Explanations:A ratio is a way to compare two quantities by division. In this problem, we are asked to find the ratio of the population of U.P. to the population of Rajasthan. The population of U.P. is given as 570 lakhs, and the population of Rajasthan is given as 1660 lakhs. To find the ratio, we divide the population of U.P. by the population of Rajasthan.
The ratio can be expressed as:
\[
\text{Ratio} = \frac{\text{Population of U.P.}}{\text{Population of Rajasthan}} = \frac{570 \text{ lakhs}}{1660 \text{ lakhs}}
\]
When we simplify this fraction, we get:
\[
\frac{570}{1660} = \frac{57}{166}
\]
Therefore, the ratio of the population of U.P. to the population of Rajasthan is 57:166. This corresponds to option B.
It is important to note that the order in which the ratio is written is crucial. The question specifically asks for the ratio of the population of U.P. to the population of Rajasthan, not the other way around. If we were to reverse the order, we would get the ratio of the population of Rajasthan to the population of U.P., which would be 166:57, corresponding to option A. This is incorrect because it does not match the order specified in the question.
Additionally, options C and D are incorrect because they do not represent the correct ratio of the populations. The numbers 19 and 80 do not correspond to the given population figures, and the order of the ratio is also incorrect in option D.
In summary, the correct answer is option B, which is 57:166. This is the ratio of the population of U.P. to the population of Rajasthan, as specified in the question.