a decimal expansion that never terminates. When people represent the value of Pi through the fraction 22/7, it is only precise to two decimal places. In reality, the fraction 355/113 is a better representation because it is precise to six decimal places. However, there is no fraction that exactly equals the decimal expansion of Pi.
Where Did the Word “Fraction” Come from?
People started using the term “fraction” as a math term sometime after the early fifteenth century. The word originated from the Latin word “fractio,” which means “a breaking.”
Fractional Representations of Long Ago Were Quite Different Than the Ones Used in the Present
The concept of fractions originated in Egypt and was displayed in the form of hieroglyphs. An example is the unit fraction, which was demonstrated by a picture of a mouth with a number below it.
The Fraction 1/998001 Is Mind-Blowing
When the fraction 1/998001 is converted to its decimal equivalent, the result is every three-digit number except 998. For example:
1/998001 = 0.00000100200300 … 99599699799900000100 …
Fractions All Have Certain Characteristics
You can explain fractions in the following way:
- A/B where A<B is a proper fraction.
- A/B where A>B is an improper fraction.
- A fraction that is written with a whole number is a mixed fraction.
- You can write an improper fraction as a mixed fraction, and vice versa.
Fractions Can Never Have the Number 0 in the Denominator
Under no circumstances can you put a 0 as a denominator. It just isn’t done.
Glossary of Terms Related to Fractions
Benchmark Fractions: These are fractions such as ¼, 1/3, ½, 2/3, and ¾.
Cancelling: This is the process of removing common factors from fractions. For example, since 2 is the common factor in the numerator and denominator of 4/6, it can be cancelled. In other words, 4/6 is equal to 2 x 2 divided by 2 x 3, which is equal to 2/3.
Complex Fraction: This is a fraction where the numerator and/or the denominator are a fraction.
Decimal: This is a number based on the number of ten. In other words, it is a special type of fraction which has a denominator in the power of ten.
Decimal Point: A decimal point is a dot or period which is considered part of a decimal number. It signals where the whole number stops and the fraction part starts.
Denominator: Refers to the bottom part of a fraction. The denominator tells you how many equal parts the item has been divided into. In the fraction 7/8, 8 is the denominator.
Equivalent Fractions: Fractions that look different but which have the same value. For example, 2/8 is the same as, or equal to 25/100.
Factors: Whole numbers can be multiplied to equal another number.
Fraction: A fraction is a part of a whole. Common fractions are made up of a numerator and a denominator, with the numerator located on top of the line and the denominator underneath the line. The numerator shows the number of parts of the whole, while a denominator represents the number of parts into which the whole has been divided.
Half: This is a common fraction that can be written in several ways, including 50%, .5, and ½.
Higher Term Fraction: This refers to a fraction in which the numerator and the denominator have a factor in common other than one. Another way to explain a higher-term fraction is that it can be reduced further.
For example, 2/8 is a higher-term fraction because both the numerator and the denominator have the factor two and therefore, the fraction can be reduced to ¼.
Improper Fraction: This is a fraction in which the numerator is greater than the denominator, which means it always has a value that is greater than one. An example would be the fraction 7/3.
Lowest Common Denominator (LCD): This refers to the lowest common multiple of the denominators of two or more fractions.
Lowest Term Fraction: This refers to a fraction that cannot be reduced any further. In addition, the only common factor between the top and bottom numbers is number one.
Mixed Number: This is a number made up of a whole number plus a fraction. An example would be 5 7/8.
Numerator: The top part of a fraction. This number shows how many equal parts of the denominator are represented.
Percent: In a fraction with a denominator of 100, the percent represents a special type of fraction, and it is written using the “%” sign. For example, 50% is the same as 50/100.
Prime Number: A number that has only two factors, the number 1 and itself. Prime numbers include 1, 2, 3, 5, 7, and 11.
Proper Fraction: This refers to a fraction with a numerator that is less than the denominator. The fraction 9/10 is a proper fraction.
Proportion: A proportion refers to an equation which states that two ratios are equal. For example, 1/3 = 2/6 is a proportion.
Ratio: This refers to a comparison of two numbers and can be written in a variety of ways. In other words, 1:2 is the same as 1 of 2.
Reciprocal: This is a fraction whereby the numerator and denominator are switched. If you multiply the reciprocal with the original number, the number 1 is always the result. Because of this rule, all numbers have a reciprocal, with the exception of 0.
Simplest Form: This means the numerator and the denominator have no common factors other than the number 1.
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