How do you find the LcD of the fractions with the following denominators: 30, 18, and 15?

Question

Julian Avila · Accepted Answer

#LCD=90#

I like to perform a prime factorization first in order to find the LCD:

#30=2xx15=2xx3xx5# #18=2xx9=color(white)(0)2xx3xx3# #15=color(white)(000000000)3xx5#

Every element present in each of the denominators will be on the LCD.

Initially, we have twos: the thirty and the eighteen have twos, so we enter one:

#LCD=2xx?#

The 18 has two of them next to the 3's, so we added two:

#LCD=2xx3xx3xx?#

Moving on to the 5's, we included one since the 30 and the 15 both have one:

#LCD=2xx3xx3xx5#

We can now multiply it out because there are no more primes to include:

#LCD=2xx3xx3xx5=90#

So let's give it a go. Suppose we're performing:

#1/30+1/18+1/15#

The LCD should be 90:

#1/30(1)+1/18(1)+1/15(1)#

#1/30(3/3)+1/18(5/5)+1/15(6/6)#

#3/90+5/90+6/90=14/90#

Skylar Adams · Answer

undefined To find the least common denominator (LCD) of fractions with the denominators 30, 18, and 15:

1. List the prime factors of each denominator:
   - $30 = 2 \times 3 \times 5$
   - $18 = 2 \times 3^2$
   - $15 = 3 \times 5$

2. Identify the highest power of each prime factor occurring in any of the denominators:
   - Prime factor 2 occurs in $30$ and $18$ with the highest power $2^1$.
   - Prime factor 3 occurs in $18$ and $15$ with the highest power $3^2$.
   - Prime factor 5 occurs in $30$ and $15$ with the highest power $5^1$.

3. Multiply these highest powers of prime factors to find the LCD:
   $$ \text{LCD} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 $$

So, the least common denominator (LCD) of fractions with the denominators 30, 18, and 15 is 90.