Essential Knowledge:   

(1) If the fractions have the same denominator, we only need to compare the numerators.

  Eg. Compare 17/23 to 19/23.  

  Since 19 ＞17 it follows that the answer is 17/23 < 19/23.

(2) If the fractions have different denominators, we need to convert them to an equivalent fraction with the same denominator before comparing the numerators.

Eg.Compare 3/4 to 13/16.

3/4=12/16, since 12 <13 it follows that the answer is  3/4<13/16.

Problem&Solving:

Which of the following is the correct order of the fractions 15/11, 19/15, and 17/13 from least to greatest? (2019 AMC 8)

Solution 1

When comparing fractions with different denominators, we should first convert them to an equivalent fractions with the same denominator.So we have

2717 < 2805 < 2925, so the answer is 19/15 <17/13  <15/11 .

Solution 2 (Cross-multiplication) 

1)Basic knowledge:

Compare b/a to d/c  (a,c≠0)

Converting them into fractions with the same denominator,we have

So it's only necessary to compare the numerators, that is, b·c and d·a.

If b·c < d·a, b/a＜d/c;

If b·c = d·a, b/a=d/c;

If b·c > d·a, b/a>d/c;

2)Solution:

Compare these fractions pairwise.

Step1:

Compare 15/11 to 19/15, 15×15=225,11×19=209, 225 > 209, so 15/11>19/15.

Step 2:

Compare 15/11 to 17/13, 15×13=195,11×17=187, 195 > 187, so 15/11 > 17/13.

Step 3:

Compare  19/15 to 17/13,19×13=247,17×15=255, 247 < 255, so 19/15 < 17/13 .

In summary, the answer is 19/15 < 17/13 < 15/11.

Solution 3

We found that the difference between the numerator and denominator of these fractions is the same,so we can split the fraction into the form of “1+__” or “1－__” .

15/11 = 1+4/11 ,  19/15= 1+4/15 ,  17/13= 1+4/13

Now, we just to compare 4/11, 4/15, and 4/13, 4/11> 4/13> 4/15, so the answer is  19/15< 17/13 < 15/11.   

Tips: If we change this problem to comparing 7/11, 11/15, and 9/13, we can split the fraction into the form of “1－__”.

 7/11= 1－4/11, 11/15 =1－4/15,9/13  =1－ 4/13

At this point, we only need to compare 4/11 , 4/15, and 4/13,it is worth noting that in this case, the larger the fraction, the smaller its original value, 4/11>4/13>4/15, so the answer is 7/11<9/13<11/15.

Now let’s have a try！ 

(1)Which of the following is the correct order of the fractions 13/9 ,17/13, and 15/11,from least to greatest?

(2)Which of the following is the correct order of the fractions 5/9, 9/13, and 7/11 from least to greatest?

Solution:

(1) 17/13 ＜15/11 ＜13/9

(2)  5/9    ＜7/11   ＜ 9/13