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. FRIDAY 1. In a photo studio, 200 people were employed. (1)/(4) of them were men, (1)/(2) of the remainder were women and the rest were children. How many children were employed? Step 1: Calculate the number of men. Number of men = (1)/(4) × 200 = 50 men. Step 2: Calculate the remainder after men. Remainder = 200 - 50 = 150 people. Step 3: Calculate the number of women. Number of women = (1)/(2) × 150 = 75 women. Step 4: Calculate the number of children. Number of children = Remainder - Number of women Number of children = 150 - 75 = 75 children. There were 75 children employed. 2. There were 280 people attending a conference. The ratio of men and women was 3:4. Later, 20 men and 40 women left the conference. What was the ratio of men to women who remained? Step 1: Calculate the initial number of men and women. Total ratio parts = 3 + 4 = 7. Value of one ratio part = (280)/(7) = 40. Initial number of men = 3 × 40 = 120 men. Initial number of women = 4 × 40 = 160 women. Step 2: Calculate the number of men and women who remained. Men remaining = 120 - 20 = 100 men. Women remaining = 160 - 40 = 120 women. Step 3: Express the new ratio of men to women and simplify it. New ratio = 100 : 120. Divide both sides by their greatest common divisor, which is 20. (100)/(20) : (120)/(20) = 5 : 6 The ratio of men to women who remained was 5:6. 3. Simplify the following ratios: a) 27:9 Step 1: Find the greatest common divisor (GCD) of 27 and 9. The GCD is 9. Step 2: Divide both parts of the ratio by 9. (27)/(9) : (9)/(9) = 3 : 1 The simplified ratio is 3:1. b) 81:42 Step 1: Find the greatest common divisor (GCD) of 81 and 42. Factors of 81: 1, 3, 9, 27, 81 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 The GCD is 3. Step 2: Divide both parts of the ratio by 3. (81)/(3) : (42)/(3) = 27 : 14 The simplified ratio is 27:14. c) 90:45 Step 1: Find the greatest common divisor (GCD) of 90 and 45. The GCD is 45. Step 2: Divide both parts of the ratio by 45. (90)/(45) : (45)/(45) = 2 : 1 The simplified ratio is 2:1. 4. A fruit seller has 10 apples, 15 bananas and 16 mangoes. Express this as a ratio. Step 1: Write the ratio of apples to bananas to mangoes. Ratio = Apples : Bananas : Mangoes Ratio = 10 : 15 : 16 Step 2: Find the greatest common divisor (GCD) of 10, 15, and 16. Factors of 10: 1, 2, 5, 10 Factors of 15: 1, 3, 5, 15 Factors of 16: 1, 2, 4, 8, 16 The only common factor for all three numbers is 1. Therefore, the ratio cannot be simplified further. The student's simplification "2:3:16" is incorrect because all parts of a ratio must be divided by the same common factor. In this case, 10 and 15 were divided by 5, but 16 was not. The ratio is 10:15:16. That's 2 down. 3 left today — send the next one.