Applications and Abuses of Percents

$ Tip = 12\% \:\:of\:\:\$34.99 \\[3ex] = \dfrac{12}{100} * 34.99 \\[5ex] = 4.1988 \approx 4.20 \\[3ex] $ The tip is $\$4.20$

$ Sales\:\: tax\:\: = 7\% \:\:of\:\: 220.98 \\[3ex] = \dfrac{7}{100} * 220.98 \\[5ex] = 15.4686 \approx 15.47 \\[3ex] $ Sales tax = $\$15.47$

You may begin with a preview to ensure students understand it.
Preview is in black color.
Assume the cost of a phone is $\$200.00$
Include a $5\%$ sales tax
How much is the sales tax?

$ 5\% = \dfrac{5}{100} = 0.05 \\[5ex] 5\%\:\:sales\:\:tax = 0.05(200) = \$10 \\[3ex] Checkout\:\:price = \$200 + \$10 = \$210 \\[3ex] $ Ask students this question:
Given: $\$210$, with $5\%$ sales tax (included), how can we get $\$10$?
In other words, Given the checkout price which includes the sales tax, how do we figure out the sales tax?
Introduce and discuss the concept of Algebra (using variables).
First, we need to find the cost of the phone without the sales tax.
This is also known as the initial cost of the phone.

$ Let\:\:the\:\:initial\:\:cost\:\:of\:\:the\:\:phone = x \\[3ex] 5\%\:\:sales\:\:tax = 0.05(x) = 0.05x \\[3ex] Initial\:\:cost + Sales\:\:tax = Checkout\:\:price \\[3ex] x + 0.05x = 210 \\[3ex] 1.05x = 210 \\[3ex] x = \dfrac{210}{1.05} \\[5ex] x = 200 \\[3ex] Sales\:\:tax = 0.05x = 0.05(200) = \$10 \\[3ex] OR \\[3ex] Sales\:\:tax = Checkout\:\:price - Initial\:\:cost \\[3ex] Sales\:\:tax = 210 - 200 = \$10 $

Let the initial cost of the phone = $p$

$ 7\%\:\: sales\:\: tax\:\: of\:\: p = 7\% * p \\[3ex] = \dfrac{7}{100} * p \\[5ex] = 0.07 * p \\[3ex] = 0.07p \\[3ex] $ The initial price of the phone and the tax equals $\$220.98$

$ p + 0.07p = 220.98 \\[3ex] 1.07p = 220.98 \\[3ex] p = \dfrac{220.98}{1.07} \\[5ex] p = 206.5233645 \\[3ex] 7\%\:\: sales\:\: tax = 0.07p \\[3ex] = 0.07 * 206.5233645 \\[3ex] = 14.45663551 \\[3ex] $ The initial cost of the phone = $\$206.52$
The sales tax = $\$14.46$

The initial price of the laptop = $960$

$ Discount = 9\%\:\:of\:\: 960 \\[3ex] = 9\% * 960 \\[3ex] = 0.09 * 960 \\[3ex] = 86.4 \\[3ex] Discount = \$86.40 \\[3ex] Sale\:\: price\:\:means\:\: 9\%\:\:off\:\: 960 \\[3ex] Sale\:\:price = Initial\:\:price - Discount \\[3ex] = 960 - 86.4 \\[3ex] = 873.60 \\[3ex] Sale\:\:price = \$873.60 $

You may begin with a preview to ensure students understand it.
Preview is in black color.
Assume the initial cost of a phone is $\$200.00$
Assume a $5\%$ discount (Assume it is on sale for $5\%$ off)
How much is the sale price?

$ 5\% = \dfrac{5}{100} = 0.05 \\[5ex] Discount = 5\%\:\:of\:\:200 = 0.05(200) = \$10 \\[3ex] Sale\:\:price = 5\%\:\:off\:\:200 = \$200 - \$10 = \$190 \\[3ex] Sale\:\:price = \$190 \\[3ex] $ Ask students this question:
Given: $\$190$, with $5\%$ discount (included), how can we get $\$200$?
In other words, Given the sale price which includes the discount, how do we figure out the actual price (initial price)?
Introduce and discuss the concept of Algebra (using variables).

$ Let\:\:the\:\:initial\:\:cost\:\:of\:\:the\:\:phone = x \\[3ex] Discount = 5\%\:\:of\:\:x = 0.05(x) = 0.05x \\[3ex] Sale\:\:price = 5\%\:\:off\:\:x = x - 0.05x = 0.95x \\[3ex] Sale\:\:price = 190 \\[3ex] \rightarrow 0.95x = 190 \\[3ex] x = \dfrac{190}{0.95} \\[5ex] x = 200 \\[3ex] Initial\:\:price = \$200 $

$ Let\:\:the\:\:initial\:\:price = p \\[3ex] 12\% = \dfrac{12}{100} = 0.12 \\[5ex] Sale\:\:price = \$550 \\[3ex] Discount = 12\%\:\:of\:\:p = 0.12p \\[3ex] Sale\:\:price = 12\%\:\:off\:\:p = p - 0.12p = 0.88p \\[3ex] \rightarrow 0.88p = 550 \\[3ex] p = \dfrac{550}{0.88} \\[5ex] p = 625 \\[3ex] $ The initial price of the laptop is $\$625.00$

$ \underline{Check} \\[3ex] Initial\:\:price = \$625.00 \\[3ex] 12\%\:\:discount\:\:of\:\:625 \\[3ex] = \dfrac{12}{100} * 625 \\[5ex] = 0.12 * 625 \\[3ex] = 75 \\[3ex] Sale\:\:price = 625 - 75 = \$550 $

$ Tip = 12\% \:\:of\:\:\$34.99 \\[3ex] = \dfrac{12}{100} * 34.99 \\[5ex] = 4.1988 \\[3ex] Total\:\:cost = Cost\:\:of\:\:meal + Tip \\[3ex] Total\:\:cost = 34.99 + 4.1988 \\[3ex] Total\:\:cost = 39.1888 \approx 39.19 \\[3ex] $ The total cost is $\$39.19$

We can do this question in two ways.
Use any method you prefer.

$ Let\:\:the\:\:initial\:\:price = p \\[3ex] \underline{First\:\:Method:\:\:Quantitative\:\:Literacy} \\[3ex] Marked\:\:down = 20\%\:\:off \\[3ex] Remaining = 100\% - 20\% = 80\% \\[3ex] Exclude\:\:tax \\[3ex] 80\% \:\:of\:\:what\:\:is\:\: 588 \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100}...Percent-Proportion \\[5ex] \dfrac{588}{p} = \dfrac{80}{100} \\[5ex] Cross\:\:Multiply \\[3ex] p * 80 = 588 * 100 \\[3ex] p = \dfrac{588 * 100}{80} \\[5ex] p = \dfrac{5880}{8} \\[5ex] p = 735 \\[3ex] $ The initial price = $\$735.00$

$ \underline{Second\:\:Method:\:\:Algebraic\:\:Thinking} \\[3ex] Selling\:\:price = \$588.00 \\[3ex] Discount = 20\%\:\:of\:\: p \\[3ex] = 20\% * p \\[3ex] = \dfrac{20}{100} * p \\[5ex] = 0.2 * p \\[3ex] = 0.2p \\[3ex] Sale\:\:price = Initial\:\:price - Discount \\[3ex] Sale\:\: price = p - 0.2p \\[3ex] = 0.8p \\[3ex] Exclude\:\:tax \\[3ex] Sale\:\:price = Selling\:\:price \\[3ex] \therefore 0.8p = 588 \\[3ex] p = \dfrac{588}{0.8} \\[5ex] p = 735 \\[3ex] $ The initial price = $\$735.00$

You may begin with a preview to ensure students understand it.
Preview is in black color.

Assume the initial cost of a phone is $\$200.00$
Assume a $5\%$ discount (Assume it is on sale for $5\%$ off)
Include a $10\%$ sales tax
How much is the checkout price?

$ 5\% = \dfrac{5}{100} = 0.05 \\[5ex] Discount = 5\%\:\:of\:\:200 = 0.05(200) = \$10 \\[3ex] Sale\:\:price = 5\%\:\:off\:\:200 = \$200 - \$10 = \$190 \\[3ex] Sale\:\:price = \$190 \\[3ex] 10\% = \dfrac{10}{100} = 0.1 \\[5ex] Tax = 10\%\:\:of\:\:190 = 0.1(190) = \$19 \\[3ex] Checkout\:\:price = Sale\:\:price + Tax \\[3ex] Checkout\:\:price = 190 + 19 = \$209 \\[3ex] $ Ask students this question:
Given: $\$209$, with $5\%$ discount (included) and $10\%$ tax (included), how can we get $\$200$?
In other words, Given the checkout price which includes the discount and the tax, how do we figure out the actual price (initial price)?
Introduce and discuss the concept of Algebra (using variables).

