Which of the following are in descending order of their value?

We are asked to arrange the given fractions in descending order based on their value. To compare the fractions, we will convert them to a common denominator or use approximate decimal values. Here's the analysis of each option:

1. Option A - 1/3, 2/5, 3/7, 4/5, 5/6, 6/7:
   • Converting the fractions to decimal form:
   
   
   • 1/3 = 0.3333
   
   
   • 2/5 = 0.4
   
   
   • 3/7 ≈ 0.4286
   
   
   • 4/5 = 0.8
   
   
   • 5/6 ≈ 0.8333
   
   
   • 6/7 ≈ 0.8571
   
   
   • The order of fractions in descending order is: 6/7, 5/6, 4/5, 3/7, 2/5, 1/3.
   
   
   • This matches Option D.


2. Option B - 1/3, 2/5, 3/5, 4/7, 5/6, 6/7:
   • Converting to decimal values as shown previously:
   
   
   • 1/3 = 0.3333
   
   
   • 2/5 = 0.4
   
   
   • 3/5 = 0.6
   
   
   • 4/7 ≈ 0.5714
   
   
   • 5/6 ≈ 0.8333
   
   
   • 6/7 ≈ 0.8571
   
   
   • The order of fractions in descending order is: 6/7, 5/6, 3/5, 4/7, 2/5, 1/3.
   
   
   • This order does not match Option B.


3. Option C - 1/3, 2/5, 3/5, 4/6, 5/7, 6/7:
   • Converting to decimal values:
   
   
   • 1/3 = 0.3333
   
   
   • 2/5 = 0.4
   
   
   • 3/5 = 0.6
   
   
   • 4/6 ≈ 0.6667
   
   
   • 5/7 ≈ 0.7143
   
   
   • 6/7 ≈ 0.8571
   
   
   • The order of fractions in descending order is: 6/7, 5/7, 4/6, 3/5, 2/5, 1/3.
   
   
   • This order does not match Option C.


4. Option D - 6/7, 5/6, 4/5, 3/7, 2/5, 1/3:
   • Converting to decimal values as shown previously:
   
   
   • 6/7 ≈ 0.8571
   
   
   • 5/6 ≈ 0.8333
   
   
   • 4/5 = 0.8
   
   
   • 3/7 ≈ 0.4286
   
   
   • 2/5 = 0.4
   
   
   • 1/3 = 0.3333
   
   
   • The order of fractions in descending order is: 6/7, 5/6, 4/5, 3/7, 2/5, 1/3.
   
   
   • This matches Option D perfectly.


5. Final Answer:
   • The correct answer is D. 6/7, 5/6, 4/5, 3/7, 2/5, 1/3.