You can do the following …
1. Compare pairwise, but select appropriate pairs to eliminate a few
2. Among the ones left, judge which of the following two approaches seems easier …
… is % changes in numerators (and denominators) easier to estimate
… or is numerator, as a % of denominator easier to estimate.
Follow the easier out of the two.
3. Then there are some cool manipulations you can do (explained as the last point in this post)
I will use the given ratios to explain the above.
Say we need to find the greatest of 1089/1375, 1086/1430, 1144/1562 and 1110/1636
Step 1: Between 1089/1375 and 1086/1430, the first fraction has higher numerator (than second) and smaller denominator (than second). Thus, first fraction > second one. So second one is eliminated.
By the same reasoning, 1144/1562 > 1110/1636 (bigger numerator and smaller denominator)
We are left with comparing 1089/1375 and 1144/1562
Step 2: Comparing numerators amongst themselves & denominators amongst themselves ….
Try to go from lower number to higher number, it helps
1089 —-> 1144
is increase of 55, on a base of 1089.
10% of 1089 = 108.9. Thus, 55 will be very near to 5%
So, numerator of second fraction is 5% more than numerator of 1st fraction.
All the above should be done in mind. Try the same, in your mind, for denominator ….
1375 —-> 1562
is an increase of 187, on a base of 1375
10% of 1375 = 137. Thus, 187 will be more than 10%
So, denominator of second fraction is 10+% more than denominator of 1st fraction,
IF NUMERATOR INCREASES BY SMALLER % COMPARED TO % INCREASE IN DENOMINATOR, the fraction reduces in value.
Thus, the second fraction is less in value than the first fraction.
Thus, 1089/1375 is greatest.

However ……

…… the above may not work best for finding second-smallest. Thus, the approaches to be followed changes based on specific scenarios & values used. Had they been two digit numbers, it would have been a totally different approach (mostly of cross-multiplying)
Let’s find the second-smallest, right from scratch.
Even though by mere observation, we can get …
1086/1430 < 1089/1375
1110/1636 < 1144/1562
… it will still require, if you are lucky, one more pair to be compared. And if you are unlucky, you may need to compare two more pairs.
Thus, the best way is to estimate all the four fractions themselves …
Since the numerators are closer to the denominators, a nice trick you can use is to invert the ratios …
… dont write them again, just imagine the inverse, and remove the integral 1 …. and write the residuals
1 (268/1089)
1 (344/1086)
1 (418/1144)
1 (526/1110)
The residuals are usually much easier to approximate, atleast the fist decimal place
268/1089 is surely between 0.2 to 0.3, somewhere in the middle.
344/1086 is surely between 0.3 to 0.4, closer to 0.3
418/1144 is surely between 0.3 to 0.4, closer to 0.4
526/1110 is surely between 0.4 to 0.5, closer to 0.5
(For all the above, we just find 0.1 times the denominator and then check what multiple of it is close to the numerator e.g. in the third fraction … 0.1 × 1144 = 114.4. What multiple of 114.4 is close to 418?
114.4 × 3 = 343 and 114.4 × 4 = 456.
418 is surely between 343 and 456 and is closer to 456 than to 343.)
And since we have inversed the ratios, we now need to find the second-largest. And the answer will be the third fraction.
Thus, 1144/1562 will be the second-smallest amongst the four ratios.