Imagine you are having a Thanksgiving dinner, and everyone decides to contribute a fraction of their pizza to you. Can you determine how much pizza is in your possession? In order to achieve this, the concept of the lowest common denominator must be understood. Hereafter, you shall be learning this.
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Jetzt kostenlos anmeldenImagine you are having a Thanksgiving dinner, and everyone decides to contribute a fraction of their pizza to you. Can you determine how much pizza is in your possession? In order to achieve this, the concept of the lowest common denominator must be understood. Hereafter, you shall be learning this.
This article will explore the definition of the lowest common denominator, and examples of the application of this concept.
Given a list of fractions, the lowest common denominator (LCD) is the smallest multiple that is common to the denominators of the fractions. It is often referred to as the Lowest Common Multiple (LCM) of denominators.
This sounds complicated, but it's really not. Let's have a look at an example:
The Lowest Common Denominator of \( \frac{2}{3}\) and \( \frac{3}{4}\) is \(12\), since \(12\) is a multiple of both the denominators \((3\) and \(4),\) and it's the smallest multiple of these numbers you can find.
When a list of fractions is given, and the LCD is to be determined, we look for the lowest number large enough to be divided by each denominator of the list without a remainder.
A common denominator is a number that can be divided by other denominators without a remainder. This number is a multiple of other denominators.
You consider the fractions and : their common denominator must be divisible by both denominators (4 and 6) without a remainder. Thus, in order to find their common denominator, you have to consider multiples of each denominator.
Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, ...
Amongst the multiples of both denominators listed above, the common multiples are 12, 24, 36… This is true because when these numbers are divided by the denominators 4 and 6, they do not have a remainder.
However, the multiples of a number are infinite. Imagine how many numbers would be written as multiples of 4 and 6, we probably would run out of papers. So, mathematicians came up with the lowest common denominator or lowest common multiple, which uses only the smallest number which is a multiple among a set of numbers.
Hence, the LCD between 4 and 6 is 12.
There are four methods that can be used in finding the LCD:
LCD rules
Multiples' listing
Prime factor product
Combined prime factor product.
One way to solve either LCM or LCD is by applying the LCD rules. Note that these rules are best applied only among a set of two numbers. It can also be used in a list of numbers above two but may be more complex if not well applied.
The following are the LCD rules:
When two fractions have the same denominator, the LCD is that denominator.
If a number among a set of denominators is a multiple of all other numbers in that set, the LCM is that multiple.
The LCD of a set of denominators which are prime numbers is the product of those prime numbers. You must ensure that these denominators are prime numbers.
The LCD of a list of denominators that have no common factor is the product of those denominators. You must be sure that no number can divide any pair without a remainder.
If there is a common factor between a set of two numbers, the LCD is obtained by the following two steps:
Multiply your first result with the other number. This answer becomes the LCD.
For instance, the LCD between fractions and is 2 because the denominator of the fractions is the same.
The LCD between , , and is 20. This is because, in the denominators list of 2, 4 and 20, we have that 2 and 4 are factors of 20. Thus, 20 is a multiple of 2 and 4, so, 20 is the LCD.
The LCD between and is 15. These denominators 3 and 5 are prime numbers. Thus,
15 is the LCD here.
The LCD between and is 90. Denominators 9 and 10 are not prime numbers. Most importantly, there is no common factor between these two numbers. So the LCD is
The LCD between and is 40. The highest common factor between 8 and 20 is 4. Divide 8 by 4 and your first answer is 2. Then multiply your first result which is two by the other number 20,
So the LCD, in this case, is 40.
In some scenarios, you may not be able to understand which LCD rule you should apply. In these cases, the following methods to find the lowest common denominator of a list of fractions come in handy.
In this method, you are to list some multiples of each denominator. Afterward, you pick the lowest common multiple among all multiples listed - that is the left-most number that appears in both lists. To apply this method, you should be familiar with the multiplication table
Find the LCD of , and .
Solution:
Step 1. What are the denominators?
The denominators are 2, 4 and 6
Step 2. Write out the multiple of these numbers.
Multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
Multiples of 4 are 0, 4, 8, 12, 16, 20, 24 , 28, 32, 36, 40, 44, 48, ...
Multiples of 6 are 0, 6, 12, 18, 24 , 30, 36, 42, 48, 54, 60, 66, 72, ...
You would notice that 12 and 24 are common multiples of 2, 4 and 6. But, we want the lowest common multiple. So, our lowest common multiple is 12. This means that our LCD is 12 .
Note that the Multiples' listing method is best used when
The numbers involved are small, something between 2 to 12 eg, 2, 3, 4, 5, 6, 7... 12. Above 12 it gets tricky to write multiples down!
There are no more than 3 fractions.
More technically, the number of elements in a set is called the cardinality of the set. So the listing method is best applied on sets of fractions with cardinality 3 or less.
E.g, finding the LCD between and or between , and . This would reduce mistakes and time spent in finding the LCD.
Find the LCD between and
Solution:
Step 1. What are the denominators?
The denominators are 5 and 7
Step 2. Write out the multiple of these numbers.
Multiples of 5 are 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...
Multiples of 7 are 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ...
From the above, it is clear that the LCD is 35.
Find the LCD between, and .
Solution:
Step 1. What are the denominators?
The denominators are 2, 3 and 4
Step 2. Write out the multiple of these numbers.
Multiples of 2 are 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
Multiples of 3 are 0, 3, 6, 9, 12, 15, 18, 21, 24 , 27, 30, 33, 36, ...
Multiples of 4 are 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...
You would notice that 12 and 24 are common multiples of 2, 4 and 6. But, we want the lowest common multiple. So, our lowest common multiple is 12. This means that our LCD is 12.
3. The difference between any two numbers in a list is not more than 10.
All these must be considered before applying this method because it helps you stay within the range of your multiplication table.
The prime factor product method is done by writing each number of a set as a product of its prime factors. Remember that a prime factor is a prime number that divides another number without a remainder.
Find the LCD between and
Solution:
Step 1: Write out the denominators.
The denominators are; 24 and 10.
Step 2: Express each denominator as a product of its prime factor. To achieve this, you have to follow some simple sub-steps.
Sub-step A: Draw a column table with the denominator in the second column.
| 24 |
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Sub-step B: Use the smallest factor of the denominator and divide the number gradually until you arrive at 1. Remember that the factor is written in the first column.
2 | 24 |
| 12 |
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2 | 24 |
2 | 12 |
| 6 |
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2 | 24 |
2 | 12 |
2 | 6 |
| 3 |
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2 | 24 |
2 | 12 |
2 | 6 |
3 | 3 |
| 1 |
Sub-step C: Write all factors in the first column as a product.
2 | 24 |
2 | 12 |
2 | 6 |
3 | 3 |
| 1 |
Repeat the same step for the other denominator and you would have:
2 | 10 |
5 | 5 |
| 1 |
Step 3: Arrange these prime factors being multiplied and circle out similar factors.
This means that the common factor between 24 and 10 is 2.
Step 4: Multiply the rest of the factors which are not circled with your common factor.
So, our LCD is;
As a variation on the Prime Factor Product Method, the Combined Prime Factor Product Method requires that you find the prime factors of the denominators in just a single table. Once you are familiar with the technique, this method has the advantage of being more concise.
Find the LCD between and
Solution:
Step 1. Write out the denominators.
The denominators are 24 and 10.
Step 2. Create a table with the number of columns according to the number of denominators. In this case, the number of columns is 3 (one more than the number of denominators). Leave the first column for the factors.
| 24 | 10 |
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Step 3. Use the smallest prime factor to divide the denominators. Note that a factor may only be able to divide one denominator but fail to divide the rest. In such cases where the factor cannot divide one of the numbers in the denominator column, just write the same number in the cell below. You just need that factor to divide at least one of the denominators. It is easier by firstly trying out the smaller numbers like, 2,3, ... Remember that the factor is written in the first column.
2 | 24 | 10 |
| 12 | 5 |
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Note that in this case 2 can divide both 24 and 10 without a remainder.
