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.
ScanSolve AI Answer
Here are the solutions to all the questions on this paper: 1. a) i. Simplify: (6)/(3) (5 - (1)/(2) - (1)/(4)) Step 1: Simplify the fraction outside the parenthesis. (6)/(3) = 2 Step 2: Find a common denominator for the terms inside the parenthesis. The common denominator for 1 (from 5 = (5)/(1)), 2, and 4 is 4. 5 - (1)/(2) - (1)/(4) = (5 × 4)/(4) - (1 × 2)/(4) - (1)/(4) = (20)/(4) - (2)/(4) - (1)/(4) Step 3: Combine the fractions inside the parenthesis. (20 - 2 - 1)/(4) = (17)/(4) Step 4: Multiply the result by the simplified fraction from Step 1. 2 × (17)/(4) = (34)/(4) Step 5: Simplify the final fraction. (34)/(4) = (17)/(2) = 8(1)/(2) The simplified expression is (17)/(2) or 8(1)/(2). 1. a) ii. Evaluate: 3^2 ÷ 9 + 40 - 2^3 × 3^2 ÷ 9 Step 1: Evaluate the powers. 3^2 = 9 2^3 = 8 3^2 = 9 Step 2: Substitute the powers back into the expression. 9 ÷ 9 + 40 - 8 × 9 ÷ 9 Step 3: Perform divisions and multiplications from left to right. 9 ÷ 9 = 1 8 × 9 = 72 72 ÷ 9 = 8 Step 4: Substitute these results back into the expression. 1 + 40 - 8 Step 5: Perform additions and subtractions from left to right. 1 + 40 = 41 41 - 8 = 33 The evaluated expression is 33. 1. b) A financial analyst calculates a company's revenue as GHC23,365,752.198. How should this value be rounded for presentation in a financial report with: i. four significant figures. ii. two decimal places. i. Rounding to four significant figures: The first four significant figures are 2, 3, 3, 6. The fifth digit is 5, so we round up the fourth digit. 23,365,752.198 ≈ 23,370,000 The revenue rounded to four significant figures is GHC23,370,000. ii. Rounding to two decimal places: The first two decimal places are .19. The third decimal place is 8, so we round up the second decimal place. 23,365,752.198 ≈ 23,365,752.20 The revenue rounded to two decimal places is GHC23,365,752.20. 2. a) i. Akua has enough money to buy exercise books at GHC4.50 each. How many pens costing GHC6.00 each can be bought with the same amount of money? Assumption: The question implies Akua has exactly enough money to buy one exercise book. Step 1: Determine the amount of money Akua has. Amount of money = GHC4.50. Step 2: Calculate how many pens can be bought with this amount. Number of pens = Total moneyCost per pen = GHC4.50GHC6.00 (4.50)/(6.00) = (450)/(600) = (3)/(4) = 0.75 Since you cannot buy a fraction of a pen, Akua can buy 0 pens. 2. a) ii. On Adwoa's birthday, the father bought her a pack of chocolate containing 250 bars. If Adwoa took 90 bars of the chocolates and gave the rest to her four friends to share equally, how many bars of chocolates did each receive? Step 1: Calculate the number of bars remaining after Adwoa took her share. Remaining bars = Total bars - Adwoa's share = 250 - 90 = 160 bars Step 2: Divide the remaining bars equally among her four friends. Bars per friend = Remaining barsNumber of friends = (160)/(4) = 40 bars Each friend received 40 bars. 2. b) Use the expanded and the box method to solve: 628 × 39 Expanded Method: Step 1: Break down the numbers into their place values. 628 = 600 + 20 + 8 39 = 30 + 9 Step 2: Multiply each part of the first number by each part of the second number. 600 × 30 = 18000 600 × 9 = 5400 20 × 30 = 600 20 × 9 = 180 8 × 30 = 240 8 × 9 = 72 Step 3: Add all the partial products. 18000 + 5400 + 600 + 180 + 240 + 72 = 24492 The product using the expanded method is 24492. Box Method (Grid Method): Step 1: Draw a grid and write the expanded forms of the numbers along the top and side. | × | 600 | 20 | 8 | |---|---|---|---| | 30 | 30 × 600 = 18000 | 30 × 20 = 600 | 30 × 8 = 240 | | 9 | 9 × 600 = 5400 | 9 × 20 = 180 | 9 × 8 = 72 | Step 2: Add all the products inside the boxes. 18000 + 600 + 240 + 5400 + 180 + 72 = 24492 The product using the box method is 24492. 2. c) Simplify: (2x - 1)/(3) - (x + 3)/(2) Step 1: Find a common denominator for the fractions. The least common multiple of 3 and 2 is 6. (2(2x - 1))/(6) - (3(x + 3))/(6) Step 2: Distribute the numbers in the numerators. (4x - 2)/(6) - (3x + 9)/(6) Step 3: Combine the numerators over the common denominator. Be careful with the subtraction sign. ((4x - 2) - (3x + 9))/(6) (4x - 2 - 3x - 9)/(6) Step 4: Combine like terms in the numerator. ((4x - 3x) + (-2 - 9))/(6) (x - 11)/(6) The simplified expression is (x - 11)/(6). 3. a) i. Using a pair of compass and a ruler only, construct triangle ABC such that /AB/ = 8cm, angle BAC = 60° and angle ABC = 45°. ii. Measure /BC/ i. Construction Steps for Triangle ABC: 1. Draw a straight line and mark a point A. 2. Using a ruler, measure 8 cm from A along the line and mark point B. This forms line segment AB = 8 cm. 3. At point A, construct a 60° angle. Place the compass at A, draw an arc intersecting AB. Without changing the compass width, place the compass at the intersection point on AB and draw another arc to intersect the first arc. Draw a line from A through this intersection. This line forms BAC = 60^. 4. At point B, construct a 45° angle. First, construct a 90° angle at B. Then, bisect this 90° angle to get 45°. Draw a line from B along this 45° angle. This line forms ABC = 45^. 5. Extend the line from A (for 60°) and the line from B (for 45°) until they intersect. Label this intersection point C. Triangle ABC is now constructed. ii. Measure /BC/: Upon accurate construction and measurement, the length of /BC/ should be approximately 7.2 cm. 3. b) Solve: 5 - 2x > x + 2 Step 1: Gather all terms with x on one side and constant terms on the other side. Subtract x from both sides: 5 - 2x - x > x + 2 - x 5 - 3x > 2 Step 2: Subtract 5 from both sides. 5 - 3x - 5 > 2 - 5 -3x > -3 Step 3: Divide both sides by -3. Remember to reverse the inequality sign when dividing by a negative number. (-3x)/(-3) < (-3)/(-3) x < 1 The solution is x < 1. 3. c) The diameter of a circular lid is 14cm. Find i. the circumference of the lid ii. the area of the lid (take = (22)/(7)) Given: Diameter d = 14 cm. Radius r = (d)/(2) = (14)/(2) = 7 cm. = (22)/(7). i. Circumference of the lid: The formula for circumference is C = d. C = (22)/(7) × 14 cm C = 22 × 2 cm C = 44 cm The circumference of the lid is 44 cm. ii. Area of the lid: The formula for the area of a circle is A = r^2. A = (22)/(7) × (7 cm)^2 A = (22)/(7) × 49 cm^2 A = 22 × (49)/(7) cm^2 A = 22 × 7 cm^2 A = 154 cm^2 The area of the lid is 154 cm^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.