You might be familiar with fractions. If not, then you might have encountered situations where you would have seen fractions. For example, while eating a piece of cake or pizza. The slice you grab in the first place is a fraction of the entire cake or pizza. Hence, a fraction is a numerical representation to show a part of a whole. The top portion of the fraction is the numerator, and the bottom is the denominator. Then what is a unit fraction?
Unit Fraction Definition
Now, coming to what is a unit fraction. Unit fractions are those in which the numerator is always equal to 1. The denominator can vary and be any other integer, but the numerator has to be 1 to be called a unit fraction. A good way to remember what is a unit fraction is to remember that unit means 1. Hence, the fraction with 1 at the top is a unit fraction. Unit fraction example – ½, ⅓, ⅕, ⅙, and many others.
According to Merriam Webster, a unit fraction is a fraction whose numerator is unity and whose denominator is an integer. The look-up popularity for this word is in the top 26% of words. The word unit fraction is placed near the words unit factor and unit heater.
Unit fractions are rational numbers. 1/25, 1/100, 1/200, and many others are all rational numbers. However, the numbers 7/33, 9/10, 67/96, and many others are rational numbers, too, but they are not unit fractions.
Each unit fraction is a part of 1. Half, third, fourth, or any other number of 1. Therefore, you can call them ‘one-fifths’, ‘one-fourths’, and so on. Unit fractions can be represented on a number line, as shown below.
⅖ = ⅕ + ⅕. Similarly, 5/5 = ⅕ + ⅕ + ⅕ + ⅕ + ⅕. Hence, every fraction is a number of unit fractions.
Non Unit Fractions
The fractions whose numerator is not equal to one, those are known as non unit fractions. Non unit fractions example – 2/6, ⅜, 5/7, and others.
Let us take another example to understand unit fractions. Consider a rectangle, as shown below, with 10 small squares.
If you take one square out of those ten, the fraction will be 1/10. If you take 5 squares out of those 10, the fraction will be 5/10, which can also be simplified as ½. Again, this is a unit fraction. Hence, simplifying the fractions will yield unit fractions in many cases.
Properties of Unit Fractions
There are various properties of unit fractions. Some of those are listed below. It will help if you memorize these properties as it will help you speed up the calculations and uplevel your preparedness.
- Unit fractions will always have the numerator equal to unity. The denominator can be any positive integer.
- Division of unit fractions with whole numbers results in a unit fraction. For example, if a unit fraction ¼ is divided by a whole number, say 5, the resulting fraction will be 1/20, a unit fraction.
- Dividing unit fractions with rational numbers will yield non unit fractions. For example, if you divide ⅙ with ⅝, the resulting fraction will be 8/30, not a unit fraction.
- Multiplication of two unit fractions will always result in a unit fraction. For instance, 1/31 multiplied by 1/5 will yield 1/155, which is a unit fraction.
- Multiplication of unit fractions with a whole number will not result in unit fractions. For example, 1/7 multiplied with a whole number, say 4, will result in 4/7, a non unit fraction.
Elementary Arithmetic of Unit Fractions
Here is the arithmetic representation of unit fractions for your reference that will assist you to boost your mathematical calculations:
- 1xx 1y = 1xy
- 1x + 1y = x + yxy
- 1x – 1y= y – xxy
- 1x ÷ 1y= yx
Applications of Unit Fractions
There are various applications of unit fractions. Some of them are:
- Unit fractions are frequently used in probability and statistics. All probabilities are equal to unit fractions in a uniform distribution on a discrete space. Probabilities of this form often arise in statistical calculations due to the principle of indifference.
- According to Zipf’s law, many observed phenomena involve selecting items from an ordered sequence. The probability of selecting the nth term is proportional to the unit fraction 1/n.
- Unit fractions are mostly used in physics as well. The energy levels of photons that can be absorbed or emitted by a hydrogen atom are proportional to the differences of two unit fractions. This law is stated by Rydberg and is known as the Rydberg formula.
- Bohr’s model also uses unit fractions. According to Bohr’s law, energy levels of electron orbitals in a hydrogen atom are inversely proportional to square unit fractions. The photon’s energy is quantized to the difference between the two levels.
- Arthur Eddington stated that the fine-structure constant was a unit fraction. The first was 1/136, and the second was 1/137.
Unit Rates With Fractions
You have seen what unit fractions are in the above section. In that, the numerator was equal to 1. What if the denominator is equal to 1? Will that be called a unit fraction? Well, no!
When the fractions have denominators equal to 1, those are known as unit rates with fractions. Hence, the fraction with the denominator equal to 1 is the unit rate of that fraction. For example, 60/6 can be written as 10, representing a unit rate. Similarly, 57/8 is a unit rate and can be written as 7.125.
Unit rates are helpful in everyday life to find relative speed, calculate distance and find out any change in a unit. Unit rates are beneficial to find which commodity changes concerning another commodity. For example, to find time, distance changes concerning speed. Hence, time is the unit rate change, or simply the rate of change, of distance with respect to speed.
Unit Fraction Example
Let us try a few examples to understand unit fractions deeply. Make sure to practice these questions to make your fundamentals strong.
Example 1: In the fraction ⅚, how many unit fractions are there?
Solution: To find the unit fraction, make the numerator equal to 1 and keep the denominator as it is. Therefore, in this case, the unit fraction will be ⅙. Since the numerator in the question is 5, therefore, the total number of unit fractions will be 5.
⅚ = ⅙ + ⅙ + ⅙ + ⅙ + ⅙.
Example 2: How will you count ⅓ to 2 ⅓?
Solution: As seen from the number line below, the numbers go in order,
⅓, ⅔, 3/3 = 1, 4/3 = 1 ⅓, 5/3 = 1 ⅔, and so on.
Again, this is the sum of unit fractions. Hence, you can count them as:
⅔ = 2 x ⅓ = ⅓ + ⅓
⅜ = 3 x ⅛ = ⅛ + ⅛ + ⅛
Example 3: Add 2/8 and ⅜.
Solution: To add these, follow the pattern shown below:
2/8 = ⅛ + ⅛
⅜ = ⅛ + ⅛ + ⅛
Adding all the ⅛ given above, you will get, ⅛ + ⅛ + ⅛ + ⅛ + ⅛ + ⅛ + ⅛ + ⅛ = ⅝.
Alternatively, the answer can be found directly by adding the numerators. That is because the denominators are equal so that the numerators can be added directly. Therefore, the answer will be ⅝.
Example 4: 1 in how many fifths?
Solution: To find the answer to this questions, check the number line shown below:
From the above number line, you can see five fifths in one. Therefore, there are five ⅕ in 1.
Similarly, you can find the rest of the numbers. There are 3 thirds in 1, four fourths in 1, and so on.
Example 5: Add the given fraction and express it in improper fraction: 5/9 + 1.
Solution: To solve this, convert 1 into unit fractions. Therefore, 1 will be 9/9.
Now, adding both of these fractions, you will get,
5/9 + 9/9 = 14/9, which is an improper fraction. Here, you have expressed nine ninths to make it a unit fraction.
Example 6: Add 428 and ⅜.
Solution: Convert the mixed fraction into an improper fraction, you will get 34/8. Now, add 34/8 and ⅜, you will get 37/8. Lastly, convert them into mixed fraction as 458.
Alternatively, when the denominators of a mixed fraction, and proper fraction are the same, you can add them directly. Hence, 2/8 + ⅜ will give ⅝ and 4 will not change. Therefore, 428+ ⅜ = 458.