$ Let\:\:the\:\:initial\:\:cost\:\:of\:\:the\:\:phone = x \\[3ex] Discount = 5\%\:\:of\:\:x = 0.05(x) = 0.05x \\[3ex] Sale\:\:price = 5\%\:\:off\:\:x = x - 0.05x = 0.95x \\[3ex] Tax = 10\%\:\:of\:\:0.95x = 0.1(0.95x) = 0.095x \\[3ex] Checkout\:\:price = Sale\:\:price + Tax \\[3ex] \rightarrow 209 = 0.95x + 0.095x \\[3ex] 209 = 1.045x \\[3ex] 1.045x = 209 \\[3ex] x = \dfrac{209}{1.045} \\[5ex] x = 200 \\[3ex] Initial\:\:price = \$200 $

$ Let\:\:the\:\:initial\:\:price = p \\[3ex] Selling\:\:price = \$588.00 \\[3ex] Discount = 20\%\:\:of\:\: p \\[3ex] = 20\% * p \\[3ex] = \dfrac{20}{100} * p \\[5ex] = 0.2 * p \\[3ex] = 0.2p \\[3ex] Sale\:\:price = Initial\:\:price - Discount \\[3ex] Sale\:\: price = p - 0.2p \\[3ex] = 0.8p \\[3ex] Include\:\:tax \\[3ex] 5\%\:\: sales\:\: tax\:\: of\:\: 0.8p \\[3ex] = 0.05 * 0.8p \\[3ex] = 0.04p \\[3ex] Sale\:\:price + Tax = Selling\:\:price \\[3ex] \therefore 0.8p + 0.04p = 588 \\[3ex] 0.84p = 588 \\[3ex] p = \dfrac{588}{0.84} \\[5ex] p = 700.00 \\[3ex] $ The initial price = $$700.00$

$ \underline{Check} \\[3ex] Initial\:\:price = \$700.00 \\[3ex] 20\%\:\:discount\:\:of\:\:700 \\[3ex] = \dfrac{20}{100} * 700 \\[5ex] = 20 * 7 \\[3ex] = 140 \\[3ex] Sale\:\:price = 700 - 140 = 560 \\[3ex] 5\%\:\:sales\:\:tax\:\:of\:\:560 \\[3ex] = \dfrac{5}{100} * 560 \\[5ex] = \dfrac{56}{2} \\[5ex] = 28 \\[3ex] Selling\:price = 560 + 28 = 588 \\[3ex] Selling\:\:price = \$588.00 $

$ Initial\:\:Price = 59.95 \\[3ex] 20\%\:\:Discount = \dfrac{20}{100} * 59.95 = 0.2 * 59.95 = 11.99 \\[3ex] Sale\:\:Price = Initial\:\:Price - Discount \\[3ex] Sale\:\:Price = 59.95 - 11.99 \\[3ex] Sale\:\:Price = \$47.96 $

$ Initial\:\:Price = 80 \\[3ex] Sale\:\:Price = 60 \\[3ex] Discount = Initial\:\:Price - Sale\:\:Price \\[3ex] Discount = 80 - 60 = 20 \\[3ex] \%Discount = \dfrac{Discount}{Initial\:\:Price} * 100 \\[5ex] \%Discount = \dfrac{20}{80} * 100 \\[5ex] \%Discount = 25\% $

We can do a basic part of this question in two ways.
Use any method you prefer.

$ Let\:\:the\:\:original\:\:price = x \\[3ex] \underline{First\:\:Method:\:\:Quantitative\:\:Literacy} \\[3ex] Discount = 20\%\:\:off \\[3ex] Remaining = 100\% - 20\% = 80\% \\[3ex] Exclude\:\:tax \\[3ex] 80\% \:\:of\:\:what\:\:is\:\: 30.40 \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100}...Percent-Proportion \\[5ex] \dfrac{30.4}{x} = \dfrac{80}{100} \\[5ex] Cross\:\:Multiply \\[3ex] x * 80 = 30.4 * 100 \\[3ex] x = \dfrac{30.4 * 100}{80} \\[5ex] x = \dfrac{304}{8} \\[5ex] x = 38 \\[3ex] $ The original price = $\$38.00$

$ \underline{Second\:\:Method:\:\:Algebraic\:\:Thinking} \\[3ex] 20\%\:\:Discount = \dfrac{20}{100} * x \\[5ex] = 0.2 * x = 0.2x \\[3ex] Sale\:\:Price = Original\:\:Price - Discount \\[3ex] = x - 0.2x \\[3ex] = 0.8x \\[3ex] Sale\:\:Price = \$30.40 \\[3ex] \rightarrow 0.8x = 30.40 \\[3ex] x = \dfrac{30.40}{0.8} \\[5ex] x = 38 \\[3ex] x = \$38.00 \\[3ex] $ How much less than $\$38.00$ is $\$30.40$

$ Difference = 38.00 - 30.40 \\[3ex] Difference = \$7.60 $

Let:
SM ORNGE BA = Item 1
ENH SERIES = Item 1
There are two items but only 1 tax rate of 1%
Let us find the tax amount of 1% tax for each item

$ 1\% = \dfrac{1}{100} = 0.01 \\[5ex] \underline{Item\;1} \\[3ex] Cost = 4.98 \\[3ex] Tax = 0.01(4.98) \\[3ex] = 0.0498 \\[3ex] \approx \$0.05 \\[5ex] \underline{Item\;2} \\[3ex] Cost = 5.47 \\[3ex] Tax = 0.01(5.47) \\[3ex] = 0.0547 \\[3ex] \approx \$0.05 \\[3ex] $ As we can see, the tax amount is approximately​ the same for both items.
So, how can we figure out the item?
Note the responses of your students.
Advise them accordingly.

This is the receipt:

Let:
EL 6PK REPE = Item

$ (a.) \\[3ex] 10\% = \dfrac{10}{100} = 0.1 \\[5ex] \underline{Item} \\[3ex] Cost = 23.96 \\[3ex] Tax = 0.1(23.96) \\[3ex] = 2.396 \\[3ex] \approx \$2.40 \\[3ex] (b.) \\[3ex] Total\;\;price = 23.96 + 2.396 \\[3ex] = 26.356 \\[3ex] \approx \$26.36 $

$ Live\:\:south\:\:of\:\:Highway\:\:R = 45\%\:\:of\:\:900 \\[3ex] = \dfrac{45}{100} * 900 \\[5ex] = 45 * 9 \\[3ex] = 405 \\[3ex] Do\:\:NOT\:\:ride\:\:the\:\:bus\:\:to\:\:school = 20\%\:\:of\:\:405 \\[3ex] = \dfrac{20}{100} * 405 \\[5ex] = \dfrac{405}{5} \\[5ex] = 81 \\[3ex] Rides\:\:the\:\:bus\:\:to\:\:school \\[3ex] \underline{First\:\:Method} \\[3ex] 405 - 81 = 324 \\[3ex] \underline{Second\:\:Method} \\[3ex] Rides\:\:the\:\:bus\:\:to\:\:school = 80\%\:\:of\:\:405 \\[3ex] = \dfrac{80}{100} * 405 \\[5ex] = \dfrac{4}{5} * 405 \\[5ex] = 4 * 81 \\[3ex] = 324 $

$ Hitchcock's\:\:Vertigo = 128\:\:minutes \\[3ex] 20\%\:\:of\:\:128 = \dfrac{20}{100} * 128 = 0.2 * 128 = 25.6 \\[5ex] 20\%shorter = 128 - 25.6 = 102.4 \\[3ex] $ The new version of Vertigo is about $102$ minutes in length.

There are:
12 months in a year
52 weeks in a year

(a.)
Money spent on lottery tickets per year = 20(52) = 1040
Money spent on food per year = 125(12) = 1500
1040 is what % of 1500

$ \dfrac{is}{of} = \dfrac{what}{100}...Percent-Proportion \\[5ex] \dfrac{1040}{1500} = \dfrac{what}{100} \\[5ex] 1500 * what = 1040 * 100 \\[3ex] what = \dfrac{1040 * 100}{1500} \\[5ex] what = 69.33333333\% \\[3ex] $ On an annual basis, the money spent on lottery tickets is approximately 69% of the money spent to buy food.

(b.)
Money spent on cell phone bill per year = 76(12) = 912
Money spent on student health insurance per year = 235
912 is what % of 235

$ 912 = \dfrac{what}{100} * 235 ...Percent-Equation \\[5ex] \dfrac{what}{100} * 235 = 912 \\[5ex] what = \dfrac{912 * 100}{235} \\[5ex] what = 388.0851064\% \\[3ex] $ On an annual basis, the first set of expenses is approximately 388% of the second set of expenses.

(c.)
Money spent on lottery tickets per year = 9(3)(52) = 1404
Money spent on textbooks per year = 900
1404 is what % of 900

$ \dfrac{is}{of} = \dfrac{what}{100}...Percent-Proportion \\[5ex] \dfrac{1404}{900} = \dfrac{what}{100} \\[5ex] 900 * what = 1404 * 100 \\[3ex] what = \dfrac{1404 * 100}{900} \\[5ex] what = 156\% \\[3ex] $ On an annual basis, the first set of expenses is 156% of the second set of expenses.

Please be careful of this common mistake!

$ 20\% + 30\% - 20\% = 30\% \\[3ex] $ This is one of the common misuses of Percents.
Please take note!
If you are unsure of how to begin, try Arithmetic (with a number).
It is okay to try with a number in this case because the initial population was not given.
So, you can assume a number.
Then, you can do it with Algebra (with a variable)
You are encouraged to do with a variable so you can get used to Algebra (working with variables)

$ \underline{Arithmetic} \\[3ex] Assume\:\:Initial\:\:Population = 100 \\[3ex] 20\%\:\:of\:\:100 = \dfrac{20}{100} * 100 = 0.2 * 100 = 20 \\[3ex] 20\%\:Increase = 100 + 20 = 120 \\[3ex] 30\%\:\:of\:\:120 = \dfrac{30}{100} * 120 = 0.3 * 120 = 36 \\[3ex] 30\%\:Increase = 120 + 36 = 156 \\[3ex] 20\%\:\:of\:\:156 = \dfrac{20}{100} * 156 = 0.2 * 156 = 31.2 \\[3ex] 20\%\:Decrease = 156 - 31.2 = 124.8 \\[3ex] New\:\:Population = 124.8 \\[3ex] Change = New - Initial \\[3ex] Change = 124.8 - 100 = 24.8 \\[3ex] Change\:\:is\:\:an\:\:increase \\[3ex] \%Increase = \dfrac{Increase}{Initial} * 100 \\[5ex] \%Increase = \dfrac{24.8}{100} * 100 \\[5ex] \%Increase = 24.8\% \\[3ex] \%Increase \approx 25\% \\[3ex] \underline{Algebra} \\[3ex] Let\:\:Initial\:\:Population = x \\[3ex] 20\%\:\:of\:\:x = \dfrac{20}{100} * x = 0.2 * x = 0.2x \\[3ex] 20\%\:Increase = x + 0.2x = 1.2x \\[3ex] 30\%\:\:of\:\:1.2x = \dfrac{30}{100} * 1.2x = 0.3 * 1.2x = 0.36x \\[3ex] 30\%\:Increase = 1.2x + 0.36x = 1.56x \\[3ex] 20\%\:\:of\:\:1.56x = \dfrac{20}{100} * 1.56x = 0.2 * 1.56x = 0.312x \\[3ex] 30\%\:Decrease = 1.56x - 0.312x = 1.248x \\[3ex] New\:\:Population = 1.248x \\[3ex] Change = New - Initial \\[3ex] Change = 1.248x - x = 0.248x \\[3ex] Change\:\:is\:\:an\:\:increase \\[3ex] \%Increase = \dfrac{Increase}{Initial} * 100 \\[5ex] \%Increase = \dfrac{0.248x}{x} * 100 \\[5ex] \%Increase = 24.8\% \\[3ex] \%Increase \approx 25\% $

