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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Step 1: Define initial quantities and variables for goats sold. Let A_0 be the initial number of goats Abdul has, and C_0 be the initial number of goats Chebet has. A_0 = 32 goats C_0 = 56 goats Let x be the number of goats Abdul sold. Chebet sold twice as many goats as Abdul, so Chebet sold 2x goats. Step 2: Express the number of goats remaining after the sale. Abdul's remaining goats: A_r = A_0 - x = 32 - x Chebet's remaining goats: C_r = C_0 - 2x = 56 - 2x Step 3: Set up the ratio of remaining goats and solve for x. The ratio of goats remaining for Abdul to Chebet is 3:5. (32 - x)/(56 - 2x) = (3)/(5) Cross-multiply: 5(32 - x) = 3(56 - 2x) 160 - 5x = 168 - 6x Rearrange the terms to solve for x: 6x - 5x = 168 - 160 x = 8 Step 4: Calculate the total number of goats sold. Number of goats Abdul sold = x = 8 goats. Number of goats Chebet sold = 2x = 2 × 8 = 16 goats. Total number of goats sold = 8 + 16 = 24 goats. The total number of goats sold by Abdul and Chebet is 24 goats. Step 1: Define the four consecutive odd numbers. Let the first odd number be n. Since the numbers are consecutive odd numbers, they differ by 2. The four consecutive odd numbers are n, n+2, n+4, and n+6. Step 2: Set up the equation for their sum. The sum of these four numbers is 64. n + (n+2) + (n+4) + (n+6) = 64 Combine like terms: 4n + 12 = 64 Step 3: Solve for n. Subtract 12 from both sides: 4n = 64 - 12 4n = 52 Divide by 4: n = (52)/(4) n = 13 Step 4: Determine the four integers. The first number is n = 13. The second number is n+2 = 13+2 = 15. The third number is n+4 = 13+4 = 17. The fourth number is n+6 = 13+6 = 19. The first four such integers are 13, 15, 17, 19. Step 1: Define the number of sides for each polygon and the formula for the sum of interior angles. Let N_1 be the number of sides of the first polygon, so N_1 = n-1. Let N_2 be the number of sides of the second polygon, so N_2 = n. The sum of interior angles of a polygon with N sides is S = (N-2) × 180^. Step 2: Write the sum of interior angles for each polygon. Sum of interior angles for the first polygon (S_1): S_1 = ((n-1)-2) × 180^ = (n-3) × 180^ Sum of interior angles for the second polygon (S_2): S_2 = (n-2) × 180^ Step 3: Use the given ratio to find n. The ratio of the sums of interior angles is 4:5. (S_1)/(S_2) = ((n-3) × 180^)/((n-2) × 180^) = (4)/(5) (n-3)/(n-2) = (4)/(5) Cross-multiply: 5(n-3) = 4(n-2) 5n - 15 = 4n - 8 Solve for n: 5n - 4n = 15 - 8 n = 7 Step 4: Calculate the number of sides for the first polygon. The polygon with sides (n-1) has N_1 = n-1 = 7-1 = 6 sides. i) The size of the interior angle of the polygon with sides (n-1). For a regular polygon with N sides, the size of one interior angle is ((N-2) × 180^)/(N). For N_1 = 6 sides: Interior Angle = ((6-2) × 180^)/(6) = (4 × 180^)/(6) = (720^)/(6) = 120^ The size of the interior angle of the polygon with sides (n-1) is 120^. ii) The size of the exterior angle of the polygon with sides (n-1). For a regular polygon with N sides, the size of one exterior angle is (360^)/(N). For N_1 = 6 sides: Exterior Angle = (360^)/(6) = 60^ Alternatively, the interior angle and exterior angle sum to 180^. Exterior Angle = 180^ - Interior Angle = 180^ - 120^ = 60^ The size of the exterior angle of the polygon with sides (n-1) is 60^. Got more? Send 'em! 📸