Multiplication - Least Common Denominator
- Least Common Multiple
- Method to find the Least Common Multiple
- Applications of the Least Common Multiple
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Least Common Multiple
The smallest multiple that is common between two or more numbers is called their Least Common Multiple. It is abbreviated as LCM.
For example: The LCM of `21` and `42` is `42` as `42` is the smallest multiple that is common between `21` and `42`.
Method to find the Least Common Multiple (LCM)
There are several methods to find the Least Common Multiple, one of the most commonly used, is called the listing method.
Listing method: In this method, we first need to list the multiples of the numbers for which we are finding the Least Common Multiple. We can list the multiples in the table form. Let’s explore it with an example.
Example: LCM of `3` and `4`.
In order to find the LCM of `3` and `4`, we will follow a few steps.
Step `1`: Make the list of multiples of `3` and `4` separately. Observe the multiples list given below.
Step `2`: Circle the common multiples as shown in the below image.
Step `3`: Now look for the smallest multiple out of all the common (circled) multiples. This smallest circled number is called the Least Common Multiple.
Therefore the Least Common Multiple of `3` and `4` is `12`.
Applications of the Least Common Multiple
We usually use the Least Common Multiple when we do algebraic operations or comparisons using fractions.
Let’s explore them.
`1`. Addition of Fractions: We should find the LCM before adding the fractions with different denominators. Let’s understand this with an example.
Example: `5/6 + 7/9`
Solution: Since this calculation involves different denominators, we can determine the least common multiple of `6` and `9` to make both denominators the same.
Step `1`: Identify the denominators of the fractions (`6` and `9`).
Step `2`: List the multiples of each denominator as shown in the image below.
Step `3`: Circle the common multiples of `6` and `9`.
Step `4`: The least common multiple of `6` and `9` is `18`.
To add the fractions, rewrite the given fractions, changing their denominators to LCM and accordingly changing their numerator.
Look at the below image for equivalent fractions.
`5/6 + 7/9 = 15/18 + 8/18 = (15+8)/18 = 23/18`
Therefore, `5/6 + 7/9 = 23/18`.
`2`. Subtraction of the Fractions: For subtraction follow the same rules as for addition.
Example: `2/3 - 4/7`
Solution: Since the given fractions involve different denominators, we can determine the Least Common Multiple (LCM).
For LCM, we will list the multiples of each denominator.
Least common multiple (LCM) of `3` and `7 = 21`
Equivalent fractions of the above fractions are,
`2/3 = 14/21` and `4/7 = 12/21`
`2/3 - 4/7 = 14/21 - 12/21 = (14-12)/21 = 2/21`.
Therefore, `2/3 - 4/7 = 2/21`.
`3`. Comparing the fractions: To compare the fractions with different denominators we require the Least Common Multiple. Let’s understand this through the example given below.
Example: Compare `2/5` and `3/10`.
Solution: To compare, `2/5` and `3/10`, we will find the LCM of the denominators `(5, 10)`.
LCM of `5` and `10 = 10`
Equivalent fractions with LCM `10` are:
`2/5 = 4/10` and `3/10 = 3/10`
Since `4/10 > 3/10`, we can say `2/5 > 3/10`.
Please note that while finding LCM of `2` or more numbers, if one of the number is a multiple of the other numbers, then that number is the LCM of all the numbers.
In the example above, since `10` is a multiple of `5`, `10` becomes the LCM of `5` and `10`.
`4`. Ordering fractions: We can simplify the process of arranging the fractions of different denominators by finding their Least Common Multiple. Ordering fractions that involve different denominators is quite difficult. To make it easier we can find the LCM of the denominators and then we can convert those fractions into equivalent fractions that have the same denominators.
Example: Arrange the fractions `2/3, 4/5` and `2/15` in ascending order.
Solution: To arrange these fractions of different denominators, we first find the LCM of denominators `(3, 5, 15)`.
Least common multiples or LCM `(3, 5, 15) = 15`.
`2/3 = 10/15, 4/5 = 12/15` and `2/15`
`2/15 < 10/15 < 12/15`
`2/15 < 2/3 < 4/5`
Thus, the ascending order of the given fractions is `2/15, 2/3 , 4/5`.
Solved Examples
Example `1`: Find the LCM of `5` and `25`.
Solution:
To find the LCM of the given fractions we will list the multiples of `5` and `25`.
LCM or `5` and `25 = 25`
Example `2`: Find the LCM of `5, 9, 15`.
Solution:
List of the multiples of `5, 9, 15` are:
The Least Common Multiple of `5, 9, 15` is `45`.
Practice Problems
Q`1`. Which of the following is correct?
- `2/15 < 2/3`
- `2/3 > 3/2`
- `2/15 > 2/3`
- `2/15 > 4/15`
Answer: a
Q`2`. To find the LCM of the fractions we need to list the ……………… of the denominators.
- Denominators
- Multiples
- Dividers
- Fractions
Answer: b
Q`3`. The LCM of the denominators of the fractions `2/7, 4/5` and `2/35` is ………….
- `35`
- `25`
- `21`
- `30`
Answer: a
Q`4`. Shelby ate `1/6` of the cake and her brother Henry ate `5/36` the cake. Who ate less cake?
- Shelby
- Henry
- Both ate an equal amount of the cake
Answer: b
Frequently Asked Questions
Q`1`. How does the Least Common Multiple (LCM) help with adding or subtracting fractions?
Answer: The LCM makes different fractions work together by giving them a common "base," so we can add or subtract them more easily.
Q`2`. Why is the LCM handy when putting fractions in order?
Answer: The LCM simplifies the process of arranging fractions, providing a common ground for easy comparison and ordering.
Q`3`. Can you find the LCM for more than two numbers?
Answer: Yes, the LCM can be determined for any set of numbers.