This is the receipt:


Let the subtotal (sum of the initial prices) of the items = p

$ (a.) \\[3ex] 5.3\% = \dfrac{5.3}{100} = 0.053 \\[5ex] \underline{Item} \\[3ex] Subtotal = p \\[3ex] Tax = 0.053(p) = 0.053p...Calculate \\[3ex] Tax = 1.29638 ... Given \\[3ex] \implies \\[3ex] 0.053p = 1.29638 \\[3ex] p = \dfrac{1.29638}{0.053} \\[5ex] p = 24.46 \\[3ex] Subtotal \approx \$24.46 \\[3ex] (b.) \\[3ex] Total\;\;price = 24.46 + 1.29638 \\[3ex] = 25.75638 \\[3ex] \approx \$25.76 $

This is the receipt:


Let the subtotal (sum of the initial prices) of the items = p

$ (a.) \\[3ex] 5.3\% = \dfrac{5.3}{100} = 0.053 \\[5ex] \underline{Item} \\[3ex] Subtotal = p \\[3ex] Tax = 0.053(p) = 0.053p \\[3ex] Total\;\;price = p + 0.053p \\[3ex] = 1.053p...Calculated \\[3ex] = 34.64 ...Given \\[3ex] \implies \\[3ex] 1.053p = 34.64 \\[3ex] p = \dfrac{34.64}{1.053} \\[5ex] p = 32.89648623 \\[3ex] Subtotal \approx \$32.90 \\[3ex] (b.) \\[3ex] Tax = 0.053(32.89648623) = 1.74351377 \\[3ex] Tax \approx \$1.74 $

This is the receipt:


Let the subtotal (sum of the initial prices) of the items = p

$ (a.) \\[3ex] 1\% = \dfrac{1}{100} = 0.01 \\[5ex] \underline{Item} \\[3ex] Subtotal = p \\[3ex] Tax = 0.01(p) = 0.01p \\[3ex] Total\;\;price = p + 0.01p \\[3ex] = 1.01p...Calculated \\[3ex] = 23.03 ...Given \\[3ex] \implies \\[3ex] 1.01p = 23.03 \\[3ex] p = \dfrac{23.03}{1.01} \\[5ex] p = 22.8019802 \\[3ex] Subtotal \approx \$22.80 \\[3ex] (b.) \\[3ex] Tax = 0.01(22.8019802) = 0.228019802 \\[3ex] Tax \approx \$0.23 \\[3ex] (c.) \\[3ex] \underline{Four\;\;items} \\[3ex] Initial\;\;price = 4.98(4) = \$19.92 \\[3ex] \underline{Five\;\;items} \\[3ex] Subtotal = \$22.8019802 \\[3ex] Initial\;\;price\;\;of\;\;5th\;\;item \\[3ex] = 22.8019802 - 19.92 \\[3ex] = 2.8819802 \\[3ex] \approx \$2.88 \\[3ex] (d.) \\[3ex] Let\;\;unit\;\;price\;\;of\;\;5th\;\;item/pound = k \\[3ex] 4.96 * k = 2.8819802 \\[3ex] k = \dfrac{2.8819802}{4.96} \\[5ex] k = 0.5810443952 \\[3ex] Unit\;\;price \approx \$0.58 $

This is the receipt:


$ (a.) \\[3ex] Sam's\;\;club\;\;credit = \$4.96 \\[3ex] Remaining\;\;balance = \$11.1798 \\[3ex] Total\;\;price = 4.96 + 11.1798 \\[3ex] = 16.1398 \\[3ex] \approx \$16.14 \\[3ex] (b.) \\[3ex] Subtotal = Total\;\;price - Tax \\[3ex] = 16.1398 - 0.16 \\[3ex] = 15.9798 \\[3ex] \approx \$15.98 \\[3ex] (c.) \\[3ex] What\;\;\%\;\;of\;\;15.9798\;\;is\;\;0.16? \\[3ex] \underline{Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{what\%}{100} \\[5ex] \dfrac{0.16}{15.9798} = \dfrac{what\%}{100} \\[5ex] what\% = \dfrac{0.16 * 100}{15.9798} \\[5ex] what\% = 1\% \\[3ex] Tax\;\;percent = 1\% $

Let the subtotal (sum of the initial prices) of the items = p

$ (a.) \\[3ex] \underline{1st\;\;Order} \\[3ex] What\;\;\%\;\;of\;\;47.92\;\;is\;\;2.38? \\[3ex] \underline{Percent-Equation} \\[3ex] \dfrac{what}{100} * 47.92 = 2.38 \\[5ex] what = \dfrac{2.38 * 100}{47.92} \\[5ex] what = 4.966611018 \\[3ex] Tax\;\;rate = 4.966611018\% \\[3ex] (b.) \\[3ex] \underline{2nd\;\;Order} \\[3ex] What\;\;\%\;\;of\;\;45.78\;\;is\;\;2.43? \\[3ex] \underline{Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{what\;\%}{100} \\[5ex] \dfrac{2.43}{45.78} = \dfrac{what\;\%}{100} \\[5ex] what\;\% = \dfrac{2.43 * 100}{45.78} \\[5ex] what\;\% = 5.307994758\% \\[3ex] Tax\;\;rate = 5.307994758\% $

$ Initial\:\:Price = 1000 \\[3ex] New\:\:Price = 900 \\[3ex] Decrease = 900 - 1000 = - 100 \\[3ex] \%Decrease = \dfrac{Decrease}{Initial\:\:Price} * 100 \\[5ex] = \dfrac{100}{1000} * 100 \\[5ex] = 10\% $

We can reword this question as:
$5\%$ of what (the entire book) is $15$?
We can solve this in two ways.
Use any method you prefer.

$ Let\:\:the\:\:pages\:\:of\:\:the\:\:book = x \\[3ex] \underline{First\:\:Method:\:\:Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] \dfrac{15}{x} = \dfrac{5}{100} \\[5ex] Cross\:\:Multiply \\[3ex] x * 5 = 15 * 100 \\[3ex] x = \dfrac{15 * 100}{5} \\[5ex] x = 3 * 100 \\[3ex] x = 300\:\:pages \\[3ex] \underline{First\:\:Method:\:\:Percent-Equation} \\[3ex] 5\%\:\:of\:\:what\:\:is\:\:15? \\[3ex] \dfrac{5}{100} * x = 15 \\[5ex] \dfrac{100}{5} * \dfrac{5}{100} * x = \dfrac{100}{5} * 15 \\[5ex] x = 100 * 3 \\[3ex] x = 300\:\:pages $

This is the receipt:


$ (a.) \\[3ex] Subtotal = 4.98 + 7.47 \\[3ex] Subtotal = \$12.45 \\[3ex] (b.) \\[3ex] What\;\;\%\;\;of\;\;7.47\;\;is\;\;0.39591? \\[3ex] \underline{Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{what\;\%}{100} \\[5ex] \dfrac{0.39591}{7.47} = \dfrac{what\;\%}{100} \\[5ex] what\;\% = \dfrac{0.39591 * 100}{7.47} \\[5ex] what\;\% = 5.3\% \\[3ex] 1st\;\;tax\;\;rate = 5.3\% \\[3ex] (c.) \\[3ex] What\;\;\%\;\;of\;\;7.47\;\;is\;\;0.5229? \\[3ex] \underline{Percent-Equation} \\[3ex] \dfrac{what}{100} * 7.47 = 0.5229 \\[5ex] what = \dfrac{0.5229 * 100}{7.47} \\[5ex] what = 7 \\[3ex] 2nd\;\;tax\;\;rate = 5.3\% \\[3ex] (d.) \\[3ex] Total\;\;price = 12.45 + 0.39591 + 0.05 + 0.5229 \\[3ex] = 13.41881 \\[3ex] \approx \$13.42 $

$ Let\:\:the\:\:\%\:\:number\:\:of\:\:women = w \\[3ex] 22\dfrac{1}{2} + 47\dfrac{1}{2} + w = 100...Total\:\:Percent \\[5ex] 22 + \dfrac{1}{2} + 47 + \dfrac{1}{2} + w = 100 \\[5ex] 69 + 1 + w = 100 \\[3ex] 70 + w = 100 \\[3ex] w = 100 - 70 \\[3ex] w = 30\% \\[3ex] Let\:\:the\:\:number\:\:of\:\:people\:\:in\:\:the\:\:hall = p \\[3ex] \rightarrow 30\%\:\:of\:\:x = 84...number\:\:of\:\:women \\[3ex] \dfrac{30}{100} * x = 84 \\[5ex] \dfrac{100}{30} * \dfrac{30}{100} * x = \dfrac{100}{30} * 84 \\[5ex] x = \dfrac{10 * 84}{3} \\[5ex] x = 10 * 28 \\[3ex] x = 280\:\:people \\[3ex] Number\:\:of\:\:men = 47\dfrac{1}{2}\% \:\:of\:\:280 \\[5ex] = 47\dfrac{1}{2} \div 100 * 280 \\[5ex] = \dfrac{95}{2} \div \dfrac{100}{1} * 280 \\[5ex] = \dfrac{95}{2} * \dfrac{1}{100} * 280 \\[5ex] = \dfrac{95 * 1 * 280}{2 * 100} \\[5ex] = 19 * 7 \\[3ex] = 133\:\:men $

