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: SECTION E - PROBABILITY 20. A bag contains 5 red balls and 3 blue balls. What is the probability of picking a red ball? Step 1: Calculate the total number of balls. Total balls = 5 (red) + 3 (blue) = 8 Step 2: Determine the number of favorable outcomes (red balls). Red balls = 5 Step 3: Calculate the probability. Probability (Red) = Number of red ballsTotal number of balls = (5)/(8) The answer is (5)/(8). 21. A fair die is rolled. What is the probability of getting an even number? Step 1: List all possible outcomes when rolling a die. Outcomes = \1, 2, 3, 4, 5, 6\ Total outcomes = 6 Step 2: List the even numbers among the outcomes. Even numbers = \2, 4, 6\ Number of even outcomes = 3 Step 3: Calculate the probability. Probability (Even) = Number of even outcomesTotal outcomes = (3)/(6) = (1)/(2) The answer is (1)/(2). 22. A box contains 10 apples, 3 of which are rotten. If an apple is picked at random, what is the probability that it is not rotten? Step 1: Determine the total number of apples. Total apples = 10 Step 2: Determine the number of apples that are not rotten. Not rotten apples = Total apples - Rotten apples = 10 - 3 = 7 Step 3: Calculate the probability. Probability (Not rotten) = Number of not rotten applesTotal apples = (7)/(10) The answer is (7)/(10). 23. A coin is tossed twice. What is the probability of getting two heads? Step 1: List all possible outcomes when tossing a coin twice. Outcomes = \HH, HT, TH, TT\ Total outcomes = 4 Step 2: Identify the outcome with two heads. Two heads = \HH\ Number of two heads outcomes = 1 Step 3: Calculate the probability. Probability (Two heads) = Number of two heads outcomesTotal outcomes = (1)/(4) The answer is (1)/(4). SECTION F - MENSURATION 24. The area of a square is 36 cm^2. Find its perimeter. Step 1: Find the side length (s) of the square using the area formula A = s^2. s^2 = 36 cm^2 s = sqrt(36) cm = 6 cm Step 2: Calculate the perimeter (P) of the square using the formula P = 4s. P = 4 × 6 cm = 24 cm The answer is 24 cm. 25. The length of a rectangle is 10 cm and its width is 5 cm. Find its area. Step 1: Use the formula for the area of a rectangle, A = length × width. A = 10 cm × 5 cm = 50 cm^2 The answer is 50 cm^2. 26. Find the circumference of a circle with radius 7 cm. (Use = (22)/(7)) Step 1: Use the formula for the circumference of a circle, C = 2 r. C = 2 × (22)/(7) × 7 cm Step 2: Simplify the expression. C = 2 × 22 cm = 44 cm The answer is 44 cm. SECTION G - WORD PROBLEMS 27. A car travels 180 km in 3 hours. What is its average speed? Step 1: Use the formula for average speed, Speed = DistanceTime. Speed = 180 km3 hours Step 2: Calculate the speed. Speed = 60 km/h The answer is 60 km/h. 28. A sum of money is shared between John and Mary in the ratio 2:3. If John gets 40, what is Mary's share? Step 1: Determine the value of one ratio part from John's share. John's share = 2 parts = \40 1 part = (\40)/(2) = \20 Step 2: Calculate Mary's share based on her ratio. Mary's share = 3 parts = 3 × \20 = \60 The answer is \60. 29. If 15% of a number is 30, what is the number? Step 1: Let the number be x. Set up the equation. 0.15x = 30 Step 2: Solve for x by dividing both sides by 0.15. x = (30)/(0.15) x = (30)/(15)100 = (30 × 100)/(15) x = 2 × 100 = 200 The answer is 200. 30. Find the area of a circle with radius 7 cm. (Use = (22)/(7)) Step 1: Use the formula for the area of a circle, A = r^2. A = (22)/(7) × (7 cm)^2 Step 2: Calculate the area. A = (22)/(7) × 49 cm^2 = 22 × 7 cm^2 = 154 cm^2 The answer is 154 cm^2. 31. A ship sails on a bearing of 045^. What bearing should it sail on to return directly to its starting point? Step 1: To return to the starting point, the ship must sail on the back bearing. Step 2: Since the initial bearing (045^) is less than 180^, add 180^ to find the back bearing. Back bearing = 045^ + 180^ = 225^ The answer is 225^. 32. In a right-angled triangle, the opposite side is 3 and the hypotenuse is 5. Find . Step 1: Recall the definition of sine in a right-angled triangle. = OppositeHypotenuse Step 2: Substitute the given values. = (3)/(5) The answer is (3)/(5)$. That's 2 down. 3 left today — send the next one.