Here are now 45 minutes of exact concentration that we expect you to be able to exercise. You are in Year 9 now, ideally 15 years old. You should feel accountable and comfortable to think in silence for forty-five consequtive minutes. Goodluck.

\begin{theorembox}
  \textbf{Important:} You should really take this opportunity to become a master of your calculator. You have an expert at the front of the room. Use them!

\end{theorembox}

\begin{problembox}
  \begin{questions}
    \question[2] Sarah decides to spend \(40 \%\) of her weekly earnings on social activities, give \(15 \%\) to her mother to repay a loan, and save the rest. She earns \(\$ 84\) a week.
    \begin{parts}
      \part How much does Sarah spend each week on social activities?
          \begin{solutionordottedlines}[1in]
            \( \$ 84 \times 40 \% = \$ 33.60 \)
          \end{solutionordottedlines}
      \part What percentage of her weekly earnings does she save?
          \begin{solutionordottedlines}[1in]
            \( 100 \% - 40 \% - 15 \% = 45 \% \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] Nick's share portfolio consists of shares, with value \(\$ 10000\), in the banking industry, \(\$ 3000\) in mining shares and \(\$ 15000\) in the gold market.
    \begin{parts}
      \part What percentage of his share portfolio is made up of shares in the mining sector?
          \begin{solutionordottedlines}[1in]
            Total value = \( \$ 10000 + \$ 3000 + \$ 15000 = \$ 28000 \)
            Percentage in mining = \( \frac{\$ 3000}{\$ 28000} \times 100 \% = 10.71 \% \)
          \end{solutionordottedlines}
      \part What percentage of Nick's share portfolio are not banking shares?
          \begin{solutionordottedlines}[1in]
            Not banking shares = \( \$ 3000 + \$ 15000 = \$ 18000 \)
            Percentage not banking = \( \frac{\$ 18000}{\$ 28000} \times 100 \% = 64.29 \% \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] A real estate agent charges a commission of \(8.9 \%\) on every property sale.
    \begin{parts}
      \part If a house sells for \(\$ 540000\), how much commission will the real estate agent receive, and how much is left for the seller?
          \begin{solutionordottedlines}[1in]
            Commission = \( \$ 540000 \times 8.9 \% = \$ 48060 \)
            Seller receives = \( \$ 540000 - \$ 48060 = \$ 491940 \)
          \end{solutionordottedlines}
      \part If the real estate agent receives a commission of \(\$ 8455\) for selling a house, what was the selling price of the house, and what did the seller actually receive?
          \begin{solutionordottedlines}[1in]
            Selling price = \( \frac{\$ 8455}{8.9 \%} = \$ 95000 \)
            Seller receives = \( \$ 95000 - \$ 8455 = \$ 86545 \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] It cost the owners of the Corner Newsagency \(\$ 3500000\) to run their business last year. They recorded a profit of \(4.5 \%\).
    \begin{parts}
      \part What was their profit last year?
          \begin{solutionordottedlines}[1in]
            Profit = \( \$ 3500000 \times 4.5 \% = \$ 157500 \)
          \end{solutionordottedlines}
      \part What was the total of their sales?
          \begin{solutionordottedlines}[1in]
            Sales = \( \$ 3500000 + \$ 157500 = \$ 3657500 \)
          \end{solutionordottedlines}
      \part In the previous year, their costs were \(\$ 2750000\) and their sales were only \(\$ 2635000\). What percentage loss did they make on their costs?
          \begin{solutionordottedlines}[1in]
            Loss = \( \$ 2750000 - \$ 2635000 = \$ 115000 \)
            Percentage loss = \( \frac{\$ 115000}{\$ 2750000} \times 100 \% = 4.18 \% \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] Grant earned \$1260 interest on money he had invested four years ago at a simple interest rate of \(4.5 \%\) p.a. How much did Grant originally invest?
          \begin{solutionordottedlines}[1in]
            Principal = \( \frac{\$ 1260}{4 \times 4.5 \%} = \$ 7000 \)
          \end{solutionordottedlines}
    \question[2] A country is experiencing inflation of \(12 \%\) p.a.
    \begin{parts}
      \part If the price of bread is adjusted in line with inflation, what will an annual bread bill of \(\$ 2500\) become in the next year?
          \begin{solutionordottedlines}[1in]
            New bread bill = \( \$ 2500 \times (1 + 12 \%) = \$ 2800 \)
          \end{solutionordottedlines}
      \part If Janienne earns \(\$ 72000\) in one year and \(\$ 78000\) the next, is her salary increase keeping pace with inflation?
          \begin{solutionordottedlines}[1in]
            Salary increase = \( \$ 78000 - \$ 72000 = \$ 6000 \)
            Percentage increase = \( \frac{\$ 6000}{\$ 72000} \times 100 \% = 8.33 \% \)
            No, the salary increase is not keeping pace with inflation.
          \end{solutionordottedlines}
    \end{parts}
    \question[2] A regional country medical centre has lost the services of one of its 16 doctors due to retirement, and has been unable to replace her.
    \begin{parts}
      \part What percentage loss is represented by the retiring doctor?
          \begin{solutionordottedlines}[1in]
            Percentage loss = \( \frac{1}{16} \times 100 \% = 6.25 \% \)
          \end{solutionordottedlines}
      \part If the medical centre treated 400 patients per day last year and need to reduce this number by the percentage found in part a, how many patients per day will they be able to treat in the coming year?
          \begin{solutionordottedlines}[1in]
            Reduced number of patients = \( 400 \times (1 - 6.25 \%) = 375 \) patients per day
          \end{solutionordottedlines}
    \end{parts}