We shall use two different depreciation methods to solve this question

The Declining Balance Method and the Straight-Line Depreciation Method

$ \underline{Declining\:\:Balance\:\:Method:\:\:Manually} \\[3ex] 8\% = \dfrac{8}{100} \\[5ex] At\:\:the\:\:end\:\:of\:\:1st\:\:year \\[3ex] Depreciation = 8\%\:\:of\:\:65000 = \dfrac{8}{100} * 65000 = 8 * 650 = 5200 \\[5ex] Value = 65000 - 5200 = 59800 \\[3ex] At\:\:the\:\:end\:\:of\:\:2nd\:\:year \\[3ex] Depreciation = 8\%\:\:of\:\:59800 = \dfrac{8}{100} * 59800 = 8 * 598 = 4784 \\[5ex] Value = 59800 - 4784 = 55016 \\[3ex] Value = \$55,016.00 \\[3ex] \underline{Declining\:\:Balance\:\:Method:\:\:Formula} \\[3ex] V = N(1 - r)^t \\[4ex] N = \$65000 \\[3ex] r = 8\% = \dfrac{8}{100} = 0.08 \\[5ex] t = 2\:years \\[3ex] V = 65000(1 - 0.08)^2 \\[4ex] V = 65000(0.92)^2 \\[4ex] V = 65000(0.8464) \\[3ex] V = 55016 \\[3ex] Value = \$55,016.00 \\[5ex] \underline{Straight-Line\:\:Depreciation\:\:Method:\:\:Formula} \\[3ex] V = N(1 - rt) \\[4ex] N = \$65000 \\[3ex] r = 8\% = \dfrac{8}{100} = 0.08 \\[5ex] t = 2\:years \\[3ex] V = 65000(1 - 0.08(2)) \\[3ex] V = 65000(1 - 0.16) \\[3ex] V = 65000(0.84) \\[3ex] V = 54600 \\[3ex] Value = \$54,600.00 $

$ Let\:\:Mary's\:\:paycheck = p \\[3ex] Let\:\:Martha's\:\:paycheck = k \\[3ex] 30\% = \dfrac{30}{100} = 0.3 \\[5ex] 30\%\:\:of\:\:p = 0.3 * p = 0.3p \\[3ex] Remaining:\:\: p - 0.3p \\[3ex] \rightarrow k = p - 0.3p \\[3ex] k = 0.7p \\[3ex] k + p = 665 \\[3ex] \rightarrow 0.7p + p = 665 \\[3ex] 1.7p = 665 \\[3ex] p = \dfrac{665}{1.7} \\[5ex] p = 391.176471 \\[3ex] p \approx 391.18 \\[3ex] k = 0.7p = 0.7(391.176471) \\[3ex] k = 273.82353 \\[3ex] k \approx 273.82 \\[3ex] $ Mary's weekly paycheck is $\$391.18$ while Martha's weekly paycheck is $\$273.82$

Based on the histogram:

$2$ adults had no child

$8$ adults had one child

$5$ adults had $2$ children

$2$ adults had $3$ children

No adult had $4$ children

$2$ adults had $5$ children

$ (a.) \\[3ex] Total\:\:Number\:\:of\:\:adults = 2 + 8 + 5 + 2 + 0 + 2 = 19\:\:adults \\[3ex] (b.) \\[3ex] \%\:\:of\:\:adults\:\:with\:\:one\:\:child \\[3ex] = \dfrac{Number\:\:of\:\:adults\:\:with\:\:one\:\:child}{Total\:\:number\:\:of\:\:adults} * 100 \\[5ex] = \dfrac{8}{19} * 100 \\[5ex] = \dfrac{800}{19} \\[5ex] = 42.1052630\% $

$ Amount\:\:spent\:\:on\:\:Fun \\[3ex] = 16\%\:\:of\:\:2500 \\[3ex] = \dfrac{16}{100} * 2500 \\[5ex] = 16 * 25 \\[3ex] = 400 \\[3ex] $ Judith spent $\$400.00$ on Fun activities.

Less than $\$100$ "on average" includes the percent of adults who spend less than $\$50$ "on average" as well as the percent of adults who spend no less than $\$50$ and no more than $\$99$ "on average"

$ Less\:\:than\:\:\$100\:\:on\:\:average \\[3ex] = 8\% + 17\% \\[3ex] = 25\% \\[3ex] = \dfrac{25}{100} \\[5ex] = 0.25...Option\:\:E \\[3ex] $ NOTE:

$ 25 \ne 25\% \\[3ex] 0.25 = 25\% $

$ At\:\:least\:\:\$150\:\:means \ge \$150 \\[3ex] \%\:\:of\:\:adults\:\:that\:\:spend\:\:\ge \$150 \\[3ex] = 15\% + 21\% + 10\% \\[3ex] = 46\% $

$ Frequency\:\:for\:\:\$300\:\:or\:\:more = 10\% \\[3ex] Total\:\:Frequency, \Sigma F = 100\% \\[3ex] Sectorial\:\angle \:\:for\:\:\$300\:\:or\:\:more \\[3ex] = \dfrac{Frequency\:\:for\:\:\$300\:\:or\:\:more}{Total\:\:Frequency} * 360^\circ \\[5ex] = \dfrac{10}{100} * 360 \\[5ex] = 1 * 36 \\[3ex] = 36\^circ $

This is a somewhat "tricky" question.
We cannot just assume that the trader will make a loss by selling the $20$ oranges for $₦1.10$
This is because we do not know how much he bought those $20$ oranges
So, we need to find the cost of those $20$ oranges first.
We find the cost price based on the first sentence that was given...based on what we know...that he made an $8\%$ profit by selling the $20$ oranges for $₦1.35$
Then, we determine if that cost is greater than or less than $₦1.10$ to find if he made a profit or loss.
Then, we calculate the percentage gain or loss.

$ \underline{First:\:\:Determine\:\:the\:\:Cost\:\:Price} \\[3ex] Let\:\:C = cost\:\:price \\[3ex] Let\:\:S = selling\:\:price \\[3ex] \%Profit = \dfrac{S - C}{C} \\[5ex] S = ₦1.35 \\[3ex] \%Profit = 8\% \\[3ex] 8\% = \dfrac{1.35 - C}{C}...\%\:\:is\:\:given \\[5ex] \dfrac{8}{100} = \dfrac{1.35 - C}{C} \\[5ex] Cross\:\:Multiply \\[3ex] 8 * C = 100(1.35 - C) \\[3ex] 8C = 135 - 100C \\[3ex] 8C + 100C = 135 \\[3ex] 108C = 135 \\[3ex] C = \dfrac{135}{108} = \dfrac{15}{12} = \dfrac{5}{4} \\[5ex] C = ₦1.25 \\[3ex] S = ₦1.10 \\[3ex] \underline{Second:\:\:Profit\:\:or\:\:Loss} \\[3ex] ₦1.25 \gt ₦1.10...he\:\:made\:\:a\:\:loss \\[3ex] He\:\:bought\:\:it\:\:for\:\:more\:\:than\:\:what\:\:he\:\:sold\:\:it \\[3ex] \underline{Third:\:\:Calculate\:\:the\:\:\%\:\:Loss} \\[3ex] \%Loss = \dfrac{C - S}{C} * 100...\%\:\:needs\:\:be\:\:found \\[5ex] \%Loss = \dfrac{1.25 - 1.10}{1.25} * 100 \\[5ex] = \dfrac{0.15}{1.25} * 100 \\[5ex] = \dfrac{15}{125} * 100 \\[5ex] = \dfrac{3}{25} * 100 \\[5ex] = 3 * 4 \\[3ex] = 12\% \\[3ex] $ Be selling the $20$ oranges for $₦1.10$, he made a loss of $12\%$

We can do this question in two ways.
Choose any method you prefer.

$ \underline{First\:\:Method:\:\:Quantitative\:\:Reasoning} \\[3ex] Mark\:\:obtained\:\:for\:\:01\:\:Paper \\[3ex] = 55\%\:\:of\:\:30 \\[3ex] = \dfrac{55}{100} * 30 \\[5ex] = \dfrac{11 * 3}{2} \\[5ex] = 16.5 \\[3ex] Mark\:\:obtained\:\:for\:\:02\:\:Paper \\[3ex] = 60\%\:\:of\:\:50 \\[3ex] = \dfrac{60}{100} * 50 \\[5ex] = 6 * 5 \\[3ex] = 30 \\[3ex] Mark\:\:obtained\:\:for\:\:03\:\:Paper \\[3ex] = 80\%\:\:of\:\:20 \\[3ex] = \dfrac{80}{100} * 20 \\[5ex] = 8 * 2 \\[3ex] = 16 \\[3ex] Final\:\:Mark = 16.5 + 30 + 16 = 62.5\:\:out\:\:of\:\:100 \\[3ex] \dfrac{62.5}{100} = 62.5\% \\[3ex] Final\:\:Mark = 62.5\% \\[3ex] \underline{Second\:\:Method:\:\:Weighted\:\:Average\:\:Method} \\[3ex] $ This is similar to the Grading Method Mr. C used for your class.

Assessment	Weight	Your Score	Weighted Score
$01$	$30$	$55$	$30 * 55 = 1650$
$02$	$50$	$60$	$50 * 60 = 3000$
$03$	$20$	$80$	$20 * 80 = 1600$

$ \Sigma Weighted\:\:Scores = 1650 + 3000 + 1600 = 6250 \\[3ex] \Sigma Weights = 30 + 50 + 20 = 100 \\[3ex] Final\:\:Grade = \dfrac{\Sigma Weighted\:\:Scores}{\Sigma Weights} \\[5ex] = \dfrac{6250}{100} \\[5ex] = 62.5\% $

$ Let\:\:R = red \\[3ex] Let\:\:B = blue \\[3ex] Let\:\:G = green \\[3ex] Let\:\:P = purple \\[3ex] Let\:\: K = Other\:\:sector \\[3ex] S = sample\:\:space \\[3ex] n(S) = 100\% \\[3ex] n(R) = 25\% \\[3ex] n(B) = 30\% \\[3ex] n(G) = 20\% \\[3ex] n(P) = 10\% \\[3ex] n(K) = 100 - (25 + 30 + 20 + 10) \\[3ex] n(K) = 100 - 85 = 15\% \\[3ex] Sectorial\:\angle \:\:for\:\:K \\[3ex] \dfrac{n(K)}{n(S)} * 360 \\[5ex] = \dfrac{15}{100} * 360 \\[5ex] = \dfrac{3}{2} * 36 \\[5ex] = 3 * 18 \\[3ex] = 54^\circ \\[3ex] $ The sectorial angle for the Other sector is $54^\circ$

$ Cost\:\:price\:\:for\:\:250\:\:oranges = D1000 \\[3ex] Cost\:\:price\:\:for\:\:1\:\:orange = \dfrac{1000}{250} = D4 \\[5ex] Kept\:\:for\:\:himself:\:\: 20\% \:\:of\:\: 250 \\[3ex] = \dfrac{20}{100} * 250 \\[5ex] = \dfrac{250}{5} \\[5ex] = 50\:\:oranges \\[3ex] Remaining:\:\: 250 - 50 = 200\:\:oranges \\[3ex] Sold:\:\: 115\:\:oranges\:\:@\:\:D6.50\:\:each = 115(6.5) = D747.5 \\[3ex] Remaining:\:\: 200 - 115 = 85\:\:oranges \\[3ex] Sold:\:\: 85\:\:oranges\:\:@\:\:D5.00\:\:each = 85(5) = D425 \\[3ex] \underline{For\:\:the\:\:oranges\:\:sold} \\[3ex] 200\:\:oranges \\[3ex] Cost\:\:price\:\:for\:\:200\:\:oranges\:\:@\:\:D5.00\:\:each = 200(4) = D800 \\[3ex] Selling\:\:price = D747.5 + D425 = D1172.5 \\[3ex] Profit = Selling\:\:price - Cost\:\:price \\[3ex] = D1172.5 - D800 = D372.5 \\[3ex] \%\:\:Profit = \dfrac{Profit}{Cost\:\:price} * 100 \\[5ex] = \dfrac{372.5}{800} * 100 \\[5ex] = \dfrac{372.5}{8} \\[5ex] = 46.5625\% $

$ Bill = \$32 \\[3ex] Tip\:\:amount \\[3ex] = 20\%\:\:of\:\:32 \\[3ex] = \dfrac{20}{100} * 32 \\[5ex] = \dfrac{32}{5} \\[5ex] = 6.4 \\[3ex] Check-out\:\:bill = 32 + 6.4 = 38.4 \\[3ex] Check-out\:\:bill = \$38.40 \\[3ex] $ Lucy has to pay $\$38.40$

We have several approaches to solve this question.
Use any approach you prefer.

