Fraction comparison

This one is traditionally known as a hurdle for teachers, parents and students. When we were kids, we were taught to compare fractions in mechanical way. We were taught LCM/HCF before that. So, it was like 'follow the given process, write exam and show us your score' :-) I hated this approach. While teaching fraction comparison to my kids, I tried to be as natural as possible and tried to bring out the concept from hands-on techniques. Hope you find this useful:

2/5th: 2 out of 5 equal slices

3/5th: 3 out of 5 equal slices

 1. Same denominator fractions: compare 2/5 and 3/5

   * Resolution: Oh, this one is real simple. We have a pizza that is cut into five equal slices. If I eat 2 of these slices and you eat 3 of these slices, who eats more? Very simple! You eat more. So, 3/5 (3 out of 5) is more than 2/5 (2 out of 5). No brainer, right?

2/5th: 2 out of 5 slices

2/7th: 2 out of 7 slices

2. Same numerator fractions: compare 2/5 and 2/7
   * Resolution: This one is slightly trickier than the earlier problem, but not by much and can be solved by sheer intuition and visualization. There are two equal pizzas here. The first pizza is cut into five equal slices. The second pizza is cut into seven equal slices. So naturally, the slices of the first pizza is fatter than those of the second one. If I eat two of the fatter slices and you eat two of the thinner slices, who eats more? Definitely, it's me! Because the number of slices eaten are same, but the sizes of the slices are different. So 2/5 is larger than 2/7
      * So, in general, whenever I cut the same pizza into more number of slices, I reduce the proportions of the slices. This concept is VERY IMPORTANT. I strongly advise parents to get pizzas and cut in front of the kids to get this concept inside their system This will be very helpful throughout the logical part of their education in future :-)

3. Different numerators and different denominators: compare 2/5 and 1/3

 1/3rd: 1 out of the 3 slices shown in different color

• Resolution: This is the challenging one. But something like this may help:
• There are two things for the pizza slices: size of the slices and the number of slices. To compare the slices, we need either the size of the slices (case 1 above) or the number of slices (case 2 above) to be same. Here the problem comes because both the dimensions are different. We need to bring one dimension same for both the pizzas.
• Let us try to make the size of the slices to be same. To so that, we just need to cut both the pizzas into SAME number of slices. The size of each of the slices will automatically become same, right?
• The problem reduces to: how to cut the pizzas to equal number of slices?
• The first pizza was cut into 5 slices and hence could be equally distributed into 5 people.
• The first pizza was cut into 3 slices and hence could be equally distributed into 3 people.
• To make each of the pizza with equal number of slices, we need to ensure than each of them is distributable to 5 as well as 3 people. This is IMPORTANT.
• If the pizza can be equally distributed to 5 as well as 3 people, it should be possible to distribute to 3 groups of 5 people each (or 5 groups of 3 people). That means, it should be possible to distribute the pizza to 15 (= 5 x 3) people.
• What is the easiest way to cut the pizzas to equally distribute to 15 people? Well, it is 15 slices!



The first pizza was cut into 5 slices earlier, now it is  cut into 15 slices

• First pizza, already cut into 5 equal slices: To make it 15 slices, just cut each of the 5 slices into 3 thinner (and equally thin) slices. So each of the 5 fat slices become 3 thinner slices. So 2 out of the fat slices is actually 6 (= 2 x 3) thinner slices! The total number of slices is 15. So our earlier 2/5 is same as 6/15 after further cutting the pizza!

The second pizza was cut into 3 slices earlier, now it is cut into 15 slices. Each of the 3 slices has been cut into 5 thinner slices.

• Second pizza, already cut into 3 equal slices: To make it 15 slices, just cut each of the 3 slices into 5 thinner (and equally thin) slices. So each of the 3 fat slices become 5 thinner slices. So 1 out of the fat slices is actually 5 (= 1 x 5) thinner slices! The total number of slices is 15. So our earlier 1/3 is same as 5/15 after further cutting the pizza!

•  Now we have an easy problem in hand! Two equal size pizzas, both cut into 15 equal slices. I eat 6 of the slices (6/15) from the first pizza and you eat 5 of the slices (5/15) from the 2nd pizza. Who eats more? Obviously I eat more! So 6/15 is more than 5/15.
• Our current 6/15 is actually 2/5 in the problem. I mean 6 out of the 15 thin slices is same as 2 out of the 5 fat slices of the same pizza.
• Great! We are done, 2/5 is more than 1/3 :-)
• Let us see if we can arrive at a rule to do this quick:
• STEP 1: Find out the LCM of the denominators. 
• STEP 2: Multiply the numerators and the denominators of the fractions by the relevant numbers to arrive at the equivalent fractions so that the denominators of the fractions are same!
• STEP 3: Just compare the numerators!