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What is the sum of all positive common fractions (in simplest terms) less than $5$ whose denominator is 3?

 Feb 9, 2024

Best Answer 

 #1
avatar+1632 
+2

You have 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, 10/3, 11/3, 13/3, 14/3.

You have to skip 3/3, 6/3, 9/3, and 12/3 because they are not in simplest form, as they simplify to 1, 2, 3, and 4 respectively (in the same order). And you have to skip 15/3 = 5 because the problem states "less than 5".

 

Group by 2's for adding, like how 1/3 + 2/3 = 3/3 = 1

1/3 + 2/3 + 4/3 + 5/3 + 7/3 + 8/3 + 10/3 + 11/3 + 13/3 + 14/3 = 1 + 3 + 5 + 7 + 9 = 25.

 

Just a note: Notice how when I add the first 5 positive odd numbers, I get 5^2- can you generalize this?

 Feb 9, 2024
 #1
avatar+1632 
+2
Best Answer

You have 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, 10/3, 11/3, 13/3, 14/3.

You have to skip 3/3, 6/3, 9/3, and 12/3 because they are not in simplest form, as they simplify to 1, 2, 3, and 4 respectively (in the same order). And you have to skip 15/3 = 5 because the problem states "less than 5".

 

Group by 2's for adding, like how 1/3 + 2/3 = 3/3 = 1

1/3 + 2/3 + 4/3 + 5/3 + 7/3 + 8/3 + 10/3 + 11/3 + 13/3 + 14/3 = 1 + 3 + 5 + 7 + 9 = 25.

 

Just a note: Notice how when I add the first 5 positive odd numbers, I get 5^2- can you generalize this?

proyaop Feb 9, 2024

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