Finding The Least Common Denominator
The least common denominator of two or more non-zero denominators is actually the smallest whole number that is divisible by each of the denominators. There are two widely used methods for finding the least common denominator.
Actually, this is the same basic idea behind finding the Least Common Multiple (LCM) for whole numbers (without the fractional parts).
Note: In the examples below, we’ll be adding three fractions instead of the usual two because the principles are the same. This will give you a better understanding of the process. And in the “Pulling Everything Together” section, we will be adding four fractions.
To find the least common denominator, simply list the multiples of each denominator (multiply by 2, 3, 4, etc. out to about 6 or seven usually works) then look for the smallest number that appears in each list.
Example: Suppose we wanted to add 1/5 + 1/6 + 1/15. We would find the least common denominator as follows…
- First we list the multiples of each denominator.
Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,…
Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,…
Multiples of 15 are 30, 45, 60, 75, 90,….
- Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
- Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.
This method works pretty good. But, adding fractions with larger numbers in the denominators it can get pretty messy.
So hold that thought for a moment, as we look at another way to find a least common denominator for adding these same fractions.
To find the least common denominator using this method, factor each of the denominators into primes. Then for each different prime number in all of the factorizations, do the following…
- Count the number of times each prime number appears in each of the factorizations.
- For each prime number, take the largest of these counts.
- Write down that prime number as many times as you counted for it in step #2.
- The least common denominator is the product of all the prime numbers written down.
Example: We’ll use the same fractions as above: 1/5, 1/6 and 1/15.
Factor into primes (Click here to see our table of prime numbers.)
- Prime factorization of 5 is 5 (5 is a prime number)
- Prime factorization of 6 is 2 x 3
- Prime factorization of 15 is 3 x 5
- Notice that the different primes are 2, 3 and 5.
- Now, we do Step #1 – Count the number of times each prime number appears in each of the factorizations…
- The count of primes in 5 is one 5
- The count of primes in 6 is one 2 and one 3
- The count of primes in 15 is one 3 and one 5
- Step #2 – For each prime number, take the largest of these counts. So we have…
- The largest count of 2s is one
- The largest count of 3s is one
- The largest count of 5s is one
- Step #3 – Since we now know the count of each prime number, you simply – write down that prime number as many times as you countedfor it in step #2.Here are the numbers… 2, 3, 5
- Step #4 – The least common denominator is the product of all the prime numbers written down.
2 x 3 x 5 = 30
// Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.
As you can see, both methods end up with the same results.
The reason we might want to use the different methods is because Method #1 works great for small numbers. But when the numbers get bigger, Method #2 is the ONLY way to go.
Now let’s make the tricky part, really easy — convert each fraction to an equivalent fraction using the newly found least common denominator, which is 30.
Remember our problem: Add: 1/5 +1/6 + 1/15
Step #2 for adding fractions with different denominators says – “Re-write each equivalent fraction using the least common denominator as the denominator.” So let’s do it!
This is going to get a little detailed, so hang in there!
Re-write Each Fraction As An Equivalent Fraction
The Rule to re-write a fraction as an equivalent fraction using the least common denominator says…
- Divide the least common denominator by the denominator of the fraction.
- Multiple the answer times the numerator of the fraction.
- Re-write the fraction using the least common denominator as the denominator.
So, if we write 1/5 as an equivalent fraction using 30 as our denominator, we have 30 divided by the denominator “5″, which equals 6. We then multiple that 6 times the numerator “1″ which gives us the new numerator of 6.
Finally, we re-write the equivalent fraction using the 30 as our denominator, therefore our equivalent fraction is 6/30.
The Rule actually looks like this…
New Numerator = (LCD ÷ Denominator) x Numerator
New Denominator = LCD
Now we repeat the process for 1/6 and 1/15
Using 1/6 next, (30 ÷ 6 = 5, and 5 x 1 = 5), so 1/6 is equivalent to 5/30
And for 1/15, (30 ÷ 15 = 2, and 2 x 1=2), so 1/15 is equivalent to 2/30
Now then, at long last we can add our fractions…
1/5 + 1/6 + 1/15 = 6/30 + 5/30 + 2/30 = 13/30
Pulling Everything Together
We are going to use a little tougher problem for adding fractions with different denominators to illustrate that you CAN do it. Also, we will use Method #2 to find the least common denominator because it works best in almost every case.
If you have problems with any parts of this exercise, re-read the section above that covers it. We are going to talk through each step for adding these fractions without citing the rules exactly, just like in the “real world”.
Look at how everything works and you will be just fine!
We’ll be adding these fractions…
1/9 + 1/8 + 5/12 + 7/18
Since our denominators are 9, 8, 12, and 18, we need to find the least common denominator. So we factor each number into primes.
- Prime Factorization of 9 is 3 x 3
- Prime Factorization of 8 is 2 x 2 x 2
- Prime Factorization of 12 is 2 x 2 x 3
- Prime Factorization of 18 is 2 x 3 x 3
When we do our largest count of the prime numbers, we find three2s, and two 3s (do you see them?), so we re-write the count and find the product.
2 x 2 x 2 x 3 x 3 = 72
Now we have our least common denominator of 72
Next… we convert each fraction to an equivalent fraction using 72 as our new denominator. So, let’s convert…1/9 + 1/8 + 5/12 + 7/18
- 1/9 = 8/72
- 1/8 = 9/72
- 5/12 = 30/72
- 7/18 = 28/72
Mission accomplished… all of our denominators are the same, so we can just add up the numerators.
Our new equivalent fractions are 8/72, 9/72, 30/72 and 28/72
That’s what changes our problem to…
8/72 + 9/72 + 30/72 + 28/72
Now adding all of the numerators and placing the results over our common denominator, the answer is…
8/72 + 9/72 + 30/72 + 28/72 = 75/72
Since our answer is an improper fraction (the numerator is larger than the denominator), we now have three options.
- Show the answer as is, with the least common denominator.
- Reduce the fraction and show it as the lowest reduced equivalent.
- Simplify the answer, and show it as a mixed number.
Remember, always show your answer in the form asked for in your instructions.