Problem Solving

cypherskull wrote: Sharing it just in case anyone finds it useful.

Considering the 2 fractions - 2/3 (LHS) and 3/4 (RHS). We want to determine which is greater.

1) Cross-multiply the 2 fractions - 2*4 (LHS) and 3*3 (RHS), i.e., 8 (LHS) and 9 (RHS).
2) As seen from step 1, RHS (right hand side) is higher.
3) So, 3/4 > 2/3.


Consider, 7/11 (LHS) & 11/12 (RHS).

Cross-multiplying yields, 84 and 121. So, 11/12 (RHS) is greater.

Apply it to any fraction...it works!!!

It might be useful to understand just why this works - it's something you've certainly done many times (but in a slightly different way) before learning it as a 'trick'. If you take the fractions 7/11 and 11/12, then one way to compare them is to get a common denominator. We can use 11*12. Then our fractions would be (7*11)/(11*12) and (11*11)/(11*12). Now whichever fraction has the larger numerator has the larger overall value. Notice the numerators here are exactly what you got using the 'cross-multiplying trick'.

So when you use that 'cross-multiplying' trick, you're just finding a common denominator, but you aren't writing that denominator down. One reason it can be useful to understand just what you're doing in these situations is that it may let you save a lot of time. If you need to compare the fractions 13/18 and 17/24, say, cross-multiplying certainly isn't very fast. It's much faster to notice we can use a common denominator of 72:

13/18 = 52/72
17/24 = 51/72

so 13/18 is larger.