$ Initial\;\;price = 2.70\;lb * \dfrac{\$8.98}{1\;lb} = \$24.246 \\[5ex] Sale\;\;price = 2.70\;lb * \dfrac{\$6.98}{1\;lb} = \$18.846 \\[5ex] \underline{Algebraic\;\;Thinking} \\[3ex] Let\;\;\%\;discount = p \\[3ex] p\% = \dfrac{p}{100} = 0.01p \\[5ex] Discount = p\%\;\;of\;\;24.246 \\[3ex] = 0.01p(24.246) \\[3ex] = 0.24246p \\[3ex] Sale\;\;price = p\%\;\;off\;\;24.246 \\[3ex] = 24.246 - 0.24246p \\[3ex] \implies \\[3ex] 24.246 - 0.24246p = 18.846 \\[3ex] 24.246 - 18.846 = 0.24246p \\[3ex] 0.24246p = 5.4 \\[3ex] p = \dfrac{5.4}{0.24246} \\[5ex] p = 22.27171492 \\[3ex] \therefore \%\;discount = 22.27171492\% \\[5ex] \underline{Percent-Proportion} \\[3ex] What\;\;\%\;\;of\;\;24.246\;\;is\;\;18.846? \\[3ex] \dfrac{is}{of} = \dfrac{what\;\%}{100} \\[5ex] \dfrac{18.846}{24.246} = \dfrac{what\;\%}{100} \\[5ex] what\;\% = \dfrac{18.846 * 100}{24.246} \\[5ex] what\;\% = 77.72828508\% \\[3ex] \%\;discount = 100\% - 77.72828508\% \\[3ex] \%\;discount = 22.27171492\% $

Let the levels of carbon IV oxide in parts per million be $C$
Let parts per million = $ppm$

$ \underline{1959} \\[3ex] Initial\:\:C = 304ppm \\[3ex] \underline{2005} \\[3ex] New\:\:C = 379ppm \\[3ex] Increase = New\:\:C - Initial\:\:C \\[3ex] = 379 - 304 \\[3ex] = 75ppm \\[3ex] \%Increase = \dfrac{Increase}{Initial\:\:C} * 100 \\[5ex] = \dfrac{75}{304} * 100 \\[5ex] = \dfrac{7500}{304} \\[5ex] = 24.6710526\% \\[3ex] \approx 24.7\% \\[3ex] $ The increase is $75$ parts per million
The percent increase is about $24.7\%$

$ Cost\:\:Price\:\:for\:\:10\:\:dozen\:\:eggs\:\:@\:\:₦300\:\:per\:\:dozen \\[3ex] = 10(300) \\[3ex] = ₦3000 \\[3ex] Number\:\:for\:\:10\:\:dozen\:\:eggs\:\:@\:\:12\:\:eggs\:\:per\:\:dozen \\[3ex] = 10(12) \\[3ex] = 120\:\:eggs \\[3ex] But:\:\: 20\:\:eggs\:\:were\:\:broken \\[3ex] Remaining:\:\: 120 - 20 = 100\:\:eggs \\[3ex] Profit = 10\% = \dfrac{10}{100}...already\:\:given \\[5ex] \%Profit = \dfrac{Selling\:\:Price - Cost\:\:Price}{Cost\:\:Price} \\[5ex] \dfrac{10}{100} = \dfrac{Selling\:\:Price - 3000}{3000} \\[5ex] Cross\:\:Multiply \\[3ex] 100(Selling\:\:Price - 3000) = 10(3000) \\[5ex] Selling\:\:Price - 3000 = \dfrac{10 * 3000}{100} \\[5ex] Selling\:\:Price - 3000 = 10 * 30 \\[3ex] Selling\:\:Price - 3000 = 300 \\[3ex] Selling\:\:Price = 300 + 3000 \\[3ex] Selling\:\:Price = 3300 \\[3ex] $ He sold the remaining $100$ eggs ($8$ dozen eggs and $4$ eggs) for $₦3,300$ in order to make a profit of $10\%$

$ Let\:\:her\:\:salary\:\:last\:\:year = p \\[3ex] 3\%\:\:raise\:\:of\:\:p \\[5ex] = \dfrac{3}{100} * p \\[5ex] = \dfrac{3p}{100} = 0.03p \\[5ex] Current\:\:salary = p + 0.03p = 1.03p \\[3ex] \rightarrow 1.03p = 68804 \\[3ex] p = \dfrac{68804}{1.03} \\[5ex] p = 66800 \\[3ex] $ Hannah's salary last year was $\$66,800.00$

$ \underline{Publisher} \\[3ex] Number\:\:of\:\:copies\:\:printed = 30000 \\[3ex] Cost\:\:price = 30000\:\:@\:\:GH¢2.00\:\:each = 30000(2) = GH¢60000 \\[3ex] But\:\:25,380\:\:copies\:\:were\:\:sold\:\:@\:\:GH¢2.76\:\:each \\[3ex] Selling\:\:price = 25380\:\:@\:\:GH¢2.76\:\:each = 25380(2.76) = GH¢70048.8 \\[3ex] \underline{Paid\:\:to\:\:Author} \\[3ex] First\:\:Payment:\:\:10\%\:\:of\:\:selling\:\:price\:\:of\:\:first\:\:6000\:\:copies \\[3ex] Selling\:\:price\:\:for\:\:first\:\:6000\:\:copies = 6000\:\:@\:\:GH¢2.76\:\:each = 6000(2.76) = GH¢16560 \\[3ex] First\:\:Payment = 10\%\:\:of\:\:16560 \\[3ex] = \dfrac{10}{100} * 16560 \\[5ex] = 1 * 1656 \\[3ex] = GH¢1656 \\[3ex] Second\:\:Payment:\:\:12\dfrac{1}{2}\%\:\:of\:\:selling\:\:price\:\:of\:\:all\:\:copies\:\:in\:\:excess\:\:of\:\:6000\:\:copies \\[3ex] All\:\:copies\:\:in\:\:excess\:\:of\:\:6000 = Remaining\:\:copies \\[3ex] Remaining\:\:copies = 25380 - 6000 = 19380 \\[3ex] Selling\:\:price\:\:for\:\:remaining\:\:19380\:\:copies = 19380\:\:@\:\:GH¢2.76\:\:each = 19380(2.76) = GH¢53488.8 \\[3ex] Second\:\:Payment = 12\dfrac{1}{2}\%\:\:of\:\:53488.8 \\[5ex] = 12.5\%\:\:of\:\:53488.8 \\[3ex] = \dfrac{12.5}{100} * 53488.8 \\[5ex] = 0.125 * 53488.8 \\[3ex] = GH¢6,686.1 \\[3ex] (i) \\[3ex] Total\:\:amount\:\:received\:\:by\:\:Author = 1656 + 6686.1 \\[3ex] = GH¢8342.10 \\[3ex] (ii) \\[3ex] Net\:\:profit\:\:made\:\:by\:\:Publisher = Selling\:\:price - (Cost\:\:price + Total\:\:amount\:\:paid\:\:to\:\:Author) \\[3ex] = 70048.8 - (60000 + 8342.1) \\[3ex] = 70048.8 - 68342.1 \\[3ex] = GH¢1,706.70 \\[3ex] (b.) \\[3ex] What\:\:\%\:\:of\:\:the\:\:Author's\:\:total\:\:receipt\:\:is\:\:the\:\:Publisher's\:\:net\:\:profit? \\[3ex] Let\:\:the\:\:\% = p \\[3ex] \underline{Percent-Proportion\:\:Method} \\[3ex] \dfrac{is}{of} = \dfrac{p}{100} \\[5ex] \dfrac{1706.7}{8342.1} = \dfrac{p}{100} \\[5ex] Cross\:\:multiply \\[3ex] 8342.1 * p = 1706.7 * 100 \\[3ex] p = \dfrac{1706.7 * 100}{8342.1} \\[5ex] p = \dfrac{170670}{8342.1} \\[5ex] p = 20.4588773\% \\[3ex] p \approx 20.5\% \\[3ex] $ The publisher's net profit as a percentage of the author's total receipt is about $20.5\%$

$ Oxygen\:\:saturation\:\:level \\[3ex] = \dfrac{7.3}{9.8} * 100 \\[5ex] = \dfrac{7.3 * 100}{9.8} \\[5ex] = \dfrac{730}{9.8} \\[5ex] = 74.4897959 \\[3ex] \approx 74\% \\[3ex] $ The oxygen saturation level is approximately $74\%$.

