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01-Preliminaries.Rmd
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# (PART) Essentials {-}
# Preliminaries
## 1.2 Order of Operations {-}
* Simplify $\$250 ( 1 + 0.065 × 172/365)$
* $257.66 [Click here for full solution](https://www.youtube.com/watch?v=bpgDtSvmLt0)
<p> </p>
* Simplify $\$1775 ( 1 + 0.6/2)^3( 1 + 7.84/4)^2$
* $34,167.39 [Click here for full solution](https://www.youtube.com/watch?v=dPtWFsmhBI0)
<p> </p>
* Simplify $\$1500/(1+ 0.08/12)^4$
* $1460.66 [Click here for full solution](https://www.youtube.com/watch?v=nAlCJJ7F3nQ)
<p> </p>
* A hockey team says that its season ticket sales represent 72% of
the stadium seating capacity of 18,500 seats. Rounded to the nearest 100,
how many seats were not sold to season ticket holders?
* 5,200 [Click here for full solution](https://www.youtube.com/watch?v=r-YA7n_3RWY)
<p> </p>
* Billy is a Realtor. He receives 44% of the 7% commission that the real estate
agency charges on sales. If his sales for the past year were $14,500,000, what was the dollar
value of his commission?
* \$446,600 [Click here for full solution](https://www.youtube.com/watch?v=qoo0mt3FgQ8)
<p> </p>
## 1.3 Rational Numbers {-}
Simplify the following:
* $3/4 - x/4 - 2x^2 - 0.5 x^2 − 10/(4x) + 3/2$
* $-2.5x^2 - x/4 - 10/(4x)+ 9/4$ [Click here for full solution](https://www.youtube.com/watch?v=tIk7EkumoAw)
<p> </p>
* $y(1 − 0.08 \times 200/365) + 2y/(1 + 0.08 \times 60/365)$
* $2.9302y$ [Click here for full solution](https://www.youtube.com/watch?v=u3EWjZLyUvU&t=5s)
<p> </p>
* $x/(1 + 0.04/2)^2 − 3x (1 + 0.04/2)^3$
* $-2.2225x$ [Click here for full solution](https://www.youtube.com/watch?v=aev4KtTo99c)
<p> </p>
* $500((1 + 0.04/2)^25 - 1)/(0.04/2)$
* $105,050$ [Click here for full solution](https://www.youtube.com/watch?v=uTtT9hn4rmA&t=2s)
<p> </p>
* $500/(0.06/4) + 500$
* $33,833.33$ [Click here for full solution](https://www.youtube.com/watch?v=FLYaSFsqH5A)
<p> </p>
## 1.4 Expanding and Factoring {-}
Simplify the following:
* $12x^2y + (−5x^2y) − (−7x^2y)$
* $14x^2y$ [Click here for full solution](https://www.youtube.com/watch?v=PMlvcOMKILc)
<p> </p>
* $(8m^3 − 3m − 4m^2 + 14) − (2m^3 − 6 + 8m − 14m^2)$
* $6m^3 + 10m^2 - 11m + 20$ [Click here for full solution](https://www.youtube.com/watch?v=poKHlgg34pU)
<p> </p>
* $10x− [8 − 4(3x − 4)]$
* $22x - 24$ [Click here for full solution](https://www.youtube.com/watch?v=LOTdpEzEVrk)
<p> </p>
* $−2xy(6x^2 − 2xy − 5y^2)$
* $-12x^3y + 4x^2y^2 + 10xy^3$ [Click here for full solution](https://www.youtube.com/watch?v=QCpCmtvZrr8)
<p> </p>
* $(4a^2b^7 - 8a^5b^3)/(2a^2b^2)$
* $2b^5 - 4a^3b$ [Click here for full solution](https://www.youtube.com/watch?v=Zb8b77JlH_o)
<p> </p>
## 1.5 Solving Systems of Equations {-}
Solve the following systems of equations by any method:
* $9x-6y = 18$ and $2y = 2x - 2$
* $x = 4, y = 3$ [Click here for full solution](https://www.youtube.com/watch?v=OOgqVeiXDZQ&t=11s)
<p> </p>
* $2x - 2y = 2$ and $6x + 8y = 40$
* $x = 24/7, y = 17/7$ [Click here for full solution](https://www.youtube.com/watch?v=FXnmALrqdoE&t=6s)
<p> </p>
* $x - y = 20$ and $(4/3)x + (5/2)y = 0$
* $x = 13.04, y = -6.96$ [Click here for full solution](https://www.youtube.com/watch?v=Rrcd9LUWSk0&t=44s)
<p> </p>
* A car rental company charges an hourly rate plus a per kilometre charge. Jack paid \$60 for a two-hour rental during which he drove 50 km. Jules paid \$145 for five hours and 100 km driven. What rate is charged per hour and per kilometre?
