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Calculate the percentage of the $360 that she spent on presents for her sisters.

Mathematics

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2 parts
i

\7,680$

ii

\34,240$

\5,306.04$

26.7%26.7\%

Step-by-step solution

11 steps

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}O = \{1, 3, 5, 7, 9, 11, 13, 15\}
F={1,2,3,4,6,12}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}\{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\boxed{5}.

17. (b) (iii) Find n(F∪O)n(F \cup O).

Step 1: Determine the union of set F and set O.
The union F∪OF \cup 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}F \cup O = \{1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15\}

Step 2: Count the number of elements in F∪OF \cup O.
n(F∪O)n(F \cup O) is the number of elements in the union.
Counting the elements: 1,2,3,4,5,6,7,9,11,12,13,151, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15 are 12 elements.
n(F∪O)=12n(F \cup O) = \boxed{12}.

18. (a) In 2018, Gretal earned $32 000.

Part i· 1 step

She paid tax of 24% on these earnings. Work out the amount she paid in tax in 2018.

  1. Calculate 24% of her earnings.

    Tax amount = 24%24\% of \32,000Taxamount= Tax amount =0.24 \times $32,000Taxamount= Tax amount =$7,680Theamountshepaidintaxin2018is The amount she paid in tax in 2018 is\boxed{$7,680}$.

Part ii

\34,240$

In 2019, Gretal's earnings increased by 7%. Work out her earnings in 2019.

  1. Calculate the increase in earnings.

    Increase = 7%7\% of \32,000Increase= Increase =0.07 \times $32,000Increase= Increase =$2,240$

  2. Add the increase to her 2018 earnings.

    Earnings in 2019 = Earnings in 2018 + Increase
    Earnings in 2019 = \32,000 + $2,240Earningsin2019= Earnings in 2019 =$34,240Herearningsin2019are Her earnings in 2019 are\boxed{$34,240}$.

    18. (b) Gretal invests $5 000 at a rate of 2% per year compound interest. Calculate the value of her investment after 3 years.

  3. Use the compound interest formula.

    The formula for compound interest is A=P(1+r)tA = P(1 + r)^t, where:
    AA = final amount
    PP = principal amount (5 000)5\,000) r=annualinterestrate(2= annual interest rate (2% or 0.02) t$ = number of years (3)

  4. Substitute the values into the formula and calculate.

    A=$5 000(1+0.02)3A = \$5\,000(1 + 0.02)^3 A=$5 000(1.02)3A = \$5\,000(1.02)^3 A=$5 000×1.061208A = \$5\,000 \times 1.061208 A=$5 306.04A = \$5\,306.04 The value of her investment after 3 years is \boxed{\5,306.04}$.

    18. (c) One month, Gretal spent a total of 360onpresents.∗∗∗∗Shespent360 on presents.** **She spent \frac{1}{5}ofthistotalonpresentsforherparents.∗∗∗∗Shespentof this total on presents for her parents.** **She spent\frac{2}{3}oftheremainingmoneyonpresentsforherfriends.∗∗∗∗Shespenttherestofthemoneyonpresentsforhersisters.∗∗∗∗Calculatethepercentageoftheof the remaining money on presents for her friends.** **She spent the rest of the money on presents for her sisters.** **Calculate the percentage of the360 that she spent on presents for her sisters.

  5. Calculate the amount spent on parents.

    Amount for parents = \frac{1}{5} \times \360 = $72$.

  6. Calculate the remaining money after buying presents for parents.

    Remaining money = \360 - $72 = $288$.

  7. Calculate the amount spent on friends.

    Amount for friends = \frac{2}{3} \times \288 = $192$.

  8. Calculate the amount spent on sisters.

    Amount for sisters = Remaining money - Amount for friends
    Amount for sisters = \288 - $192 = $96$.

  9. Calculate the percentage of the total spent on sisters.

    Percentage for sisters = Amount for sistersTotal spent×100%\frac{\text{Amount for sisters}}{\text{Total spent}} \times 100\%
    Percentage for sisters = \frac{\96}{$360} \times 100%Percentageforsisters= Percentage for sisters =0.2666... \times 100%Percentageforsisters= Percentage for sisters =26.66...%$

  10. Round the percentage to one decimal place.

    Percentage for sisters ≈26.7%\approx 26.7\%
    The percentage of the 360thatshespentonpresentsforhersistersis360 that she spent on presents for her sisters is \boxed{26.7%}$.

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