Adam wants to compare the fractions 2/5, 1/6, and 1/3. He wants to order them from least to greatest and rewrite them so they all have the same denominator. Explain how Adam can rewrite the fraction.

He needs to find the common denominator. In this problem, the common denominator is 30. Study this site to learn how to turn these fractions into equivalent fractions.

http://www.mathsisfun.com/equivalent_fractions.html

To compare the fractions 2/5, 1/6, and 1/3, Adam needs to rewrite them so they all have the same denominator. To do this, Adam needs to find the least common denominator (LCD) for the three fractions.

To find the LCD, Adam needs to find the least common multiple (LCM) of the denominators.

1. Start by listing the multiples of each denominator:
- For 5: 5, 10, 15, 20, 25, 30, ...
- For 6: 6, 12, 18, 24, 30, ...
- For 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

2. Identify the first common multiple found in all three lists. In this case, it is 30.

3. Therefore, 30 is the least common multiple (LCM) of 5, 6, and 3. So the least common denominator (LCD) is 30.

Now that Adam has the LCD, he can rewrite each fraction with this denominator:

- 2/5 can be rewritten as (2/5) x (6/6) = 12/30
- 1/6 can be rewritten as (1/6) x (5/5) = 5/30
- 1/3 can be rewritten as (1/3) x (10/10) = 10/30

Now all three fractions have the same denominator of 30. Adam can compare them and order them from least to greatest:

- 5/30 is the smallest
- 10/30 comes next
- 12/30 is the largest

So, the new order of the fractions from least to greatest is: 5/30, 10/30, 12/30.

To order the fractions from least to greatest and rewrite them with a common denominator, Adam can follow these steps:

Step 1: Find a common denominator for the fractions: To do this, Adam needs to find the least common multiple (LCM) of the denominators of the given fractions, which are 5, 6, and 3.

Step 2: To find the least common multiple (LCM), Adam can use various methods. One common approach is to list the multiples of each denominator until a common number appears. Alternatively, Adam can use prime factorization to find the LCM.

For example:
- The multiples of 5 are: 5, 10, 15, 20, 25, 30, ...
- The multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
- The multiples of 3 are: 3, 6, 9, 12, 15, 18, ...

From the listed multiples above, it can be seen that the least common multiple (LCM) of 5, 6, and 3 is 30.

Step 3: Now, Adam can rewrite the fractions with the common denominator of 30.

For the fraction 2/5:
Since the original denominator is 5 and we want to make it 30 (the common denominator), we multiply both the numerator and denominator by 6 (the result of 30 divided by 5) to get:
2/5 * 6/6 = 12/30

For the fraction 1/6:
Since the original denominator is 6 and we want to make it 30 (the common denominator), we multiply both the numerator and denominator by 5 (the result of 30 divided by 6) to get:
1/6 * 5/5 = 5/30

For the fraction 1/3:
Since the original denominator is 3 and we want to make it 30 (the common denominator), we multiply both the numerator and denominator by 10 (the result of 30 divided by 3) to get:
1/3 * 10/10 = 10/30

Step 4: Finally, Adam can order the rewritten fractions from least to greatest:
5/30, 10/30, 12/30