    \question[2] A shop made a profit of \(6.2 \%\) on total costs last year. If the actual profit was \(\$ 156000\), what were the total costs and what were the total sales?
          \begin{solutionordottedlines}[1in]
            Total costs = \( \frac{\$ 156000}{6.2 \%} = \$ 2516129.03 \)
            Total sales = \( \$ 2516129.03 + \$ 156000 = \$ 2672129.03 \)
          \end{solutionordottedlines}
    \question[2] In the January sales, the Best Dress shop has discounted all its prices by \(18 \%\).
    \begin{parts}
      \part What is the discounted price of a dress with a marked price of \(\$ 240\) ?
          \begin{solutionordottedlines}[1in]
            Discounted price = \( \$ 240 \times (1 - 18 \%) = \$ 196.80 \)
          \end{solutionordottedlines}
      \part What was the original price of a dress with a discounted price of \(\$ 49.20\) ?
          \begin{solutionordottedlines}[1in]
            Original price = \( \frac{\$ 49.20}{1 - 18 \%} = \$ 60 \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] The number of books in a local library varies from year to year. Three years ago, the number fell by \(25 \%\), then it rose \(41 \%\) the following year, and finally rose \(8 \%\) last year.
    \begin{parts}
      \part What is the percentage increase or decrease over the three years, correct to the nearest \(1 \%\) ?
          \begin{solutionordottedlines}[1in]
            Overall change = \( (1 - 25 \%) \times (1 + 41 \%) \times (1 + 8 \%) - 1 \)
            Overall change = \( 0.75 \times 1.41 \times 1.08 - 1 \approx 0.1419 \)
            Percentage change = \( 14.19 \% \) increase
          \end{solutionordottedlines}
      \part If there were 429000 books in the library three years ago, approximately how many books are in the library now?
          \begin{solutionordottedlines}[1in]
            Current number of books = \( 429000 \times 1.1419 \approx 489600 \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] The original asking price for a farm dropped by \(30 \%\) a year ago, but did not attract a buyer. The price has now been further reduced by \(15 \%\).
    \begin{parts}
      \part By what percentage has the original asking price been reduced?
          \begin{solutionordottedlines}[1in]
            Overall reduction = \( (1 - 30 \%) \times (1 - 15 \%) - 1 \)
            Overall reduction = \( 0.70 \times 0.85 - 1 \approx -0.405 \)
            Percentage reduction = \( 40.5 \% \)
          \end{solutionordottedlines}
      \part If the farm is now for sale at \(\$ 2677500\), what was the original asking price of the farm?
          \begin{solutionordottedlines}[1in]
            Original asking price = \( \frac{\$ 2677500}{1 - 40.5 \%} \approx \$ 4500000 \)
          \end{solutionordottedlines}
    \end{parts}
    \question[2] The height of a mature tree is measured on the same day each year. Each year for the last six years, the growth has been \(9 \%\) less than the previous year's growth.
    \begin{parts}
      \part What is the percentage decrease in growth over the six years, correct to the nearest percent?
          \begin{solutionordottedlines}[1in]
            Decrease in growth = \( 1 - (1 - 9 \%)^6 \approx 0.4353 \)
            Percentage decrease = \( 43.53 \% \) (rounded to the nearest percent: \( 44 \% \))
          \end{solutionordottedlines}
      \part If the growth this year was \(320 \mathrm{~mm}\), what was the growth six years ago, correct to the nearest millimetre?
          \begin{solutionordottedlines}[1in]
            Initial growth = \( \frac{320 \mathrm{~mm}}{(1 - 9 \%)^6} \approx 563 \mathrm{~mm} \)
          \end{solutionordottedlines}
    \end{parts}
    \question[4] Sam's investment of \(\$ 50000\) for five years earns her interest at the rate of \(6.3 \%\) p.a., compounded annually.
    \begin{parts}
      \part How much will the investment be worth at the end of six years?
          \begin{solutionordottedlines}[1in]
            Future value = \( \$ 50000 \times (1 + 6.3 \%)^6 \approx \$ 70968.77 \)
          \end{solutionordottedlines}
      \part What is the percentage increase of her original investment at the end of six years?
          \begin{solutionordottedlines}[1in]
            Percentage increase = \( \frac{\$ 70968.77 - \$ 50000}{\$ 50000} \times 100 \% \approx 41.94 \% \)
          \end{solutionordottedlines}
      \part What is the total interest earned over the six years?
          \begin{solutionordottedlines}[1in]
            Total interest = \( \$ 70968.77 - \$ 50000 = \$ 20968.77 \)
          \end{solutionordottedlines}
      \part What would the simple interest on the investment have been, assuming the same interest rate of \(6.3 \%\) p.a.?
          \begin{solutionordottedlines}[1in]
            Simple interest = \( \$ 50000 \times 6.3 \% \times 6 = \$ 18900 \)
          \end{solutionordottedlines}
    \end{parts}
    \question[4] A company buys new company cars every three years. At the end of the three years, it offers them for sale to the employees on the assumption that they have depreciated at \(30 \%\) p.a. The company is presently advertising some cars at \(\$ 30000\) each.
    \begin{parts}
      \part What did each car cost the company originally, correct to the nearest thousand dollars?
          \begin{solutionordottedlines}[1in]
            Original cost = \( \frac{\$ 30000}{(1 - 30 \%)^3} \approx \$ 78100 \)
          \end{solutionordottedlines}
      \part What is the average depreciation in dollars p.a., correct to the nearest hundred dollars, on each car over the three-year period?
          \begin{solutionordottedlines}[1in]
            Average depreciation p.a. = \( \frac{\$ 78100 - \$ 30000}{3} \approx \$ 16033.33 \)
          \end{solutionordottedlines}
    \end{parts}
    \end{questions}
\end{problembox}
