What is the Estimate the square root of 184 by finding the consecutive whole numbers it lies between?

Answer 1

132 = 169 < 184 < 196 = 142

So 13 < √184 < 14

So estimating from consecutive whole numbers would give the answer "13 and a bit". But you can do better than that by interpolating. The square-root function is, obviously, not linear but you will not go too far wrong between consecutive integers (unless you are dealing with very small number).

184 is (184-169)/(196-169) = 15/27 of the distance between 132 and 142

so the square root is approx 15/27 = 5/9 of the way from 13 to 14 ie 13+5/9 = 13.555... recurring.

In fact, the correct value is approx 13.5647 so the estimate is accurate to 2 decimal places.