Using square root table, find the square root

$\sqrt{\frac{101}{169}}=\frac{\sqrt{101}}{\sqrt{169}}$

The square root of 101 is not listed in the table. This is because the table lists the square roots of all the numbers below 100.

Hence, we have to manipulate the number such that we get the square root of a number less than 100. This can be done in the following manner:

$\sqrt{101}=\sqrt{1.01 \times 100}=\sqrt{1.01} \times 10$

Now, we have to find the square root of 1.01.

We have:

$\sqrt{1}=1$ and $\sqrt{2}=1.414$

Their difference is 0.414.

Thus, for the difference of 1 (2 -">− 1), the difference in the values of the square roots is 0.414.

For the difference of 0.01, the difference in the values of the square roots is:

0.01 ×">× 0.414 = 0.00414

$\therefore \sqrt{1.01}=1+0.00414=1.00414$

$\sqrt{101}=\sqrt{1.01} \times 10=1.00414 \times 10=10.0414$

Finally, $\sqrt{\frac{101}{169}}=\frac{\sqrt{101}}{1313}=\frac{10.0414}{13}=0.772$

This value is really close to the one from the key answer.