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 your questions: Question 1: The price of a loaf of bread last week was x. This week, the price rose by 20%. Determine the price of the loaf after the increase. Step 1: Express the percentage increase as a decimal. 20\% = (20)/(100) = 0.20 Step 2: Calculate the amount of the increase. Increase = 0.20 × x = 0.2x Step 3: Add the increase to the original price to find the new price. New Price = x + 0.2x = 1.2x The price of the loaf after the increase is 1.2x. --- Question 2: Four men takes x days to paint a room. How long will it take 16 men working at the same rate to paint the room? Step 1: Set up the inverse proportion relationship. The total work (men × days) remains constant. M_1 D_1 = M_2 D_2 Here, M_1 = 4 men, D_1 = x days, M_2 = 16 men. We need to find D_2. Step 2: Substitute the known values into the equation. 4 × x = 16 × D_2 Step 3: Solve for D_2. 4x = 16 D_2 D_2 = (4x)/(16) D_2 = (x)/(4) It will take 16 men (x)/(4) days. --- Question 3: Write 0.009 in standard form. Step 1: Move the decimal point to the right until the number is between 1 and 10. 0.009 9 The decimal point moved 3 places to the right. Step 2: Since the original number was less than 1, the exponent will be negative. 0.009 = 9 × 10^-3 In standard form, 0.009 is 9 × 10^-3. --- Question 4: Simplify the following expression: (7x-7)/(9x-9) Step 1: Factor out the common term from the numerator. 7x - 7 = 7(x-1) Step 2: Factor out the common term from the denominator. 9x - 9 = 9(x-1) Step 3: Rewrite the expression with the factored terms and simplify by canceling the common factor (x-1). (7(x-1))/(9(x-1)) = (7)/(9) The simplified expression is (7)/(9). --- Question 5: The figure at the right is a sketch of a piece of land. The lengths are in meters. a) Write and simplify an algebraic expression for perimeter of the land. b) If the shortest side is 9 m, what is the perimeter of the land? The figure shows a right-angled triangle with legs x and x+6. a) Write and simplify an algebraic expression for perimeter of the land. Step 1: Find the length of the hypotenuse using the Pythagorean theorem. Let the hypotenuse be h. h^2 = x^2 + (x+6)^2 h^2 = x^2 + (x^2 + 12x + 36) h^2 = 2x^2 + 12x + 36 h = sqrt(2x^2 + 12x + 36) Step 2: Write the expression for the perimeter by summing the lengths of all three sides. Perimeter = x + (x+6) + sqrt(2x^2 + 12x + 36) Step 3: Simplify the expression for the perimeter. Perimeter = 2x + 6 + sqrt(2x^2 + 12x + 36) The algebraic expression for the perimeter is 2x + 6 + sqrt(2x)^2 + 12x + 36 meters. b) If the shortest side is 9 m, what is the perimeter of the land? Step 1: Identify the shortest side. The sides are x, x+6, and sqrt(2x^2 + 12x + 36). Since x is a length, x > 0. Comparing x and x+6, x is clearly shorter. The hypotenuse is always the longest side in a right triangle. So, the shortest side is x. Step 2: Set the shortest side equal to 9 m. x = 9 Step 3: Substitute x=9 into the perimeter expression from part (a). Perimeter = 2(9) + 6 + sqrt(2(9)^2 + 12(9) + 36) Perimeter = 18 + 6 + sqrt(2(81) + 108 + 36) Perimeter = 24 + sqrt(162 + 108 + 36) Perimeter = 24 + sqrt(306) Step 4: Calculate the numerical value of the perimeter. sqrt(306) ≈ 17.4928 Perimeter ≈ 24 + 17.4928 Perimeter ≈ 41.4928 The perimeter of the land is approximately 41.49 meters. That's 2 down. 3 left today — send the next one.