answer:First, let's find out how many toads Jim has. Since Jim has 20 more toads than Tim, and Tim has 30 toads, we can calculate Jim's toads as follows: Jim's toads = Tim's toads + 20 Jim's toads = 30 + 20 Jim's toads = 50 Now, we know that Sarah has 100 toads. To find the ratio of the number of toads Sarah has to the number of toads Jim has, we divide Sarah's toads by Jim's toads: Ratio = Sarah's toads / Jim's toads Ratio = 100 / 50 Ratio = 2 So, the ratio of the number of toads Sarah has to the number of toads Jim has is boxed{2:1} .

answer:1. **Identify the number of people wearing both sunglasses and caps**: Given that the probability of a person wearing a cap also wearing sunglasses is frac{3}{7}, and there are 84 people wearing caps, the number of people wearing both is: [ text{Number of people wearing both} = frac{3}{7} times 84 = 36. ] 2. **Calculate the probability that a person wearing sunglasses is also wearing a cap**: There are 120 people wearing sunglasses, and from the previous step, we know 36 of these are also wearing caps. The probability is calculated by: [ text{Probability} = frac{text{Number of people wearing both}}{text{Total number of people wearing sunglasses}} = frac{36}{120}. ] 3. **Simplify the fraction**: Simplify frac{36}{120}, reducing it by dividing both the numerator and denominator by gcd(36, 120), which is 12: [ frac{36}{120} = frac{36 div 12}{120 div 12} = frac{3}{10}. ] 4. **Conclusion with the boxed answer**: The probability that a person wearing sunglasses is also wearing a cap is: [ frac{3{10}} ] The final correct answer is **boxed{textbf{(A)} frac{3}{10}}**.

answer:1. **Problem Restatement**: We are given the set ( S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ). We need to find the number of its subsets, each containing at least 2 elements and for any two elements ( a ) and ( b ) in the subset, (|a - b| > 1). 2. **General Approach**: Let ( a_n ) be the number of subsets of ( {1, 2, ldots, n} ) that satisfy the given conditions. 3. **Recurrence Relation**: To find a general formula for ( a_n ), consider the set ( T = {1, 2, ldots, n, n+1, n+2} ).. For each valid subset of ( T ): - Some may contain the element ( n+2 ). - Some may not contain ( n+2 ). 4. **Counting Subsets Containing ( n+2 )**: If a subset contains ( n+2 ), then the other elements must come from ( {1, 2, ldots, n} ) and none of these subsets can contain ( n+1 ). The number of such subsets is ( a_n + n ) (each combination of a subset of ( {1, 2, ldots, n} ) plus ( n+2 ), and ( n ) subsets where ( n+2 ) is paired with exactly one of ( 1, 2, ldots, n )). 5. **Counting Subsets Not Containing ( n+2 )**: If a subset does not contain ( n+2 ), it is a subset of ( {1, 2, ldots, n+1} ) that satisfies the given conditions. There are ( a_{n+1} ) such subsets. 6. **Combined Recurrence Relation**: Hence, the number of subsets in ( T ) is given by: [ a_{n+2} = a_n + a_{n+1} + n ] 7. **Base Cases**: - ( a_3 = 1 ): The only valid subset is ( {1, 3} ). - ( a_4 = 3 ): The valid subsets are ( {1, 3}, {2, 4}, {1, 4} ). 8. **Successively Applying the Recurrence Relation**: Using the recurrence relation, calculate ( a_n ) for ( n = 5 ) through ( n = 10 ): - ( a_5 = a_3 + a_4 + 3 = 1 + 3 + 3 = 7 ) - ( a_6 = a_4 + a_5 + 4 = 3 + 7 + 4 = 14 ) - ( a_7 = a_5 + a_6 + 5 = 7 + 14 + 5 = 26 ) - ( a_8 = a_6 + a_7 + 6 = 14 + 26 + 6 = 46 ) - ( a_9 = a_7 + a_8 + 7 = 26 + 46 + 7 = 79 ) - ( a_{10} = a_8 + a_9 + 8 = 46 + 79 + 8 = 133 ) 9. **Conclusion**: The number of subsets of ( {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ) that satisfy the given conditions is therefore ( boxed{133} ).

answer:1. For the function f(x)=sin left(frac{pi }{2}xright), the period of the sine function is 4. This means that the function repeats its values over intervals of length 4. However, we are interested in finding a "domain-equivalent interval". For A=[0,1], we have: - f(0) = sin(0) = 0 in [0,1] - f(1) = sinleft(frac{pi}{2}right) = 1 in [0,1] Similarly, for A=[-1,0], we have: - f(-1) = sinleft(-frac{pi}{2}right) = -1 in [-1,0] - f(0) = sin(0) = 0 in [-1,0] Since there are at least two different "domain-equivalent intervals", the uniqueness condition is not satisfied. 2. For the function f(x)=2x^{2}-1, when considering A=[-1,1], we have: - f(-1) = 2(-1)^2 - 1 = 1 in [-1,1] - f(1) = 2(1)^2 - 1 = 1 in [-1,1] The graph of f(x) is a parabola opening upwards with vertex at (0, -1). This means that for x in [-1,1], f(x) in [-1,1]. This interval is unique because the function's values outside this interval will not fall within [-1,1]. 3. For the function f(x)=|1-2^{x}|, when x in [0,1], we have: - f(0) = |1-2^{0}| = 0 in [0,1] - f(1) = |1-2^{1}| = 1 in [0,1] The function f(x)=2^{x}-1 for x in [0,1] is monotonically increasing, and it maps the interval [0,1] onto itself. This satisfies the condition for a "domain-equivalent interval", and it is unique since the behavior of f(x) outside this interval does not satisfy the condition. 4. For the function f(x)=log _{2}(2x-2), its domain is (1,+infty). If a "domain-equivalent interval" exists, it must satisfy: - log _{2}(2m-2)=m - log _{2}(2n-2)=n This implies: - 2m-2=2^{m} - 2n-2=2^{n} Let f(x)=2^{x}-2x+2, then f'(x)=2^{x}ln 2-2. For x>1, f'(x)>0, indicating f(x) is monotonically increasing. Therefore, the equation 2^{x}-2x+2=0 cannot have two solutions, implying that f(x)=log _{2}(2x-2) does not have a "domain-equivalent interval". Therefore, the correct choice is: boxed{text{B. 2, 3}}.