This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 1. Solve: (1)/(3)(x-2) (1)/(5)(2x+3) Step 1: Multiply both sides by the least common multiple of the denominators (3 and 5), which is 15. 15 · (1)/(3)(x-2) 15 · (1)/(5)(2x+3) 5(x-2) 3(2x+3) Step 2: Distribute the numbers on both sides. 5x - 10 6x + 9 Step 3: Gather x terms on one side and constant terms on the other. -10 - 9 6x - 5x -19 x This can also be written as x -19. The correct option is C. C. x -19 2. The ages in years of six teachers in a school are: 3x, 42, (2x+6), 28, 49, 31. If the mean age is 36 years, find the value of x. Step 1: Write the formula for the mean. Mean = Sum of agesNumber of teachers Step 2: Substitute the given ages and mean into the formula. 36 = (3x + 42 + (2x+6) + 28 + 49 + 31)/(6) Step 3: Simplify the numerator by combining like terms. 36 = ((3x + 2x) + (42 + 6 + 28 + 49 + 31))/(6) 36 = (5x + 156)/(6) Step 4: Solve for x. Multiply both sides by 6. 36 × 6 = 5x + 156 216 = 5x + 156 Subtract 156 from both sides. 216 - 156 = 5x 60 = 5x Divide by 5. x = (60)/(5) x = 12 The correct option is C. C. 12 3. Given that 2x - y = 11 and 4x + y = 1, find the value of (x+y). Step 1: Use the elimination method to solve the system of equations. Add the two equations together. (2x - y) + (4x + y) = 11 + 1 6x = 12 Step 2: Solve for x. x = (12)/(6) x = 2 Step 3: Substitute the value of x into the second equation (4x + y = 1) to find y. 4(2) + y = 1 8 + y = 1 y = 1 - 8 y = -7 Step 4: Calculate the value of (x+y). x+y = 2 + (-7) x+y = -5 The correct option is A. A. -5 4. The mean of 4, 5, 7, 8 and 11 is 7. Calculate the standard deviation. Step 1: List the data points (x_i) and the mean (x). Data points: 4, 5, 7, 8, 11 Mean (x): 7 Number of data points (n): 5 Step 2: Calculate the deviation of each data point from the mean (x_i - x) and square it (x_i - x)^2. • 4 - 7 = -3 (-3)^2 = 9 • 5 - 7 = -2 (-2)^2 = 4 • 7 - 7 = 0 (0)^2 = 0 • 8 - 7 = 1 (1)^2 = 1 • 11 - 7 = 4 (4)^2 = 16 Step 3: Sum the squared deviations. (x_i - x)^2 = 9 + 4 + 0 + 1 + 16 = 30 Step 4: Calculate the variance (^2) using the formula ^2 = (x_i - x)^2n. ^2 = (30)/(5) = 6 Step 5: Calculate the standard deviation () by taking the square root of the variance. = sqrt(6) The correct option is B. B. sqrt(6) 5. Given that OT = 3 \\ 5 , AB = -1 \\ -2 , where O is the origin, find OB. Step 1: Understand the vector relationship. The vector AB can be expressed as OB - OA. AB = OB - OA Since O is the origin, OA is the position vector of point A. The vector OT is a distractor in this problem, as it refers to point T, which is not involved in finding OB from AB. Step 2: To find OB, we need OA. Given the multiple-choice options, the most plausible interpretation is that point A coincides with the origin O. If A is the origin, then its position vector OA is the zero vector. OA = 0 \\ 0 Step 3: Substitute OA into the vector relationship. AB = OB - 0 \\ 0 AB = OB Step 4: Use the given value for AB to find OB. OB = -1 \\ -2 The correct option is A. A. -1 \\ -2 Send me the next one 📸