$ \underline{Nonstudents} \\[3ex] Initially\:\:approved = 85 \\[3ex] Error = 15 \\[3ex] Correctly\:\:approved = 85 - 15 = 70 \\[3ex] \underline{Approved} \\[3ex] Total = 129 \\[3ex] Nonstudents = 70 \\[3ex] High\:\:school\:\:and\:\:College\:\:students = 129 - 70 = 59 \\[3ex] \underline{College\:\:students} \\[3ex] Disapprove = 10 \\[3ex] No\:\:opinion = 6 \\[3ex] Approved = ???\:\:(changed\:\:because\:\:of\:\:the\:\:error\:\:with\:\:misidentification\:\:of\:\:Nonstudents) \\[3ex] Total = ???\:\:changed\:\:as\:\:well \\[3ex] Let\:\:the\:\:total\:\:number\:\:of\:\:College\:\:students = p \\[3ex] Exactly\:\:60\%\:\:of\:\:p\:\:Approved \\[3ex] This\:\:implies\:\:that\:\:(100\% - 60\%)\:\:did\:\:not\:\:Approve \\[3ex] Exactly\:\:40\%\:\:of\:\:p\:\:Disapprove\:\:and\:\:No\:\:opinion \\[3ex] Disapprove\:\:and\:\:No\:\:opinion = 10 + 6 = 16 \\[3ex] 40\%\:\:of\:\:p = 16 \\[3ex] \dfrac{40}{100} * p = 16 \\[3ex] 0.4 * p = 16 \\[3ex] p = \dfrac{16}{0.4} \\[5ex] p = 40\:\:students \\[3ex] Total = 40\:\:students \\[3ex] Approved = 40 - 16 = 24\:\:students \\[3ex] \underline{High\:\:school\:\:students} \\[3ex] High\:\:school\:\:and\:\:College\:\:students\:\:who\:\:Approved = 59 \\[3ex] College\:\:students\:\:who\:\:Approved = 24 \\[3ex] \therefore High\:\:school\:\:students\:\:who\:\:Approved = 59 - 24 = 35 \\[3ex] Disapprove = 4 \\[3ex] No\:\:opinion = 11 \\[3ex] Total = 35 + 4 + 11 = 50 \\[3ex] \%\:\:who\:\:Approved \\[3ex] = \dfrac{Approved}{Total} * 100 \\[5ex] = \dfrac{35}{50} * 100 \\[5ex] = 35(2) \\[3ex] = 70\% $

$ \underline{1-point\:\:free\:\:throw} \\[3ex] Number\:\:of\:\:points \\[3ex] = 75\%\:\:of\:\:80 \\[3ex] = \dfrac{75}{100} * 80 \\[5ex] = 0.75 * 80 \\[3ex] = 60\:points \\[3ex] \underline{2-point\:\:free\:\:throw} \\[3ex] Number\:\:of\:\:points \\[3ex] = 60\%\:\:of\:\:90 \\[3ex] = \dfrac{60}{100} * 90 \\[5ex] = 0.6 * 90 \\[3ex] = 54\:\:(2-points) \\[3ex] = 54(2) = 108\:points \\[3ex] \underline{3-point\:\:free\:\:throw} \\[3ex] Number\:\:of\:\:points \\[3ex] = 60\%\:\:of\:\:25 \\[3ex] = \dfrac{60}{100} * 25 \\[5ex] = 0.6 * 25 \\[3ex] = 15\:\:(3-points) \\[3ex] = 15(3) = 45\:points \\[3ex] Total\:\:Number\:\:of\:\:points = 60 + 108 + 45 = 213\:\:points $

$ Number\:\:of\:\:bracelets = 165 \\[3ex] Cost\:\:price\:\:of\:\:all\:\:bracelets = \$6800 \\[3ex] Customs\:\:duty = \$1360 \\[3ex] (i) \\[3ex] Total\:\:cost = \$6800 + \$1360 = \$8160 \\[3ex] (ii) \\[3ex] 165\:\:bracelets\:\:@\:\:\$68.85\:\:per\:\:bracelet \\[3ex] Selling\:\:price\:\:of\:\:all\:\:bracelets = 165(68.85) = \$11360.25 \\[3ex] (a.) \\[3ex] Total\:\:profit = \$11360.25 - \$8160 = \$3200.25 \\[3ex] (b.) \\[3ex] \%Profit\:\:based\:\:on\:\:cost\:\:price \\[3ex] \%Profit = \dfrac{Profit}{Cost\:\:price} * 100 \\[5ex] = \dfrac{3200.25}{6800} * 100 \\[5ex] = \dfrac{320025}{6800} \\[5ex] = 47.0625\% \\[3ex] \approx 47\% $

From the Bar Graph,
$1$ student earned $0$
$3$ students earned $1$
$4$ students earned $2$
$7$ student earned $3$
$10$ students earned $4$
$8$ students earned $5$
$7$ student earned $6$
$9$ students earned $7$
$8$ students earned $8$
$2$ students earned $9$
$1$ students earned $10$

$ Number\:\:of\:\:Students = 60 \\[3ex] Verify:\:\: 1 + 3 + 4 + 7 + 10 + 8 + 7 + 9 + 8 + 2 + 1 = 60 \\[3ex] \underline{Below\:\:Pass\:\:Mark} \\[3ex] Marks = 4, 3, 2, 1, 0 \\[3ex] Number\:\:of\:\:students = 10 + 7 + 4 + 3 + 1 = 25 \\[3ex] \%\:\:of\:\:students \\[3ex] = \dfrac{25}{60} * 100 \\[5ex] = \dfrac{25}{3} * 5 \\[5ex] = \dfrac{25 * 5}{3} \\[5ex] = \dfrac{125}{3} \\[5ex] = 41.6666667\% \\[3ex] \approx 41.7\% \\[3ex] $ About $41.7\%$ of students failed the test.

The refurbishing cost should be included in the cost price

Cost price = $250000 + 70000 = 320000$

Selling Price = $400000$

Gain = Selling Price - Cost Price

Gain = $400000 - 320000 = 80000$

$ \% Gain = \dfrac{Gain}{Cost\:\:Price} * 100 \\[5ex] \% Gain = \dfrac{80000}{320000} * 100 \\[5ex] \% Gain = \dfrac{100}{4} \\[5ex] \% Gain = 25\% $

Area of a rectangle = Length * Width

Let the length of the original rectangle = $L$

Let the width of the original rectangle = $W$

Area of the original rectangle = $L * W$

Length of the new rectangle/resulting rectangle = $L + 25\% = L + 0.25L = 1.25L$

Width of the new rectangle/resulting rectangle = $W - 10\% = W - 0.10W = 0.9W$

Area of the new rectangle/resulting rectangle = $(1.25L)(0.9W) = 1.125LW$

Area of resulting rectangle - Area of original rectangle = $1.125LW - LW = 0.125LW$

$0.125LW = 0.12LW * 100 = 12.5\%LW$

Area of resulting rectangle is larger than the Area of original rectangle by $12.5\%$

Student: Why did you subtract?
Teacher: Because of what the question asked: By how much is the area of the resulting rectangle larger than the area of the original rectangle?

$ Discount\:\:Price = Sale\:\:Price \\[3ex] List\:\:Price = 1000 \\[3ex] Discount = 20\%\:\:of\:\:1000 \\[3ex] = \dfrac{20}{100} * 1000 \\[5ex] = 20(10) = 200 \\[3ex] Sale\:\:Price = 20\%\:\:off\:\:1000 \\[3ex] = 1000 - 200 = 800 \\[3ex] Tax = 6\%\:\:of\:\:800 \\[3ex] = \dfrac{6}{100} * 800 \\[5ex] = 6(8) = 48 \\[3ex] Total\:\:Price = 800 + 48 = \$848 $

$ First\:\:Discount \\[5ex] = 40\%\:\:of\:\: 10 \\[5ex] = \dfrac{40}{100} * 10 \\[5ex] = 0.4(10) = 4 \\[3ex] First\:\:Sale\:\:Price = 40\%\:\:off\:\:10 \\[3ex] = 10 - 4 = 6 \\[3ex] Second\:\:Discount = Coupon \\[3ex] = 10\%\:\:of\:\:6 \\[3ex] = \dfrac{10}{100} * 6 \\[5ex] = 0.1(6) = 0.6 \\[3ex] Second\:\:Sale\:\:Price = 10\%\:\:off\:\:6 \\[3ex] = 6 - 0.6 = 5.4 \\[3ex] Final\:\:Sale\:\:Price = \$5.40 $

We can solve this question in at least two ways.
Use any method you prefer

$ Length\:\:of\:\:south\:\:side = 40\:inches \\[3ex] Length\:\:of\:\:north\:\:side = 28\:inches \\[3ex] 40\:inches\:\:is\:\:what\:\:percent\:\:of\:\:28\:inches? \\[3ex] Let\:\:the\:\:percent = p \\[3ex] \underline{First\:\:Method:\:\:Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] \dfrac{40}{28} = \dfrac{p}{100} \\[5ex] Cross\:\:Multiply \\[3ex] 28 * p = 40 * 100 \\[3ex] p = \dfrac{40 * 100}{28} \\[5ex] p = \dfrac{10 * 100}{7} \\[5ex] p = \dfrac{1000}{7} \\[5ex] p = 142\dfrac{6}{7} \\[5ex] p = 142\dfrac{6}{7}\% \\[5ex] \underline{Second\:\:Method:\:\:Percent-Equation} \\[3ex] 40\:inches\:\:is\:\:what\:\:percent\:\:of\:\:28\:inches? \\[3ex] Let\:\:the\:\:percent = p \\[3ex] 40 = p\% * 28 \\[3ex] 40 = \dfrac{p}{100} * 28 \\[5ex] Swap \\[3ex] \dfrac{p}{100} * 28 = 40 \\[5ex] Multiply\:\:both\:\:sides\:\:by\:\: \dfrac{100}{28} \\[5ex] \dfrac{100}{28} * \dfrac{p}{100} * 28 = \dfrac{100}{28} * 40 \\[5ex] p = \dfrac{100 * 40}{28} \\[5ex] p = 142\dfrac{6}{7} \\[5ex] p = 142\dfrac{6}{7}\% \\[5ex] $ The length of the south side of the park is $142\dfrac{6}{7}\%$ of the length of the north side.

