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 from the image: 17. (b) (ii) Write down one number that is in set O, but not in set F. Step 1: Recall the elements of set O and set F. From the previous step, we have: O = \1, 3, 5, 7, 9, 11, 13, 15\ F = \1, 2, 3, 4, 6, 12\ Step 2: Identify an element in O that is not in F. Any number from the set \5, 7, 9, 11, 13, 15\ is in O but not in F. For example, the number 5. One number that is in set O, but not in set F, is 5. 17. (b) (iii) Find n(F O). Step 1: Determine the union of set F and set O. The union F O includes all unique elements from both sets F and O. F O = \1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15\ Step 2: Count the number of elements in F O. n(F O) is the number of elements in the union. Counting the elements: 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15 are 12 elements. n(F O) = 12. 18. (a) In 2018, Gretal earned 32 000. (i) She paid tax of 24% on these earnings. Work out the amount she paid in tax in 2018. Step 1: Calculate 24% of her earnings. Tax amount = 24\% of \32\,000 Tax amount = 0.24 × \32\,000 Tax amount = \7\,680 The amount she paid in tax in 2018 is \7\,680. (ii) In 2019, Gretal's earnings increased by 7%. Work out her earnings in 2019. Step 1: Calculate the increase in earnings. Increase = 7\% of \32\,000 Increase = 0.07 × \32\,000 Increase = \2\,240 Step 2: Add the increase to her 2018 earnings. Earnings in 2019 = Earnings in 2018 + Increase Earnings in 2019 = \32\,000 + \2\,240 Earnings in 2019 = \34\,240 Her earnings in 2019 are \34\,240. 18. (b) Gretal invests 5 000 at a rate of 2% per year compound interest. Calculate the value of her investment after 3 years. Step 1: Use the compound interest formula. The formula for compound interest is A = P(1 + r)^t, where: A = final amount P = principal amount (5\,000) r = annual interest rate (2% or 0.02) t = number of years (3) Step 2: Substitute the values into the formula and calculate. A = \5\,000(1 + 0.02)^3 A = \5\,000(1.02)^3 A = \5\,000 × 1.061208 A = \5\,306.04 The value of her investment after 3 years is \5\,306.04. 18. (c) One month, Gretal spent a total of 360 on presents. She spent (1)/(5) of this total on presents for her parents. She spent (2)/(3) of the remaining money on presents for her friends. She spent the rest of the money on presents for her sisters. Calculate the percentage of the 360 that she spent on presents for her sisters. Step 1: Calculate the amount spent on parents. Amount for parents = (1)/(5) × \360 = \72. Step 2: Calculate the remaining money after buying presents for parents. Remaining money = \360 - \72 = \288. Step 3: Calculate the amount spent on friends. Amount for friends = (2)/(3) × \288 = \192. Step 4: Calculate the amount spent on sisters. Amount for sisters = Remaining money - Amount for friends Amount for sisters = \288 - \192 = \96. Step 5: Calculate the percentage of the total spent on sisters. Percentage for sisters = Amount for sistersTotal spent × 100\% Percentage for sisters = (\96)/(\360) × 100\% Percentage for sisters = 0.2666... × 100\% Percentage for sisters = 26.66...\% Step 6: Round the percentage to one decimal place. Percentage for sisters ≈ 26.7\% The percentage of the 360 that she spent on presents for her sisters is 26.7\%. Drop the next question.