How do you know when to multiply denominators

  • Thread starter jim1174
  • Start date
In summary, when adding or subtracting fractions with different denominators, it is necessary to change the denominators so they are the same. This can be done by finding the Least Common Multiple (LCM) of the denominators. In some cases, you may need to multiply both denominators together to get the LCM, while in other cases, the LCM can be found without multiplying. The key is to aim for the smallest number that is divisible by both denominators.
  • #1
jim1174
79
0
when you are adding or subtracting fractions with different denominators you are supposed to change the denominators so they are the same. Some times you multiply both denominators together and other times you don't. So how do you know when to multiply the denominators and when not to ? Here is a example 2/5 + 5/6 you are supposed to multiply 6 and 5 to get 30 but on this problem 5/6 7/12 you don't multiply the two bottom numbers.
 
Mathematics news on Phys.org
  • #2
What you're actually doing is finding the Least Common Multiple (LCM) of the denominators. That is, what is the smallest number that is divisible by both denominators.
Let's list out the multiples of 5 and 6 to illustrate this:

5, 10, 15, 20, 25, 30, 35, 40, ...
6, 12, 18, 24, 30, 36, 42, ...

Notice that the first number to appear in both of these lists is 30. This means that 30/5 is an integer, and so is 30/6.
Now let's look at 6 and 12

6, 12, 18, 24, 30, ...
12, 24, 36, ...

The first number is 12. See how the numbers sort of overlap? 6 goes into 12, so when you list the multiples of 6 and 12, you'll get a small LCM number (unlike with 5 and 6 which didn't happen for a little while).

If the Highest Common Factor (HCF) of two numbers is 1, then the LCM of the two numbers is their product.

Factors of 5 and 6:
5 = 1, 5
6 = 1, 2, 3, 6

The largest number in both of these lists is only 1, which means that 5 and 6 have no common factors, so you need to multiply the two numbers to make the denominator equal.

Factors of 6 and 12:
6 = 1,2,3,6
12 = 1,2,3,4,6,12

The largest in both of these lists is 6, so this means that the LCM of the two numbers is going to be less than 6*12 = 72. Since the HCF is 6, then the LCM is 6*12 / 6 = 12.

Another example:

9 and 15. The factors are
9 = 1,3,9
15 = 1,3,5,15

The HCF is 3, so the denominator you are aiming for is going to be 9*15/3 = 9*5 = 45.

Checking the multiples of 9 and 15:
9,18,27,36,45, ...
15,30,45, ...

45 is the first to appear, so it seems to work.
 
  • #3
You got a brilliant answer above. Note that you can always multiply the denominators if you want to. Using the LCM is a sort of short cut. So, for example

5/6 + 7/12 = (60 + 42)/72

Will give you the same answer as

5/6 + 7/12 = (10 + 7)/12

If you reduce both fractions to their lowest form, of course. Try simplifying those two answers above and see if you get the same result.

Which way did you find easier?
 
  • #4
jim1174 said:
when you are adding or subtracting fractions with different denominators you are supposed to change the denominators so they are the same. Some times you multiply both denominators together and other times you don't. So how do you know when to multiply the denominators and when not to ? Here is a example 2/5 + 5/6 you are supposed to multiply 6 and 5 to get 30 but on this problem 5/6 7/12 you don't multiply the two bottom numbers.
Based on your post here and another one a day or two ago, it appears to me that your arithmetic skills need some work, particularly those related to working with fractions. I would advise you to spend some time refreshing them. Khanacademy would be a good place to start -- http://www.khanacademy.org/math/arithmetic/fractions.
 
  • #5


I would approach this question by looking at the underlying mathematical principles and rules involved. In this case, the rule for adding or subtracting fractions with different denominators is to find a common denominator. This means finding a number that both denominators can divide into evenly. In some cases, this common denominator can be found by simply multiplying the two denominators together. However, in other cases, a different approach may be needed.

One method to determine when to multiply the denominators is to first look at the numbers involved and see if there is a common factor between them. For example, in the first problem given (2/5 + 5/6), both 5 and 6 have a common factor of 2. This means that we can simplify the fractions by dividing both the numerator and denominator by 2. This results in the fractions becoming 1/3 and 5/6, and the common denominator is now 6. In this case, we do not need to multiply the denominators, as a common denominator can be found by simplifying the fractions.

In the second problem given (5/6 + 7/12), there is no common factor between 6 and 12. In this case, we must find a different approach to finding a common denominator. One method is to find the least common multiple (LCM) of the two denominators. In this case, the LCM of 6 and 12 is 12. This means that we need to multiply the first fraction by 2/2 (which is equivalent to 1) to get a denominator of 12. This results in the fractions becoming 10/12 and 7/12, and the common denominator is now 12.

In summary, when adding or subtracting fractions with different denominators, it is important to first look for any common factors or multiples between the denominators. If a common factor is found, the fractions can be simplified and a common denominator can be easily found. If not, other methods such as finding the LCM may be needed. With practice and familiarity with these rules and methods, it will become easier to determine when to multiply the denominators and when not to.
 

1. How do you know when to multiply denominators?

When multiplying fractions, you will need to multiply the denominators if the denominators are different.

2. Why do you need to multiply denominators?

When multiplying fractions, the denominators represent the total number of equal parts in each fraction. In order to find the product of two fractions, you need to multiply both the numerators and denominators.

3. Can you multiply denominators if they are the same?

If the denominators are the same, you do not need to multiply them. You can simply multiply the numerators and keep the denominator the same.

4. Is there a specific order to multiply denominators?

There is no specific order to multiply denominators. You can multiply them in any order as long as you multiply the corresponding numerators as well.

5. When do you simplify before or after multiplying denominators?

You can simplify fractions at any point during a problem. However, it may be easier to simplify before multiplying denominators to avoid working with larger numbers.

Similar threads

  • General Math
Replies
2
Views
686
Replies
8
Views
14K
Replies
4
Views
806
  • General Math
Replies
7
Views
2K
  • General Math
Replies
4
Views
1K
Replies
5
Views
1K
Replies
2
Views
926
Replies
11
Views
6K
  • General Math
Replies
2
Views
929
  • Precalculus Mathematics Homework Help
Replies
21
Views
792
Back
Top