We can solve this question in at least two ways.
Use any method you prefer

$ At\:\:the\:\:beginning\:\:of\:\:2014: \\[3ex] Number\:\:of\:\:lions = 200 \\[3ex] Total\:\:number\:\:of\:\:large\:\:animals = 2400 \\[3ex] 200\:\:was\:\:p\:\:percent\:\:of\:\:2400? \\[3ex] \underline{First\:\:Method:\:\:Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] \dfrac{200}{2400} = \dfrac{p}{100} \\[5ex] Cross\:\:Multiply \\[3ex] 2400 * p = 200 * 100 \\[3ex] p = \dfrac{200 * 100}{2400} \\[5ex] p = \dfrac{100 * 2}{24} \\[5ex] p = \dfrac{200}{24} \\[5ex] p = 8.33333333 \\[3ex] p \approx 8\% \\[3ex] \underline{Second\:\:Method:\:\:Percent-Equation} \\[3ex] 200\:\:was\:\:p\:\:percent\:\:of\:\:2400? \\[3ex] 200 = p\% * 2400 \\[3ex] 200 = \dfrac{p}{100} * 2400 \\[5ex] Swap \\[3ex] \dfrac{p}{100} * 2400 = 200 \\[5ex] Multiply\:\:both\:\:sides\:\:by\:\: \dfrac{100}{2400} \\[5ex] \dfrac{100}{2400} * \dfrac{p}{100} * 2400 = \dfrac{100}{2400} * 200 \\[5ex] p = \dfrac{100 * 2}{24} \\[5ex] p = \dfrac{200}{24} \\[5ex] p = 8.33333333 \\[3ex] p \approx 8\% \\[3ex] $ The number of lions in the park was approximately $8\%$ percent of the total number of large animals.

$ Let\:\:the\:\:father's\:\:income = p \\[3ex] 20\% = \dfrac{20}{100} = \dfrac{1}{5} \\[5ex] Monthly\:\:allowance\:\:given\:\:to\:\:his\:\:three\:\:children = 20\%\:\:of\:\:p \\[3ex] = \dfrac{1}{5} * p \\[5ex] = \dfrac{1}{5}p \\[5ex] Eldest\:\:child's\:\:share = 45\%\:\:of\:\:\dfrac{1}{5}p \\[5ex] = \dfrac{45}{100} * \dfrac{1}{5}p \\[5ex] = \dfrac{9}{100}p \\[5ex] Youngest\:\:child's\:\:share = 25\%\:\:of\:\:\dfrac{1}{5}p \\[5ex] = \dfrac{25}{100} * \dfrac{1}{5}p \\[5ex] = \dfrac{5}{100}p \\[5ex] = \dfrac{1}{20}p \\[5ex] Remaining = Second\:\:child's\:\:share \\[3ex] = 100\% - (45\% + 25\%) \\[3ex] = 100\% - 70\% \\[3ex] = 30\% \\[3ex] Second\:\:child's\:\:share = 30\%\:\:of\:\:\dfrac{1}{5}p \\[5ex] = \dfrac{30}{100} * \dfrac{1}{5}p \\[5ex] = \dfrac{6}{100}p \\[5ex] = \dfrac{3}{50}p \\[5ex] Second\:\:child's\:\:share = ₦3,000 \\[3ex] \implies \dfrac{3}{50}p = 3000 \\[5ex] Multiply\:\:both\:\:sides\:\:by\:\:\dfrac{50}{3} \\[5ex] \dfrac{50}{3} * \dfrac{3}{50}p = \dfrac{50}{3} * 3000 \\[5ex] p = 50 * 1000 \\[3ex] p = ₦50,000 \\[3ex] $ The father's monthly income is $₦50,000$

Price for gas on January $1$ of Year $6$ was $3\%$ higher than the price on January $1$ of Year $5$.
This means that price for gas on January $1$ of Year $6$ is $3\%$ higher than $\$1.36$

$ 3\%\:\:of\:\:1.36 \\[3ex] = \dfrac{3}{100} * 1.36 \\[5ex] = 0.03(1.36) \\[3ex] = \$0.0408 \\[3ex] Price\:\:for\:\:gas\:\:on\:\:January\:1\:\:of\:\:Year\:6 \\[3ex] = 1.36 + 0.0408 \\[3ex] = 1.4008 \approx \$1.40 \\[3ex] $ The price for gas on January $1$ of Year $6$ was approximately $\$1.40$ higher than the price on January $1$ of Year $5$.

$ Final\:\:exam\:\:is\:\:worth\:\:20\%\:\:of\:\:the\:\:final\:\:grade \\[3ex] This\:\:implies\:\:that\:\:all\:\:other\:\:assessments\:\:besides\:\:the\:\:final\:\:exam\:\:are\:\:worth\:\:80\%\:\:(100\% - 20\%) \\[3ex] Philemon\:\:made\:\:an\:\:average\:\:of\:\:84\%\:\:on\:\:all\:\:those\:\:assessments \\[3ex] (a.) \\[3ex] \underline{All\:\:assessments\:\:besides\:\:the\:\:final\:\:exam} \\[3ex] \bar{x} = mean\:\:or\:\:average \\[3ex] \bar{x} = 84\% \\[3ex] weights = 80\% \\[3ex] \bar{x} = \dfrac{\Sigma weighted\:\:scores}{weights} \\[5ex] \Sigma weighted\:\:scores = \bar{x} * weights \\[3ex] \Sigma weighted\:\:scores = 84 * 80 = 6720 \\[3ex] \underline{Final\:\:exam} \\[3ex] weight = 20\% \\[3ex] To\:\:find\:\:the\:\:best\:\:course\:\:final\:\:grade,\:\:we\:\:should\:\:assume\:\:he\:\:makes\:\:a\:\:100\%\:\:on\:\:the\:\:final\:\:exam \\[3ex] score = 100\% \\[3ex] weighted\:\:score = 20 * 100 = 2000 \\[3ex] \underline{Course\:\:final\:\:grade:\:\:All\:\:assessments\:\:including\:\:the\:\:final\:\:exam} \\[3ex] Final\:\:grade = \dfrac{\Sigma weighted\:\:scores}{\Sigma weights} \\[5ex] Final\:\:grade = \dfrac{6720 + 2000}{80 + 20} \\[5ex] Final\:\:grade = \dfrac{8720}{100} \\[5ex] Final\:\:grade = 87.20\% \\[3ex] The\:\:best\:\:final\:\:score\:\:he\:\:can\:\:earn\:\:is\:\:87.20\% \\[3ex] The\:\:best\:\:final\:\:grade\:\:he\:\:can\:\:earn\:\:is\:\:a\:\:B \\[3ex] (b.) \\[3ex] (i) \\[3ex] He\:\:cannot\:\:make\:\:an\:\:A \\[3ex] (ii) \\[3ex] He\:\:can\:\:make\:\:a\:\:B \\[3ex] \underline{To\:\:determine\:\:the\:\:minimum\:\:score\:\:on\:\:the\:\:final\:\:exam\:\:in\:\:order\:\:to\:\:make\:\:a\:\:B} \\[3ex] Let\:\:the\:\:minimum\:\:score = p \\[3ex] weight = 20\% \\[3ex] score = p\% \\[3ex] weighted\:\:score = 20 * p = 20p \\[3ex] For\:\:a\:\:B, \\[3ex] Final\:\:grade = 80\% \\[3ex] Final\:\:grade = \dfrac{\Sigma weighted\:\:scores}{\Sigma weights} \\[5ex] 80 = \dfrac{6720 + 20p}{80 + 20} \\[5ex] 80 = \dfrac{6720 + 20p}{100} \\[5ex] 6720 + 20p = 80(100) \\[3ex] 6720 + 20p = 8000 \\[3ex] 20p = 8000 - 6720 \\[3ex] 20p = 1280 \\[3ex] p = \dfrac{1280}{20} \\[5ex] p = 64\% \\[3ex] The\:\:minimum\:\:score\:\:on\:\:the\:\:final\:\:exam\:\:needed\:\:for\:\:Philemon\:\:to\:\:make\:\:a\:\:B\:\:in\:\:the\:\:course\:\:is\:\:64\% $

$ \underline{Grade\;\;4} \\[3ex] regular\;\;enrollment\;\;fee = \$400 \\[3ex] 20\%\;\;of\;\;400 \\[3ex] = \dfrac{20}{100} * 400 \\[5ex] = \$80 \\[3ex] 20\%\;\;off = 400 - 80 = \$320 \\[3ex] \underline{Grade\;\;6} \\[3ex] regular\;\;enrollment\;\;fee = \$400 \\[3ex] 20\%\;\;of\;\;500 \\[3ex] = \dfrac{20}{100} * 500 \\[5ex] = \$100 \\[3ex] 20\%\;\;off = 500 - 100 = \$400 \\[3ex] Total\;\;amount \\[3ex] = 320 + 400 \\[3ex] = \$720 $

We do not know the final semester grade.
Let us assume it is 100

$ Homework\;\;averages = 30\%\;\;of\;\;100 \\[3ex] = 0.3(100) \\[3ex] = 30 \\[3ex] Test\;\;averages = 70\%\;\;of\;\;100 \\[3ex] = 0.7(100) \\[3ex] = 70 \\[3ex] Final\;\;exam = 20\%\;\;of\;\;70 \\[3ex] = 0.2(70) \\[3ex] = 14 \\[3ex] $ So, this is the question: what % of 100 is 14?
It is 14%
How?

$ \dfrac{is}{of} = \dfrac{\%}{100} ...Percent-Proportion \\[5ex] Let\;\;the\;\;\% = p \\[3ex] \dfrac{14}{100} = \dfrac{p}{100} \\[5ex] 14 = p \\[3ex] p = 14\% \\[3ex] $ Student: What if the final semester grade is another number besides 100?
Teacher: We shall still get the same answer.
I used 100 because it is much easier to work with.
What number would you like it to be?
Student: Assume it is 90
Teacher: Let us check out 90

Assume the final semester grade is 90

$ Homework\;\;averages = 30\%\;\;of\;\;90 \\[3ex] = 0.3(90) \\[3ex] = 27 \\[3ex] Test\;\;averages = 70\%\;\;of\;\;90 \\[3ex] = 0.7(90) \\[3ex] = 63 \\[3ex] Final\;\;exam = 20\%\;\;of\;\;63 \\[3ex] = 0.2(63) \\[3ex] = 12.6 \\[3ex] $ So, this is the question: what % of 90 is 12.6?

