Introduction:
Rational numbers are numbers that can be expressed as the ratio of two integers. For example, 3/5 and 5/12 are rational numbers. Between any two rational numbers, we can find an infinite number of other rational numbers. In this question, we are asked to find ten rational numbers between 3/5 and 5/12.

Method:
To find rational numbers between two given rational numbers, we need to find a common denominator and then divide the range between the two numbers into equal parts. We can then add these parts to the smaller number to find the required rational numbers.

Steps:
1. Find the common denominator of 3/5 and 5/12. The smallest number that both 5 and 12 divide into is 60. So, we will convert both fractions to the equivalent fractions with a denominator of 60.
- 3/5 = 36/60
- 5/12 = 25/60
2. Divide the range between 36/60 and 25/60 into 11 equal parts.
- The range between 36/60 and 25/60 is 11/60.
- Dividing this range into 11 parts gives us 1/60.
3. Add 1/60 to 36/60 to find the first rational number.
- 36/60 + 1/60 = 37/60
4. Repeat step 3 ten more times to find the remaining rational numbers.
- 37/60 + 1/60 = 38/60 = 19/30
- 19/30 + 1/60 = 20/30 = 2/3
- 2/3 + 1/60 = 41/60
- 41/60 + 1/60 = 42/60 = 7/10
- 7/10 + 1/60 = 43/60
- 43/60 + 1/60 = 44/60 = 11/15
- 11/15 + 1/60 = 47/60
- 47/60 + 1/60 = 48/60 = 4/5
- 4/5 + 1/60 = 49/60
- 49/60 + 1/60 = 50/60 = 5/6

Conclusion:
In conclusion, we have found ten rational numbers between 3/5 and 5/12. These numbers are 37/60, 19/30, 2/3, 41/60, 7/10, 43/60, 11/15, 47/60, 4/5, and 5/6. The method used to find these numbers was to find a common denominator and then divide the range between the two numbers into equal parts. We added these parts to the smaller number to find the required rational numbers.