answer:1. **Convert Decimals to Fractions**: - 0.5 as a fraction is frac{1}{2}. - 0.2 as a fraction is frac{1}{5}. 2. **Set Up the Product**: Replace the decimals with their fractional equivalents. [ 12 times 0.5 times 3 times 0.2 = 12 times frac{1}{2} times 3 times frac{1}{5} ] 3. **Multiply Numerators and Denominators**: Handle the multiplication of numerators and denominators. [ 12 times frac{1}{2} times 3 times frac{1}{5} = frac{12 times 3}{2 times 5} ] [ = frac{36}{10} ] 4. **Simplify the Expression**: Simplify the fraction. [ frac{36}{10} = frac{18}{5} ] 5. **Conclusion**: The product of 12 times 0.5 times 3 times 0.2 simplifies to frac{18}{5}. [ frac{18{5}} ] The final answer is D) boxed{frac{18}{5}}

answer:To find the coordinates of point B after the translation of point A(-2,2), we follow the given translations step by step: 1. **First Translation (4 units downward):** - The downward translation affects only the y-coordinate, decreasing it by 4 units. - Therefore, the new coordinates after this translation are: [ (-2, 2 - 4) = (-2, -2). ] 2. **Second Translation (3 units to the right):** - The rightward translation affects only the x-coordinate, increasing it by 3 units. - Thus, the coordinates after this second translation are: [ (-2 + 3, -2) = (1, -2). ] Therefore, after translating point A(-2,2) 4 units downward and then 3 units to the right, we obtain point B with coordinates boxed{(1, -2)}.

answer:Part (a) Given a cascade generated by the number ( r ), which is a sequence of natural numbers: ( r, 2r, ldots, 12r ). 1. Define the terms of any cascade: [ text{Cascade:} { r, 2r, ldots, 12r } ] 2. Consider two numbers ( a ) and ( b ) such that ( a < b ): [ text{Then} quad frac{a}{b} quad text{should be a fraction with the denominator no greater than 12.} ] 3. Recognize that to match this condition, ( frac{a}{b} ) should be one of the following fractions which can simplify to the same form: [ frac{1}{2} = frac{2}{4} = frac{3}{6} = frac{4}{8} = frac{5}{10} = frac{6}{12} ] 4. Each of these fractions implies a relation within a cascade: if ( a = 3x ) and ( b = 6x ), then the pair (3x, 6x) belongs to a cascade generated by ( x ). 5. To have such cascades for these fractions, we need ( a ) and ( b ) to take forms involving multiple possible values of ( x ): [ a = 60 quad text{and} quad b = 120. ] 6. Thus, such pairs ( (a, b) ) can indeed be part of multiple cascades. Specifically, the numbers ( r ) generating these cascades can be: [ r = 60, 30, 20, 15, 12, 10. ] Conclusion for Part (a): Yes, there exist pairs of numbers ( (a, b) ) that can be part of six different cascades. For example, let ( a = 60 ) and ( b = 120 ). Part (b) 1. To prove that it is possible to color the set of natural numbers in 12 colors such that in every cascade all elements are of different colors. 2. We define the coloring based on the properties of the numbers concerning modulo 13. Write each natural number ( n ) as: [ n = 13^s cdot m, ] where ( m ) is not divisible by 13. 3. Assign colors based on the number ( m )'s value modulo 13: [ r(m) equiv text{(remainder when ( m ) is divided by 13)}. ] 4. Since ( m ) can take 12 different remainders (from 1 to 12), we use this remainder as the color of ( n ). 5. Check that every ( r )-generated cascade will span over different colors: [ { r, 2r, 3r, ldots, 12r }. ] If ( x = 13^s cdot m ) generates a cascade, then: [ r(1 cdot x) = r left( 13^s cdot 1 cdot m right), quad r(2 cdot x) = r left( 13^s cdot 2 cdot m right), quad cdots quad r(12 cdot x) = r left( 13^s cdot 12 cdot m right). ] 6. Since each remainder is non-zero and distinct modulo 13: [ r(m), r(2m), ldots, r(12m) quad text{are distinct modulo 13}. ] 7. Hence, all numbers in the cascade will have distinct colors. Conclusion for Part (b): Yes, it is possible to color the natural numbers in 12 colors such that in every cascade, all elements are colored differently. boxed{}

answer:Since a=log_{0.5}3<0, b=2^{0.5}>1, and c=0.5^{0.3} is between (0, 1). Therefore, a<c<b. Hence, the answer is: a<c<b. **Analysis:** This conclusion can be drawn by utilizing the monotonicity of exponential and logarithmic functions. Thus, the relationship among a, b, and c is boxed{a<c<b}.