Question

When would the product of the denominators and the least common denominator of the denominators be the same?

Answer 1

The product of the denominators and least common denominator (LCD) are the same when all denominators are mutually prime, meaning they have no common factors other than 1.

Step-by-step explanation:

The product of the denominators would be the same as the least common denominator (LCD) when all the denominators are mutually prime; that is, they have no common factors other than 1. In this scenario, the LCD is simply the product of the denominators, as there are no smaller numbers by which all the denominators can be evenly divided. When working with fractions, this concept is particularly important when trying to add, subtract, or compare fractions with different denominators. It's essential to find a common denominator so that the fractions can be worked with effectively.

For instance, the fractions 1/2 and 1/3 have denominators 2 and 3, which are mutually prime. In this case, the LCD is 2 × 3 = 6, which is also the product of the denominators. However, if the denominators had a common factor (e.g., 2 and 4), we would not use their product for the LCD, as 4 can be divided by 2, making the product not the smallest number that both denominators divide into evenly.

Answer 2

Example 1:

Find the common denominator of the fractions.

16 and 38

We need to find the least common multiple of 6 and 8 . One way to do this is to list the multiples:

6,12,18,24−−,30,36,42,48,...8,16,24−−,32,40,48,...

The first number that occurs in both lists is 24 , so 24 is the LCM. So we use this as our common denominator.

Listing multiples is impractical for large numbers. Another way to find the LCM of two numbers is to divide their product by their greatest common factor ( GCF ).

Example 2:

Find the common denominator of the fractions.

512 and 215

The greatest common factor of 12 and 15 is 3 .

So, to find the least common multiple, divide the product by 3 .

12⋅153=3 ⋅ 4 ⋅ 153=60

If you can find a least common denominator, then you can rewrite the problem using equivalent fractions that have like denominators, so they are easy to add or subtract.

Example 3:

Add.

512+215

In the previous example, we found that the least common denominator was 60 .

Write each fraction as an equivalent fraction with the denominator 60 . To do this, we multiply both the numerator and denominator of the first fraction by 5 , and the numerator and denominator of the second fraction by 4 . (This is the same as multiplying by 1=55=44 , so it doesn't change the value.)

512=512⋅55=2560215=215⋅44=860

512+215=2560+860  =3360

Note that this method may not always give the result in lowest terms. In this case, we have to simplify.

=1130

The same idea can be used when there are variables in the fractions—that is, to add or subtract rational expressions .

Example 4:

Subtract.

12a−13b

The two expressions 2a and 3b have no common factors, so their least common multiple is simply their product: 2a⋅3b=6ab .

Rewrite the two fractions with 6ab in the denominator.

12a⋅3b3b=3b6ab13b⋅2a2a=2a6ab

Subtract.

12a−13b=3b6ab−2a6ab  =3b − 2a6ab

Example 5:

Subtract.

x16−38x

16 and 8x have a common factor of 8 . So, to find the least common multiple, divide the product by 8 .

16⋅8x8=16x

The LCM is 16x . So, multiply the first expression by 1 in the form xx , and multiply the second expression by 1 in the form 22 .

x16⋅xx=x216x38x⋅22=616x

Subtract.

x16−38x=x216x−616x =x2 − 616x\