benchmarking fractions

Question

Order the following fractions from least to greatest and explain your reasoning. You cannot use decimals, cross multiplication, or the common denominator.

 

7/13, 6/11, 7/16, 6/7

 

The teacher gave me an explanation that is unclear to me. It reads:

 

7/16 is the only fraction less than 1/2 so 7/16 is the smallest fraction. Both 7/13 and 6/11 are slightly over 1/2. 7/13 is 1/2/13 over 1/2 and 6/11 is 1/2/11 over 1/2. Since the 1/13 pieces are smaller than 1/11 pieces, then 1/2/13 is smaller that 1/2/11 so 7/13 is smaller than 6/11. 6/7 is close to 1 and 6 1//7 are larger than 6 1/11 pieces so 6/7 is larger than 6/11.

 

7/16, 7/13, 6/11, 6/7

Answer 1

Lets examine the denominators of all of the fractions.

 

 

7/13:

 

half of 13 is a number between 6 and 7.  7 is greater than that number, so 7/13 will be slightly over 1/2.

 

 

6/11:

 

half of 11 is a number between 5 and 6.  6 is greater than that number, so 6/11 will be slightly over 1/2.

 

 

If we compare the denominator of the first two fractions, we see that 7/13 has a denominator that is larger than the denominator of 6/11 by 2 .   The larger the denominator, the smaller the piece of something when you divide it.   The fractions numerators are not that far apart from each other.  So 7/13 is less than 6/11.

 

 

 

7/16:

 

half of 16 is 8.  7 is less than 8, so 7/16 will be less than 1/2.

 

 

6/7:

 

This fraction is closer to becoming 1.  If you add the numerator by 1, you get 7/7, which is 1.

 

 

We know that 1/2 is less than 1.  Therefore, the fractions ordered from least to greatest are

 

7/16 ,  7/13  ,  6/11   ,  6/7