You might need to consult your teacher or text to learn the details of the "ratio of perfect squares method" of determining an approximate square root. No reference to such a method can be found in an Internet search except in conjunction with problems similar to this one.
A method that can be used to find a first approximation of a square root is linear interpolation between the roots of adjacent perfect squares. For this, ...
• find the perfect squares of the consecutive integers that lie on either side of the root of interest
• form the ratio of difference between the number and the smaller square and the difference between the squares
• add this ratio to the smaller of the two integers to get an approximation of the root.
In this case, the square root of 96 lies between 9² = 81 and 10² = 100. The ratio of interest is (96 -81)/(100 -81) = 15/19. The approximate square root of 96 is then ...
√96 ≈ 9 + 15/19 ≈ 9.8
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If a = floor(√n), then this approximation to the root can be written as
√n ≈ a + (n -a²)/(2a+1)
If we define b = n - a², this looks like √n ≈ a + b/(2a+1). The approximation can be refined by replacing the 1 in the denominator with (b/(2a+1)). Repeately doing this replacement results in a "continued fraction" that converges to √n as more layers are added.
