answer:Let's denote the total number of employees before hiring the additional male workers as ( T ). Before hiring the additional male workers, the number of female workers was ( 0.60T ) because the workforce was 60% female. The number of male workers before hiring the additional ones was ( T - 0.60T = 0.40T ). After hiring the 26 additional male workers, the total number of employees became ( T + 26 ), and the number of male workers became ( 0.40T + 26 ). The new percentage of female workers is 55%, which means the percentage of male workers is 45%. Therefore, the number of male workers after hiring the additional ones can also be expressed as ( 0.45(T + 26) ). We can set up the equation: [ 0.40T + 26 = 0.45(T + 26) ] Now, let's solve for ( T ): [ 0.40T + 26 = 0.45T + 0.45 cdot 26 ] [ 26 = 0.45T + 11.7 - 0.40T ] [ 26 = 0.05T + 11.7 ] [ 26 - 11.7 = 0.05T ] [ 14.3 = 0.05T ] [ T = frac{14.3}{0.05} ] [ T = 286 ] So, before hiring the additional male workers, the company had 286 employees. After hiring the additional male workers, the company had: [ 286 + 26 = 312 ] Therefore, the company had boxed{312} employees after hiring the additional male workers.

answer:To find the highest power of 3 dividing M, we analyze its divisibility by summing its digits. 1. **Summing the digits of M:** - **Units digits:** The units digits from 15 to 95 cycle through 0 to 9 twice in every complete ten numbers (e.g., 15 to 24, ..., 85 to 94). Each cycle sums to 45. The additional digits from 95 are 5 (sum 5). The units digit 5 occurs once between 15 and 95, and it doesn't affect the determined cycle. - **Tens digits:** From 20 to 89, the tens digits 2 through 8 each appear 10 times (sum 10 times (2 + 3 + ... + 8) = 350). The numbers 15 to 19 and 90 to 95 add tens digits 1 and 9 (5 of each). The sum is 5 times (1 + 9) = 50. 2. **Total sum of digits:** - Total sum of units digits: 9 times 45 + 5 = 410 - Total sum of tens digits: 350 + 50 = 400 - Grand total sum: 410 + 400 = 810. 3. **Check divisibility by 3 and 9:** - 810 = 90 times 9, clearly divisible by 9. Since it's divisible by 9, it is also divisible by 3. 4. **Conclusion:** - Since M is divisible by 9 (and no higher powers are immediately obvious without more division checks), the highest power of 3 that divides M is 3^2. Thus, j = 2, and the correct answer is 2. The final answer is boxed{B) 2}

answer:Given x < y, we need to determine which of the inequalities is correct. Let's analyze each option step by step. **Option A: 2x < 2y** Starting from the given inequality x < y, we can multiply both sides by 2 (since 2 > 0, the inequality sign does not change): [x < y implies 2x < 2y] Therefore, Option A is correct. **Option B: -2x < -2y** Again, starting from x < y, if we multiply both sides by -2 (since -2 < 0, the inequality sign reverses): [x < y implies -2x > -2y] Thus, Option B is incorrect because the inequality sign should reverse when multiplying by a negative number. **Option C: x-2 > y-2** From x < y, subtracting 2 from both sides does not change the inequality sign: [x < y implies x-2 < y-2] Hence, Option C is incorrect because the inequality sign should remain the same when subtracting a number from both sides. **Option D: x+2 > y+2** Similarly, from x < y, adding 2 to both sides does not change the inequality sign: [x < y implies x+2 < y+2] Therefore, Option D is incorrect because the inequality sign should remain the same when adding a number to both sides. In conclusion, the only correct inequality based on the given condition x < y is Option A: 2x < 2y. Thus, the correct answer is boxed{A}.

answer:Given ( f(x) ) is a monotonically increasing function defined on ( (0, +infty) ) with the functional equation ( fleft( frac{x}{y} right) = f(x) - f(y) ). **(1)** To find ( f(1) ): 1. Substitute ( x = 1 ) and ( y = 1 ) into the functional equation: [ fleft(frac{1}{1}right) = f(1) - f(1) ] 2. Simplify the equation: [ f(1) = f(1) - f(1) implies f(1) = 0 ] Conclusion for part (1): [ boxed{0} ] **(2)** To solve the inequality ( f(x+3) - fleft( frac{1}{x} right) < 2 ): Given ( f(6) = 1 ). The goal is to manipulate the given inequality using the properties of ( f(x) ): 1. Write the given inequality: [ f(x+3) - fleft( frac{1}{x} right) < 2 ] 2. From the given functional equation, use ( x = x+3 ) and ( y = x ): [ f(x+3) = fleft( (x+3) cdot frac{1}{x} right) + f(x) = fleft( frac{x+3}{x} right) + f(x) ] 3. Use the fact ( frac{x+3}{x} = 1 + frac{3}{x} ) to transform ( fleft( frac{1}{x} right) ) using the functional equation [ fleft( frac{1}{x} right) = f(1) - f(x) = -f(x) ] 4. Substitute these transformations back into the inequality: [ f(x+3) - (-f(x)) < 2 implies f(x+3) + f(x) < 2 ] 5. Given ( f(6) = 1 ), substitute ( f(x+3) = f(6) ) for ( x = 3 ): [ f(6) + f(3) < 2 ] 6. This implies: [ 1 + f(3) < 2 implies f(3) < 1 ] Now, substitute to satisfy functional properties and solve the restriction on ( x ) given that we need ( f(x(x+3))/6 < 6 ): 7. Transform inequality ( f(x(x+3)) < 2 f(6) ) , rewrite: [ f(x(x+3)) < 2 quad text{(Since } 2f(6) = 2) ] 8. Equate function: [ fleft( frac{x(x+3)}{6} right) < f(6) ] 9. Since ( f(x) ) is increasing: [ frac{x(x+3)}{6} < 6 ] 10. Solve the quadratic inequality: [ x(x+3) < 36 ] 11. Rearrange to form and solve: [ x^2 + 3x - 36 < 0 ] 12. Find the roots of ( x^2 + 3x - 36 = 0 ) using quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ): [ x = frac{-3 pm sqrt{9 + 144}}{2} = frac{-3 pm 15}{2} ] 13. Therefore: [ x = 6 text{ or } x = -frac{9}{2} ] Working with the valid domain ( (0, +infty) ): [ 0 < x < 6 ] Conclusion for part (2): [ boxed{0 < x < frac{-3 + 3 sqrt{17}}{2}} ]