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. 2. Patrick's and James's total mass Step 1: Determine James's mass. Patrick is x kg. He is 5 kg heavier than James, so James is 5 kg lighter than Patrick. James's mass = x - 5 kg. Step 2: Write an expression for their total mass. Total mass = Patrick's mass + James's mass x + (x - 5) Step 3: Simplify the expression. x + x - 5 2x - 5 The total mass is (2x - 5) kg. 3. Brother's age in 5 years Step 1: Determine the brother's current age. Maurine is y years old. She is 6 years younger than her brother, so her brother is 6 years older than Maurine. Brother's current age = y + 6 years. Step 2: Determine the brother's age in 5 years. Brother's age in 5 years = (y + 6) + 5 y + 6 + 5 y + 11 The brother's age in 5 years will be (y + 11) years. 4. Perimeter of the rectangular paper cutout Step 1: Write the formula for the perimeter of a rectangle. The perimeter P of a rectangle with length L and width W is given by: P = 2(L + W) Step 2: Substitute the given length and width into the formula. Given length L = (3y + 6) cm and width W = (2y - 4) cm. P = 2((3y + 6) + (2y - 4)) Step 3: Simplify the expression inside the parentheses. P = 2(3y + 2y + 6 - 4) P = 2(5y + 2) Step 4: Distribute the 2. P = 2 × 5y + 2 × 2 P = 10y + 4 The simplified expression for the perimeter is (10y + 4) cm. 5. Achieng's money a) Write an expression to represent the total amount of money spent by Achieng. Step 1: Identify the amounts spent on each item. Soda: sh. P Milk: sh. 2r Cake: sh. 45 Step 2: Sum the amounts spent. Total money spent = P + 2r + 45 The total money spent is (P + 2r + 45) sh.. b) Write an expression to show the amount that Achieng remained with. Step 1: Identify the initial amount and the total spent. Initial amount: sh. 500 Total spent: (P + 2r + 45) sh. Step 2: Subtract the total spent from the initial amount. Amount remained = 500 - (P + 2r + 45) Step 3: Distribute the negative sign. 500 - P - 2r - 45 Step 4: Combine the constant terms. 455 - P - 2r The amount Achieng remained with is (455 - P - 2r) sh.. 6. Wafula's milk a) Write an expression to show the amount of milk he gets in one day. Step 1: Identify the amount of milk in the morning and afternoon. Morning milk: q litres Afternoon milk: q + 4 litres Step 2: Sum the morning and afternoon milk. Total daily milk = q + (q + 4) q + q + 4 2q + 4 The amount of milk he gets in one day is (2q + 4) litres. b) Write an expression to represent the total amount of milk per week. Step 1: Use the total daily milk from part (a). Total daily milk = (2q + 4) litres Step 2: Multiply the daily amount by 7 (for 7 days in a week). Total weekly milk = 7 × (2q + 4) Step 3: Distribute the 7. 7 × 2q + 7 × 4 14q + 28 The total amount of milk per week is (14q + 28) litres. 7. Simplify the following algebraic expressions: a) 12x + 4y + 7 + 8x - 2y Step 1: Group like terms together. (12x + 8x) + (4y - 2y) + 7 Step 2: Combine the like terms. 20x + 2y + 7 The simplified expression is 20x + 2y + 7. b) 5(4p + 4q) + 2(3p - 8q) Step 1: Distribute the numbers outside the parentheses. 5 × 4p + 5 × 4q + 2 × 3p + 2 × (-8q) 20p + 20q + 6p - 16q Step 2: Group like terms together. (20p + 6p) + (20q - 16q) Step 3: Combine the like terms. 26p + 4q The simplified expression is 26p + 4q. That's 2 down. 3 left today — send the next one.