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 math problems: 1) The numeral is 69,425,950. The digit 4 is in the hundreds of thousands place. The correct option is C. 2) To divide 1.68 by 1.2: Step 1: Write the division as a fraction. (1.68)/(1.2) Step 2: Multiply the numerator and denominator by 10 to remove the decimal from the denominator. (1.68 × 10)/(1.2 × 10) = (16.8)/(12) Step 3: Perform the division. 16.8 ÷ 12 = 1.4 The correct option is A. 3) To find the LCM of 27, 90, and 84: Step 1: Find the prime factorization of each number. 27 = 3^3 90 = 2 × 3^2 × 5 84 = 2^2 × 3 × 7 Step 2: Take the highest power of each prime factor present in any of the numbers. LCM = 2^2 × 3^3 × 5^1 × 7^1 Step 3: Calculate the product. LCM = 4 × 27 × 5 × 7 = 108 × 35 = 3780 The correct option is D. 4) Let the certain number be N. Step 1: Set up the equation based on the given information. 2\% of N = (3)/(4) Step 2: Convert the percentage to a decimal and the fraction to a decimal. 0.02 × N = 0.75 Step 3: Solve for N. N = (0.75)/(0.02) N = (75)/(2) N = 37.5 The correct option is C. 5) For a polygon with n vertices, the number of diagonals that can be drawn from a single vertex is given by the formula n-3. Step 1: Identify the number of vertices for a pentagon. A pentagon has n=5 vertices. Step 2: Apply the formula. Number of diagonals = 5 - 3 = 2 The correct option is B. 6) To simplify 8 - 6(2x + 3(4-x) - 1): Step 1: Simplify the innermost parenthesis. 8 - 6(2x + 12 - 3x - 1) Step 2: Combine like terms inside the main parenthesis. 8 - 6(-x + 11) Step 3: Distribute the -6. 8 + 6x - 66 Step 4: Combine the constant terms. 6x - 58 The correct option is A. 7) To solve for x in 3(x+4) + 5(2x-1) = 4(2x+8) + 5: Step 1: Distribute the numbers into the parentheses. 3x + 12 + 10x - 5 = 8x + 32 + 5 Step 2: Combine like terms on each side of the equation. 13x + 7 = 8x + 37 Step 3: Move terms with x to one side and constants to the other. 13x - 8x = 37 - 7 5x = 30 Step 4: Divide to solve for x. x = (30)/(5) x = 6 The correct option is B. 8) To make x the subject of ax - b = bx: Step 1: Move all terms containing x to one side of the equation. ax - bx = b Step 2: Factor out x. x(a - b) = b Step 3: Divide by (a-b) to isolate x. x = (b)/(a-b) The correct option is D. 9) To factorize U^2 - 16V^2: Step 1: Recognize this as a difference of squares, A^2 - B^2 = (A-B)(A+B). Here, A = U and B = sqrt(16V^2) = 4V. Step 2: Apply the difference of squares formula. U^2 - (4V)^2 = (U - 4V)(U + 4V) The correct option is A. 10) To factorize 3x^2+5x-2: Step 1: Look for two binomials (Ax+B)(Cx+D) that multiply to the given quadratic. Since the coefficient of x^2 is 3, the factors must be of the form (3x+B)(x+D). Step 2: Find factors of -2 (B and D) such that the sum of the inner and outer products equals 5x. Let's try (3x-1)(x+2): (3x-1)(x+2) = 3x(x) + 3x(2) - 1(x) - 1(2) = 3x^2 + 6x - x - 2 = 3x^2 + 5x - 2 This matches the original expression. The correct option is A. 11) To express the ratio 1 (1)/(2) : 2 (1)/(2) in the simplest form: Step 1: Convert the mixed numbers to improper fractions. 1 (1)/(2) = (1 × 2 + 1)/(2) = (3)/(2) 2 (1)/(2) = (2 × 2 + 1)/(2) = (5)/(2) Step 2: Write the ratio with improper fractions. (3)/(2) : (5)/(2) Step 3: Multiply both parts of the ratio by the common denominator, 2, to eliminate the fractions. ((3)/(2)) × 2 : ((5)/(2)) × 2 3 : 5 The correct option is B. 12) If y varies directly as x, the relationship is y = kx, where k is the constant of proportionality. Step 1: Use the given values (y=100 when x=20) to find k. 100 = k × 20 k = (100)/(20) k = 5 Step 2: Use the value of k to find y when x=25. y = 5 × 25 y = 125 The correct option is D. 3 done, 2 left today. You're making progress.