# How to find common denominators

The denominator is the bottom number in a fraction.

It shows how many equal parts the item is divided into

## … Common Denominator?

When the denominators of two or more fractions are the same, they have Common Denominators.

it is the smallest of all the common denominators.

### Why?

Why do we want common denominators?

Because we can't add fractions with different denominators:

 1 3 + 1 6 = ?

Before we can add them we must make the denominators the same.

### Finding a Common Denominator

• But what should the new denominator be?
• One simple answer is to multiply the current denominators together:
• 3 × 6 = 18
• So instead of having 3 or 6 slices, we will make both of them have 18 slices.

## Common Denominator – Definition with Examples

A fraction has two parts – a numerator and a denominator.

Any arithmetic operation such as addition or subtraction involving two or more fractions is possible if the denominators of both fractions are the same. This is known as the common denominator.

One can add or subtract fractions only when they have a common denominator. (see example below)

• A common denominator is a number with which both denominators share at least one factor other than 1.
• We can obtain common denominators by multiplying both numerator (top) and denominator (bottom) by the same amount.
• For example, consider the addition of two fractions that do not have the same denominator:
• Addition of 2⁄5 and 1⁄2 .
• First, multiply 2⁄5 by 2⁄2 to get 4⁄10 .
• Now multiply the second fraction 1⁄2 by 5⁄5 to get 5⁄10 .
• The new fractions have a common denominator 10.
• Add the two fractions 4⁄10 + 5⁄10 to get 9⁄10 .
 Fun fact The lowest common multiple of the denominators helps us obtain common denominator.

## Finding the LCD of Two Fractions

When we add or subtract fractions, their denominators need to be same or common. If they are different, we need to find the LCD (least common denominator) of the fractions before we add or subtract.

To find the LCD of the fractions, we find the least common multiple (LCM) of their denominators. LCD can be found by two methods.

In the first method, LCD of two or more fractions is found as the smallest of all the possible common denominators.In second method, we find the prime factors of the denominators.

Then we look for the most occurrence of each of those prime factors and then take their product. This gives the LCD of the fractions.

Formula 1:

Here is how to find out LCD of any two fractions; for example 1/3 and 1/6:

Their denominators are 3 and 6 and the multiples of 3 and 6 are

List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, …

List the multiples of 6: 6, 12, 18, 24, …

The common multiples are 6, 12, 18…The least among these common multiples is 6.
So, 6 is the Least Common Denominator of 1/3 and 1/6.

Formula 2:

• Here is how to find out LCD of any two fractions; for example 1/8 and 7/12:
• The denominators of the fractions are 8 and 12
• Their prime factorizations are
• 8 = 2 × 2 × 2
• 12 = 2 × 2 × 3
• The most occurrences of the primes 2 and 3 are 2 × 2 × 2 (in 8) and 3 (in 12).
• Their product is 2 × 2 × 2 × 3 = 24
• So, 24 is the LCD of these two fractions.

## 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.

### Method 1:

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.

### Method 2:

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…

1. Count the number of times each prime number appears in each of the factorizations.
2. For each prime number, take the largest of these counts.
3. Write down that prime number as many times as you counted for it in step #2.
4. 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.
1. Now, we do Step #1Count 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
2. 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
3. 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
4. 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

## How to Find the Least Common Denominator

1. 1

List the multiples of each denominator. Make a list of several multiples for each denominator in the equation. Each list should consist of the denominator numeral multiplied by 1, 2, 3, 4, and so on.

• Example: 1/2 + 1/3 + 1/5
• Multiples of 2: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14; etc.
• Multiples of 3: 3 * 1 = 3; 3 * 2 = 6; 3 *3 = 9; 3 * 4 = 12; 3 * 5 = 15; 3 * 6 = 18; 3 * 7 = 21; etc.
• Multiples of 5: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35; etc.
2. 2

Identify the lowest common multiple. Scan through each list and mark any multiples that are shared by all of the original denominators. After identifying the common multiples, identify the lowest multiple common to all the denominators.

• Note that if no common multiple exists at this point, you may need to continue writing out multiples until you eventually come across a shared multiple.
• This method is easier to use when small numbers are present in the denominator.
• In this example, the denominators only share one multiple and it is 30: 2 * 15 = 30; 3 * 10 = 30; 5 * 6 = 30
• The LCD = 30
3. 3

Rewrite the original equation. In order to change each fraction in the equation so that it remains true to the original equation, you will need to multiply each numerator (the top of the fraction) and denominator by the same factor used to multiply the corresponding denominator when reaching the LCD.

• Example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
• New equation: 15/30 + 10/30 + 6/30
4. 4

Solve the rewritten problem. After finding the LCD and changing the fractions accordingly, you should be able to solve the problem without further difficulty. Remember to simplify the fraction at the end.

• Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30
1. 1

List all of the factors of each denominator. The factors of a number are all of the whole numbers that are evenly divisible into that number.[3] The number 6 has four factors: 6, 3, 2, and 1. (Every number has a factor of 1, because every number can be evenly divided by 1.)

• For example: 3/8 + 5/12.
• Factors of 8: 1, 2, 4, and 8
• Factors of 12: 1, 2, 3, 4, 6, 12
2. 2

Identify the greatest common factor between both denominators. Once you have listed the factors of each denominator, circle all of the common factors. The largest of the common factors is the greatest common factor (GCF) that will be used to continue solving the problem.

• In our example, 8 and 12 share the factors 1, 2, and 4.
• The greatest common factor is 4.
3. 3

Multiply the denominators together. In order to use the greatest common factor to solve the problem, you must first multiply the two denominators together.

• Continuing our example: 8 * 12 = 96
4. 4

Divide this product by the GCF. After finding the product of the two denominators, divide that product by the GCF you found previously. This number will be your least common denominator (LCD).

5. 5

Divide the LCD by the original denominator. To determine the multiple needed to make the denominators equal, divide the LCD you determined by the original denominator. Multiply the numerator and the denominator of each fraction by this number. The denominators should now both be equal to the LCD.

• Example: 24 / 8 = 3; 24 / 12 = 2
• (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
• 9/24 + 10/24
6. 6

Solve the rewritten equation. With the LCD found, you should be able to add and subtract the fractions in the equation without further difficulty. Remember to simplify the fraction at the end, if possible.

• Example: 9/24 + 10/24 = 19/24
1. 1

Break each denominator into prime numbers. Factor each denominator digit into a series of prime numbers that multiply together to make that number. Prime numbers are numbers that cannot be divided by any other number. [5]

• Example: 1/4 + 1/5 + 1/12
• Prime factorization of 4: 2 * 2
• Prime factorization of 5: 5
• Prime factorization of 12: 2 * 2 * 3
2. 2

Count the number of times each prime appears in each factorization. Tally up the number of times that each prime number appears in the factorization of each denominator digit.

• Example: There are two 2’s in 4; zero 2’s in 5; two 2’s in 12
• There are zero 3’s in 4 and 5; one 3 in 12
• There are zero 5’s in 4 and 12; one 5 in 5
3. 3

Take the largest count for each prime. Identify the largest number of times you used each prime number for any of the denominators and note that count.

• Example: The largest count of 2 is two; the largest of 3 is one; the largest of 5 is one
4. 4

Write that prime as many times as you counted in the previous step. Do not write out the number of times each prime number appeared throughout all the original denominators. Only write out the largest count, as determined in the previous step.

5. 5

Multiply all the prime numbers written in this manner. Multiply the prime numbers together as they appeared in the previous step. The product of these numbers equals the LCD for the original equation.

• Example: 2 * 2 * 3 * 5 = 60
• LCD = 60
6. 6

Divide the LCD by the original denominator. To determine the multiple needed to make the denominators equal, divide the LCD you determined by the original denominator. Multiply the numerator and the denominator of each fraction by this number. The denominators should now both be equal to the LCD.

• Example: 60/4 = 15; 60/5 = 12; 60/12 = 5
• 15 * (1/4) = 15/60; 12 * (1/5) = 12/60; 5 * (1/12) = 5/60
• 15/60 + 12/60 + 5/60
7. 7

Solve the rewritten equation. With the LCD found, you should be able to add and subtract the fractions as usual. Remember to simplify the fraction at the end, if possible.

• Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
1. 1

Convert each integer and mixed number into an improper fraction. Convert mixed numbers into improper fractions by multiplying the integer by the denominator and adding the numerator to the product. Convert integers into improper fractions by placing the integer over a denominator of “1.”

• Example: 8 + 2 1/4 + 2/3
• 8 = 8/1
• 2 1/4; 2 * 4 + 1 = 8 + 1 = 9; 9/4
• Rewritten equation: 8/1 + 9/4 + 2/3
2. 2

Find the least common denominator. Implement any of the methods used for finding the LCD of common fractions, as explained in the previous method sections. Note that for this example, we will be using the “listing multiples” method, in which a list of multiples is created for each denominator and the LCD is identified from these lists.

• Note that you do not need to create a list of multiples for 1 since any number multiplied by 1 equals itself; in other words, every number is a multiple of 1.
• Example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12; 4 * 4 = 16; etc.
• 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12; etc.
• The LCD = 12
3. 3

Rewrite the original equation. Instead of multiplying the denominator alone, you must multiply the entire fraction by the digit required for changing the original denominator into the LCD.

