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: Problem 1: Simplify fraction expression Step 1: Convert mixed numbers to improper fractions. 6(3)/(4) = (27)/(4) 2(4)/(15) = (34)/(15) 1(2)/(3) = (5)/(3) Step 2: Evaluate the expression inside the parenthesis. 2(4)/(15) - 1(2)/(3) = (34)/(15) - (5)/(3) = (34)/(15) - (25)/(15) = (9)/(15) = (3)/(5) Step 3: Perform the "of" operation (multiplication). (2)/(3) of 6(3)/(4) = (2)/(3) × (27)/(4) = (54)/(12) = (9)/(2) Step 4: Perform the division. (9)/(2) ÷ (3)/(5) = (9)/(2) × (5)/(3) = (45)/(6) = (15)/(2) The simplified expression is (15)/(2) or 7(1)/(2). Problem 2: Evaluate and leave your answer in standard form Step 1: Perform the multiplication in the numerator. 4.56 × 3.6 = 16.416 Step 2: Perform the division. (16.416)/(0.12) = (1641.6)/(12) = 136.8 Step 3: Express the answer in standard form. 136.8 = 1.368 × 10^2 The evaluated value in standard form is 1.368 × 10^2. Problem 3: Calories burned per hour Step 1: Divide the total calories burned by the total hours. Calories per hour = Total caloriesTotal hours Calories per hour = 856 calories4 hours Step 2: Calculate the value. Calories per hour = 214 calories/hour Bongo burns 214 calories per hour. Problem 4: Simplify the algebraic expression Step 1: Factor the denominators and numerators. (5x^2)/(x^2-2x) - (x^2-4)/(x^2+2x) = (5x^2)/(x(x-2)) - ((x-2)(x+2))/(x(x+2)) Step 2: Simplify each term by canceling common factors. (5x)/(x-2) - (x-2)/(x) Step 3: Find a common denominator and combine the fractions. The common denominator is x(x-2). (5x · x)/(x(x-2)) - ((x-2)(x-2))/(x(x-2)) = (5x^2 - (x^2 - 4x + 4))/(x(x-2)) Step 4: Expand and simplify the numerator. (5x^2 - x^2 + 4x - 4)/(x(x-2)) = (4x^2 + 4x - 4)/(x(x-2)) Step 5: Factor out common terms in the numerator. (4(x^2 + x - 1))/(x(x-2)) The simplified expression is (4(x^2 + x - 1))/(x(x-2)). Problem 5: Find the value of x in the diagram Step 1: Use the property that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. (x+17^) + (2x+7^) = 9x^ Step 2: Simplify and solve for x. 3x + 24 = 9x 24 = 9x - 3x 24 = 6x x = (24)/(6) x = 4 The value of x is 4. Problem 6: Rectangle dimensions Step 1: Define the dimensions. Let the width of the rectangle be w. The length l is three times its width, so l = 3w. Step 2: Use the formula for the perimeter of a rectangle. P = 2(l+w) Given P = 24 cm. 24 = 2(3w + w) Step 3: Solve for w. 24 = 2(4w) 24 = 8w w = (24)/(8) w = 3 cm The width of the rectangle is 3 cm. Problem 7: Find the vector r Step 1: Add vectors q and p. q + p = 2 \\ 5 + 2 \\ 3 = 2+2 \\ 5+3 = 4 \\ 8 Step 2: Multiply the resulting vector by (1)/(2). r = (1)/(2)(q + p) = (1)/(2) 4 \\ 8 = (1)/(2) × 4 \\ (1)/(2) × 8 = 2 \\ 4 The vector r is 2 \\ 4 . 3 done, 2 left today. You're making progress.