Arrange the fractions in increasing order 7/15, 2/3, 11/12, 5/9

To compare fraction sizes, you have to make sure the denominators of the fractions are the same. You must find a number that is a multiple of all the denominators we currently have (15, 3, 12, 9)
The smallest number that fits this is 540
Now you must multiply the numerator and denominator of each fraction by the number that makes the denominator 540
So, for 7/15 , we multiply the top and bottom by 36, since you need to multiply 15 by 36 to get 540. This means the fraction you end up with is 252/540
For 2/3, we multiply the top and bottom by 180, since you need to multiply 3 by 180 to get 540.This means the fraction you end up with is 360/540
For 11/12, we multiply the top and bottom by 45, since you need to multiply 12 by 45 to get 540. This means that the fraction you end up with is 495/540
For 5/9, we must multiply the top and bottom by 60, since you need to multiply 9 by 60 to get 540. This means that the fraction you end up with is 300/540

Now that the hard part is out of the way, you have four fractions: 252/540, 360/540, 495/540 and 300/540. Now, you can arrange the fractions in ascending order just based on their numerator. 
So, the order would be: 252/540, 300/540, 360/540, 495/540

Now, for the final step, just convert them back to the original fractions.
So, the final answer is:
7/15, 5/9, 2/3, 11/12

Hope I helped!! xx

2 ways to do this, the official method which involves finding a common denominator or to do it through mental math

Common denominator.
You want to convert all the fractions so that they have a common denominator, once they have a common denominator, then you can just compare the numerator
If we look at the denominators given, 15,3,12,9, through prime factorization, you can determine that the least common denominator would be 3x3x4x5 which is 120

convert each fraction to have a denominator of 180
7/15 *12/12 = 84/180
2/3 *60/60 =120/180
11/12*15/15 = 165/180
5/9*20/20 = 100/180

increasing order of 84,120,165,100 would be 84,100,120,165
which is equivalent to
7/15 5/9, 2/3,11/12

now, mental math/comparison method
you need to know that 1/2 is less than 2/3 and 2/3 is less than 1

now, 7/15 is very close to 1/2 but slightly lower since 7.5/15=1/2
11/12 is very close to 1 but slightly smaller by 1/12
and 5/9 is very close to 2/3 but slightly smaller, but remember 2/3 is greater than 1/2 so 5/9 is greater than 1/2

so the order based off these comparison would be
7/5,5/9, 2/3, 11/12
because its the equivalent of comparing
1/2,2/3,2/3,1