2 | 24 | 10 |
2 | 12 | 5 |
| 6 | 5 |
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Note that 2 can divide 12 but cannot divide 5, so we still bring down 5 in the cell below.
2 | 24 | 10 |
2 | 12 | 5 |
2 | 6 | 5 |
| 3 | 5 |
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2 | 24 | 10 |
2 | 12 | 5 |
2 | 6 | 5 |
3 | 3 | 5 |
| 1 | 5 |
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2 | 24 | 10 |
2 | 12 | 5 |
2 | 6 | 5 |
3 | 3 | 5 |
5 | 1 | 5 |
| 1 | 1 |
Step 4: Write all factors in the first column as a product.
2 | 24 | 10 |
2 | 12 | 5 |
2 | 6 | 5 |
3 | 3 | 5 |
5 | 1 | 5 |
| 1 | 1 |
Once you are left with ones in the last row, you can calculate the LCD as the product of the factors in the first column.
The LCD between the denominators 24 and 10 is then
The combined prime factor product method is the best method for a large set of denominators.
As mentioned earlier, the lowest common denominator is used in simplifying fractions involving addition and subtraction. Also, it is also used to arrange fractions in ascending and descending order. Let us take some examples to best understand its applications.
Simplify
Solution:
Step 1. Find the LCD of the denominators.
Using the LCD rules method, the 2nd rule says that if one of the denominators is a multiple of the other denominators, then the LCD is such multiple.4 is a multiple of 2, so the LCD between 2 and 4 is 4.Step 2. Use the LCD as a general denominator. Then divide the LCD by each denominator and multiply by the numerator.Step 3. Solve the arithmetic.Simplify
Solution:
Step 1: Find the LCD of the denominators.
Use the combined prime factor product method.
2 | 5 | 7 | 28 |
2 | 5 | 7 | 14 |
5 | 5 | 7 | 7 |
7 | 1 | 7 | 7 |
| 1 | 1 | 1 |
The LCD of , and is the product of the numbers in the first column, that is
.
Step 2: Use the LCD as a general denominator. Then divide the LCD by each denominator and multiply by the numerator.
Step 3: Solve the arithmetic.
Step 4: See if any number can divide through the numerator and denominator to simplify your fraction.
Divide the numerator and denominator by 7
id="3047903" role="math"
With the application of LCD fractions can now be compared to find out which fractions are larger or smaller. It is on this knowledge that fractions can either be arranged in ascending or descending order.
2 | 6 | 20 | 8 | 5 |
2 | 3 | 10 | 4 | 5 |
2 | 3 | 5 | 2 | 5 |
3 | 3 | 5 | 1 | 5 |
5 | 1 | 5 | 1 | 5 |
| 1 | 1 | 1 | 1 |
The LCD of denominators 6, 20, 8 and 5 is
.
Step 2: Use the LCD as a general denominator. Then divide the LCD by each denominator and multiply by the numerator separately.
Step 3: Only use the numerators in bold. Now arrange them from highest to lowest.
48, 20, 18 and 15 which is
and
The lowest common denominator (LCD) is the smallest multiple that is common between a set of numbers or denominators.
There are four methods which can be used in finding the lowest common denominator; LCD rules, multiples' listing, prime factor product and combined prime factor product.
An example of the lowest common denominator (LCD) is if you are asked the LCD between the denominators 2, 3 and 5. The LCD is 30 because 30 is the smallest multiple that is divisible by these numbers without a remainder.
Lowest common denominator is practically used in adding and substracting fractions. Also, it is used in arranging fractions in either ascending or descending order.
The LCD rules help you find the LCD in a quicker way and there are basically 5 of them.
What is a common denominator?
A common denominator is a number which can be divided by other denominators without a remainder.
What is the LCD between half and a quarter?
4
What is the LCD between one-third and one-fifth?
15
If 5, 8, 6 and 15 are respective denominators of fractions, what is their LCD?
120
What are the prime numbers in the between 5 and 20?
7, 11, 13, 17 and 19.
What are the factors of 24?
2, 3, 4, 6, 8, 12 and 24
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