answer:To solve this, we need to calculate the future value of Sam's investment in Option B using the given compound interest rates for the two periods. For the first 3 years, the interest is compounded semi-annually, which means it is compounded twice a year. The formula for compound interest is: [ A = P left(1 + frac{r}{n}right)^{nt} ] Where: - ( A ) is the amount of money accumulated after n years, including interest. - ( P ) is the principal amount (the initial amount of money). - ( r ) is the annual interest rate (in decimal). - ( n ) is the number of times that interest is compounded per year. - ( t ) is the time the money is invested for in years. For the first 3 years with interest rate ( p% ) compounded semi-annually, we have: [ P = 150 ] [ r = frac{p}{100} ] [ n = 2 ] [ t = 3 ] The amount ( A_1 ) after the first 3 years is: [ A_1 = 150 left(1 + frac{p/100}{2}right)^{2 cdot 3} ] For the next 3 years, the interest rate is ( q% ) compounded quarterly, which means it is compounded four times a year. The principal for this period is ( A_1 ), the amount we just calculated. So, we have: [ P = A_1 ] [ r = frac{q}{100} ] [ n = 4 ] [ t = 3 ] The total amount ( A_2 ) after the next 3 years is: [ A_2 = A_1 left(1 + frac{q/100}{4}right)^{4 cdot 3} ] Substituting ( A_1 ) from the first equation into the second, we get: [ A_2 = 150 left(1 + frac{p/100}{2}right)^{2 cdot 3} left(1 + frac{q/100}{4}right)^{4 cdot 3} ] Without the specific values of ( p ) and ( q ), we cannot calculate the exact amount. However, this is the formula Sam would use to determine the future value of his investment in Option B after boxed{6} years.

answer:Let's denote the initial number of men as ( M ) and their initial average age as ( A ). When two men, aged 20 and 28, are replaced by two women, each with an average age of 32, the average age of the group increases by 2 years. The total age of the initial group of men is ( M times A ). When the two men are replaced, the total age of the group changes by the difference in the ages of the outgoing men and the incoming women: Change in total age = (Age of first woman + Age of second woman) - (Age of first man + Age of second man) Change in total age = (32 + 32) - (20 + 28) Change in total age = 64 - 48 Change in total age = 16 This change in total age results in an increase of 2 years in the average age of the group. Since the number of people in the group remains the same (still ( M )), we can write the following equation: New total age = Old total age + Change in total age ( (A + 2) times M = M times A + 16 ) Now we can solve for ( M ): ( A times M + 2M = A times M + 16 ) ( 2M = 16 ) ( M = frac{16}{2} ) ( M = 8 ) So, there were initially boxed{8} men in the group.

answer:Given f(x) = x - 3, the inverse is f^{-1}(x) = x + 3 since it reverses the operation of subtracting 3. Given g(x) = x/2, the inverse is g^{-1}(x) = 2x since it reverses the operation of dividing by 2. We compute as follows: [begin{array}{rl|l} &f(g^{-1}(f^{-1}(g(f^{-1}(g(f(23))))))) &quad=f(g^{-1}(f^{-1}(g(f^{-1}(g(20)))))) & text{subtracted 3} &quad=f(g^{-1}(f^{-1}(g(f^{-1}(10))))) & text{divided by 2} &quad=f(g^{-1}(f^{-1}(g(13)))) & text{added 3} &quad=f(g^{-1}(f^{-1}(6.5))) & text{divided by 2} &quad=f(g^{-1}(9.5)) & text{added 3} &quad=f(19) & text{multiplied by 2} &quad=boxed{16} & text{subtracted 3} end{array}]

answer:Let's analyze each statement step by step: (1) It is a common misconception that all numbers with square roots are irrational. However, this is not true. For example, the square root of 4 is 2, which is a rational number. Therefore, statement (1) does not hold because not all numbers with square roots are irrational. Some square roots result in rational numbers. (2) When considering the cube roots that are equal to themselves, we typically think of 0 and 1. However, it is also true that the cube root of -1 is -1 (sqrt[3]{-1} = -1). This means there are three numbers whose cube roots are equal to themselves: 0, 1, and -1. Therefore, statement (2) does not hold because it omits -1. (3) The statement that -a definitely has no square root is incorrect. For example, if a = -1, then the square root of -(-1) = 1 is 1, which is a real number. Moreover, in the realm of complex numbers, negative numbers do have square roots. Therefore, statement (3) does not hold because negative numbers can have square roots, especially when considering complex numbers. (4) Real numbers and points on the number line have a one-to-one correspondence. This means each real number corresponds to exactly one point on the number line, and each point on the number line corresponds to exactly one real number. This is a fundamental property of real numbers and the number line. Therefore, statement (4) holds because it accurately describes the relationship between real numbers and the number line. (5) The difference between two irrational numbers is not always irrational. For example, the difference between sqrt{2} and (sqrt{2} - 1) is 1, which is a rational number. Therefore, statement (5) does not hold because the difference between two irrational numbers can be either rational or irrational. Based on the analysis above, only statement (4) holds true. Therefore, the correct answer is: boxed{A} (1 statement).