• Example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 27/12; (4/4) * (2/3) = 8/12
• 96/12 + 27/12 + 8/12
4. 4

Solve the equation. With the LCD determined and the original equation changed to reflect the LCD, you should be able to add and subtract without difficulty. Remember to simplify the fraction at the end, if possible.

• Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12

• Question What is the LCD of 1/4 and 3/8? First, you must see what lowest number that both 4 and 8 will go into evenly. Since four can go evenly into 8, and 8 goes into itself evenly, then LCD of these two fractions is 8.
• Question How do I subtract 4/5 from 8/10? Express both fractions with the same denominator. 4/5 is the equivalent of 8/10. 8/10 – 8/10 equals zero.
• Question How do I solve 5/6 -1/2 divided by 1/4? Change ½ to 3/6. Subtract 3/6 from 5/6. Then divide by ¼, which is the same as multiplying by 4.
• Question How do I find the LCM of 7, 8, 9 and 10? The easiest way to do it is to multiply any two or three of those numbers and see if the product is also a multiple of each of the other numbers. If so, that's a common multiple. Then divide that number by any small number to see if there's a lower common multiple. In this case you'll find that the lowest common multiple is the full product of 7, 8, 9 and 10, which is 5,040.
• Question How do I solve (7/9) (2/98) +3? Simply multiply the two fractions together (numerator times numerator, and denominator times denominator), reduce the fraction if possible, then put a 3 in front of the final fraction. The answer will be a mixed number. Thus: (7/9)(2/98) = 14 / 882 = 1/63. The answer is 3 1/63.
• Question What is the geometric mean of 1/16 and 4/25? Multiply the two fractions together, then find the square root of the product: (1/16)(4/25) = 4/400 = 1/100. The square root of 1/100 is 1/10.
• Question How do I find 4/7 of 14? You multiply 14 by 4/7: (14/1)(4/7) = 56/7 = 8.
• Question What is the least common denominator for 5/6 and 2/9? Multiply the smaller denominator (6) by various small integers (2, 3, 4, etc.) until you get a product that is also a multiple of the other denominator (9). Thus, 2×6=12, which is not a multiple of 9. 3×6=18, which is a multiple of 9. Therefore, 18 is the least common denominator of 5/6 and 2/9.
• Question How do I find the least common denomination of 1/24, 5/16, and 6/18? For ease, start the process with the two smallest denominators, here 16 and 18. Multiply them together: 16 x 18 = 288. Thus, 288 is evenly divisible by 16 and by 18. Divide 288 by the other denominator to see if it, too, divides evenly: 288 ÷ 24 = 12. Because it divides evenly (in other words, with a quotient that is a whole number), 288 is the smallest number evenly divisible by each of the denominators and is thus defined as the lowest (least) common denominator in this case.
• Question What is the LCD of 11/12 and 1/8? 12 x 2 = 24, and 8 x 3 = 24. 24 is the lowest number that is a multiple of both 8 and 12, so 24 is the LCD.

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## LCD Calculator – Least Common Denominator

Use this Least Common Denominator Calculator to find the lowest common denominator (LCD) of fractions, integers and mixed numbers. Finding the LCD is important because fractions need to have the same denominator when you are doing addition or subtraction math with fractions.

### What is the Least Common Denominator?

The least common denominator (LCD) is the smallest number that can be a common denominator for a set of fractions. Also known as the lowest common denominator, it is the lowest number you can use in the denominator to create a set of equivalent fractions that all have the same denominator.

### How to Find the LCD of Fractions, Integers and Mixed Numbers:

To find the least common denominator first convert all integers and mixed numbers (mixed fractions) into fractions. Then find the lowest common multiple (LCM) of the denominators. This number is same as the least common denominator (LCD).You can then write each term as an equivalent fraction with the same LCD denominator.

### Steps to find the LCD of fractions, integers and mixed numbers

1. Convert integers and mixed numbers to improper fractions
2. Find the LCD of all the fractions
3. Rewrite fractions as equivalent fractions using the LCD

### Example Using the Lowest Common Denominator Calculator

Find the LCD of: 1 1/2, 3/8, 5/6, 3

• Convert integers and mixed numbers to improper fractions. 3/8 and 5/6 are already fractions so we can use those as they are written. 1 1/2 is the same as (1/1) + (1/2). Using the formula for adding fractions, ((n1*d2)+(n2*d1)) / (d1*d2), we get ((1*2)+(1*1)) / (1*2) = 3/2. 3 can be rewritten as a fraction as 3/1
• Equivalent fractions are: 3/2, 3/8, 5/6, 3/1

• Now find the least common denominator (LCD) (or the least common multiple (LCM) of the denominators)
• Rewriting the fractions as equivalent fractions using the LCD
• 36/24, 9/24, 20/24, 72/24

### Related Calculators

We also have calculators for least common multiple, math with fractions, simplifying fractions, math with mixed numbers, and comparing fractions.