Explanation: To understand the concept of ratio and how it applies to the population of Indian states, let's break down the problem step by step.
A ratio is a comparison of two quantities of the same kind. In this case, we are comparing the populations of two Indian states, U.P (Uttar Pradesh) and Rajasthan. The population of Rajasthan is given as 1660 lakhs, and the population of U.P is given as 570 lakhs.
To find the ratio of the population of U.P to Rajasthan, we need to divide the population of U.P by the population of Rajasthan. This can be expressed as a fraction:
\[
\text{Ratio} = \frac{\text{Population of U.P}}{\text{Population of Rajasthan}} = \frac{570 \text{ lakhs}}{1660 \text{ lakhs}}
\]
Since both the numerator and the denominator are in lakhs, we can simplify the fraction by removing the common unit (lakhs):
\[
\text{Ratio} = \frac{570}{1660}
\]
Next, we simplify the fraction by finding the greatest common divisor (GCD) of 570 and 1660. The GCD of 570 and 1660 is 10. We divide both the numerator and the denominator by 10:
\[
\text{Ratio} = \frac{570 \div 10}{1660 \div 10} = \frac{57}{166}
\]
Thus, the simplified ratio of the population of U.P to Rajasthan is 57:166.
It is important to note that the order of the states in the ratio is crucial. The ratio 57:166 means that for every 57 people in U.P, there are 166 people in Rajasthan. Reversing the order would give a different ratio, which would not accurately represent the comparison of the populations.
In summary, the correct answer is (B) 57:166, as it correctly represents the ratio of the population of U.P to Rajasthan. This problem tests the understanding of basic arithmetic and the concept of ratio, which are fundamental in various competitive examinations and general knowledge contexts.