* \$25 per hour and \$0.20 per kilometre [Click here for full solution](https://www.youtube.com/watch?v=cGv30YaK8-Y&t=15s)
<p> </p>
* Hank bought 8 steaks and 30 potatoes for \$75. Kevin purchased 11 steaks and 20 potatoes for \$75. What were the prices of the steak and potatoes.
* Steak: \$4.41 and potatoes: \$1.32 [Click here for full solution](https://www.youtube.com/watch?v=tw-oi36K4os)
<p> </p>
* A company pays monthly salaries of \$5500 to each production worker and \$4600 to each assembly worker. At the beginning of COVID, the firm decides to reduce its total monthly manufacturing payroll from \$400,000 to \$347,600 by laying off 20% of its production workers and 12% of its assembly workers. How many layoffs will there be from each of the assembly and production divisions?
* 2 production and 9 assembly [Click here for full solution](https://www.youtube.com/watch?v=X-FTiqnkV0E)
<p> </p>
## 1.6 Ratios, Proportions and Percentages {-}
* Solve the ratio $165 : 245 = 65 : x$
* $x = 96.52$ [Click here for full solution](https://www.youtube.com/watch?v=-iqsQtiylrM&t=6s)
<p> </p>
* Solve the ratio $1500: 2500 : 750 = x : 750 : y$
* $x = 450$ and $y = 225$ [Click here for full solution](https://www.youtube.com/watch?v=rAFOVVD1fYw)
<p> </p>
* Jim, Karen, and Sean started their partnership with a total investment of
\$245,000 contributed in the ratio of 4 : 6 : 3. If each partner contributes another \$30,000, what will be the ratio of their total contributions? Leave your answer with the smallest term equal to 1.
* Approximately $1.22 : 1.65 : 1.00$ [Click here for full solution](https://www.youtube.com/watch?v=OLJEk7oBbO0)
<p> </p>
* The CPI was at 120 when the union signed their contract 5 years ago, the members of a union were earning $45.55 per hour. Now the current CPI at 135 and they want a new hourly rate that will restore their former purchasing power. What hourly rate are they seeking?
* $51.42 [Click here for full solution](https://www.youtube.com/watch?v=Pn6NUm0Falo)
<p> </p>
* I live in a 4-plex. We need a new roof with costs totaling $35,000 and these costs must be divided between owners A, B, C, and D based on their square-footage 1300, 1400, 1750 and 2100 square feet, respectively. How much does each owner pay?
* Owner A: \$6,947.50, owner B: \$7,479.50, owner C: \$9,352.00, and owner D: \$11,221.00. [Click here for full solution](https://www.youtube.com/watch?v=2fgmPIiv-kk&t=34s)
<p> </p>
* Jenny, Kathy, and Fred share the costs of their coffee fund in the ratio $3/2 : 5/4 : 4/3$. How
will costs of $50 be allocated among them?
* Jenny: \$18.37, Kathy: \$15.31, Fred: \$16.33. [Click here for full solution](https://www.youtube.com/watch?v=8rOt7sE9To8&t=24s)
<p> </p>
* Lisa receives a monthly base salary of \$2500. On the first \$15,000 of sales above her monthly quota of \$25,000, she is paid a commission of 6%. On any additional sales, the commission rate is 10%. What were her total earnings for a month in which she had sales amounting to \$45,000?
* \$3,900 [Click here for full solution](https://www.youtube.com/watch?v=Vcs2qqbIzn8)
<p> </p>
* The price tags in Mel’s Flower Shop include the 8.25% tax. How much tax will she report for a plant sold at \$49.55?
* \$3.78 [Click here for full solution](https://www.youtube.com/watch?v=R27fdohO1A0)
<p> </p>
* Okanagan Food-Mart usually spends half as much on radio advertising as on newspaper advertising, and 70% as much on television advertising as on radio advertising. If next year’s total advertising budget is \$225,000, how much should be allocated to each form of advertising?
* Newspaper advertising: \$121,621.62, Radio advertising: \$60,810.81, Television advertising: \$45,567.57 [Click here for full solution](https://www.youtube.com/watch?v=aOCDt4cNzlY&t=29s)
<p> </p>
* The government forecasts expenditure of $2.85 billion on education next year represents 21% of the provincial budget. Rounded to the nearest million dollars, what is the province’s total budget for the next year?
* \$13,571,000,000 [Click here for full solution](https://www.youtube.com/watch?v=OUcXu0UkvRQ)
<p> </p>