$ \dfrac{is}{of} = \dfrac{\%}{100} ...Percent-Proportion \\[5ex] Let\;\;the\;\;\% = p \\[3ex] \dfrac{12.6}{90} = \dfrac{p}{100} \\[5ex] 90(p) = 12.6(100) \\[3ex] p = \dfrac{12.6(100)}{90} \\[5ex] p = 14\% $

The World View Enterprise ride is planning to charge what percent of the charge for a Virgin Galactic flight?
In other words, $75000 is what percent of $250000
Let the percent = p

$ \dfrac{is}{of} = \dfrac{\%}{100} ...Percent\;\;Proportion \\[5ex] \dfrac{75000}{250000} = \dfrac{p}{100} \\[5ex] \dfrac{75}{250} = \dfrac{p}{100} \\[5ex] 250 * p = 75 * 100 \\[3ex] p = \dfrac{75 * 100}{250} \\[5ex] p = 30\% \\[3ex] $ The World View Enterprise ride is planning to charge thirty percent of the charge for a Virgin Galactic flight.

$ 8/24:\;\; initial = 8600 \\[3ex] 9/30:\;\; new = 7,630 \\[3ex] change = new - initial \\[3ex] change = 7630 - 8600 = -970 \\[3ex] The\;\;change\;\;is\;\;a\;\;decrease \\[3ex] \%\;decrease = \dfrac{change}{initial} * 100 \\[5ex] \%\;decrease \\[3ex] = \dfrac{970}{8600} * 100 \\[5ex] = 11.27906977\% \\[3ex] \approx 11.3\% $

In other words, which of the options best explains why the decline from the August 24 closing value to the September 30 closing value was relatively large (-970)?
All of the options are correct. However, which option is the most correct?
As observed and also written, there are more declines than advances.
But, the main reason for the large decline is because the average of the declines is much greater than the average of the advances.

Ask students to verify that statement...the last option...Option K. by calculating the:
(i.) average of the declines
(ii.) average of the advances

$ 9/13:\;\; 7945 \\[3ex] 9/14:\;\; 8020 \\[3ex] 9/15:\;\; 8090 \\[3ex] 9/16:\;\; 7870 \\[3ex] 9/17:\;\; 7895 \\[3ex] Average\;\;closing\;\;value\;\;for\;\;the\;\;5-day\;\;period \\[3ex] = \dfrac{7945 + 8020 + 8090 + 7870 + 7895}{5} \\[5ex] = \dfrac{39820}{5} \\[5ex] = 7964 $

$ marked\;\;price = p \\[3ex] 30\%\;\;of\;\;marked\;\;price = \dfrac{30}{100} * p = 0.3p \\[5ex] 30\%\;\;off\;\;marked\;\;price = p - 0.3p \\[3ex] \implies \\[3ex] sale\;\;price = p - 0.3p $

$ \underline{Q\:\:to\:\:R} \\[3ex] \%Loss = 5\% = \dfrac{5}{100} \\[5ex] Selling\:\:Price = 209 \\[3ex] Cost\:\:Price = C \\[3ex] \%Loss = \dfrac{Cost\:\:Price - Selling\:\:Price}{Cost\:\:Price} \\[5ex] \dfrac{5}{100} = \dfrac{C - 209}{C} \\[5ex] Cross\:\:Multiply \\[3ex] 5C = 100(C - 209) \\[3ex] 5C = 100C - 20900 \\[3ex] 20900 = 100C - 5C \\[3ex] 20900 = 95C \\[3ex] 95C = 20900 \\[3ex] C = \dfrac{20900}{95} \\[5ex] C = 220 \\[3ex] $ $Q$ bought the bicycle at the cost price of $₦220.00$
This cost price is the price that $P$ sold it to $Q$
This means that:
The cost price that $Q$ bought the bicycle is the selling price that $P$ sold the bicycle

$ \underline{P\:\:to\:\:Q} \\[3ex] \%Profit = 10\% = \dfrac{10}{100} \\[5ex] Selling\:\:Price = 209 \\[3ex] Cost\:\:Price = C \\[3ex] \%Profit = \dfrac{Selling\:\:Price - Cost\:\:Price}{Cost\:\:Price} \\[5ex] \dfrac{10}{100} = \dfrac{220 - C}{C} \\[5ex] 10C = 100(220 - C) \\[3ex] 10C = 22000 - 100C \\[3ex] 10C + 100C = 22000 \\[3ex] 110C = 22000 \\[3ex] C = \dfrac{22000}{110} \\[5ex] C = 200 \\[3ex] $ $P$ bought the bicycle at the cost price of $₦200.00$

$ federal\;\;debt = \$17\;trillion \\[3ex] (a.) \\[3ex] paid\;\;off = 15\%\;\;of\;\;\$17\;trillion \\[3ex] = 0.15(17\;trillion) \\[3ex] = \$2.55\;trillion \\[3ex] (b.) \\[3ex] balance = 17\;trillion - 2.55\;trillion \\[3ex] = \$14.45\;trillion $

Let us complete the two-way table

	Men	Women	Total
Yes	36	27	63
No	7	12	19
	43	39	82

(a.) What percentage of the men said Yes?
of = how many men do we have?
is = how many men said Yes?
% = what we are trying to find

$ \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] is = 36 \\[3ex] of = 43 \\[3ex] \dfrac{36}{43} = \dfrac{what}{100} \\[5ex] 43 * what = 36 * 100 \\[3ex] what = \dfrac{36 * 100}{43} \\[5ex] what = 83.72093023\% \\[3ex] what \approx 83.7\% \\[3ex] $ (b.) What percentage of the women said Yes?
of = how many women do we have?
is = how many women said Yes?
% = what we are trying to find

$ \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] is = 27 \\[3ex] of = 39 \\[3ex] \dfrac{27}{39} = \dfrac{what}{100} \\[5ex] 39 * what = 27 * 100 \\[3ex] what = \dfrac{27 * 100}{39} \\[5ex] what = 69.23076923\% \\[3ex] what \approx 69.2\% \\[3ex] $ (c.) What percentage of the people who said Yes were men?
of = how many people said Yes?
is = how many men said Yes?
% = what we are trying to find

$ \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] is = 36 \\[3ex] of = 63 \\[3ex] \dfrac{36}{63} = \dfrac{what}{100} \\[5ex] 63 * what = 36 * 100 \\[3ex] what = \dfrac{36 * 100}{63} \\[5ex] what = 57.14285714\% \\[3ex] what \approx 57.1\% \\[3ex] $ (d.) If a large group of 350 men had the same rate of responses as the men in this sample, how many men of the 350 would say Yes?
This could be reworded as:
Out of 43 men, 36 men said Yes
Out of 350 men, how many would say Yes?

Proportional Reasoning Method

$ \dfrac{43}{350} = \dfrac{36}{number} \\[5ex] 43 * number = 350 * 36 \\[3ex] number = \dfrac{350 * 36}{43} \\[5ex] number = 293.0232558 \\[3ex] number \approx 294 \\[3ex] $ If a large group of 350 men had the same rate of responses as the men in this sample, about 294 men of the 350 would say Yes.

Student: I thought the answer should be 293
Rounding the exact value to the nearest integer gives 293, not 294
Teacher: If the thing in question is an inanimate object, we can "round normal" to the nearest integer.
However, when we deal with animate objects such as human beings, we always "round up" to the nearest integer.

$ (I.) \\[3ex] Initial\;\;price = \$180 \\[3ex] 200\%\;\;of\;\;180 \\[3ex] = \dfrac{200}{100} * 180 \\[5ex] = \$360 \\[3ex] Current\;\;price = 200\%\;\;increase \\[3ex] = \$360 + \$180 \\[3ex] = \$540 \\[3ex] $ The currentr price is $540.00

$ (II.) \\[3ex] 36\;\;is\;\;what\;\;\%\;\;of\;\;30? \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] \dfrac{36}{30} = \dfrac{what}{100} \\[5ex] 30 * what = 36(100) \\[3ex] what = \dfrac{3600}{30} \\[3ex] what = 120 \\[3ex] $ 36 is 120% of 30

$ (III.) \\[3ex] \% increase = 50\% \\[3ex] New\;\;value = 30\% \\[3ex] Initial\;\;value = p\% \\[3ex] Everything\;\;is\;\;in\;\;\% \\[3ex] \%\:\: of\:\: Increase = \dfrac{New - Initial}{Initial} * 100 \\[5ex] 50 = \dfrac{30 - p}{p} * 100 \\[5ex] \dfrac{30 - p}{p} * 100 = 50 \\[5ex] \dfrac{30 - p}{p} = \dfrac{50}{100} \\[5ex] \dfrac{30 - p}{p} = \dfrac{1}{2} \\[5ex] p(1) = 2(30 - p) \\[3ex] p = 60 - 2p \\[3ex] p + 2p = 60 \\[3ex] 3p = 60 \\[3ex] p = \dfrac{60}{3} \\[5ex] p = 20 \\[3ex] $ The interest rate a decade ago was 20​% because​ 50% of 20 is 10​, and 20 + 10 = 30.

(IV.) We can solve this question using at least two approaches.
Use any approach you prefer.

$ 10\%\;\;per\;\;year \\[3ex] = \dfrac{10}{100}\;\;per\;\;year \\[5ex] = 0.1\;\;per\;\;year \\[3ex] Salary\;\;per\;\;month = \$600 \\[3ex] Salary\;\;per\;\;year = 600(12) = \$7200 \\[5ex] \underline{1st\;\;Approach:\;\;Quantitative\;\;Reasoning} \\[3ex] \underline{1st\;\;Year} \\[3ex] 10\%\;\;of\;\;7200 = 0.1(7200) = 720 \\[3ex] 10\%\;\;increase \\[3ex] = 7200 + 720 \\[3ex] = 7920 \\[3ex] Salary = \$7920.00 \\[3ex] \underline{2nd\;\;Year} \\[3ex] 10\%\;\;of\;\;7920 = 0.1(7920) = 792 \\[3ex] 10\%\;\;increase \\[3ex] = 7920 + 792 \\[3ex] = 8712 \\[3ex] Salary = \$8712.00 \\[3ex] \underline{3rd\;\;Year} \\[3ex] 10\%\;\;of\;\;8712 = 0.1(8712) = 871.2 \\[3ex] 10\%\;\;increase \\[3ex] = 8712 + 871.2 \\[3ex] = 9583.2 \\[3ex] Salary = \$9583.20 \\[3ex] \underline{4th\;\;Year} \\[3ex] 10\%\;\;of\;\;9583.2 = 0.1(9583.2) = 958.32 \\[3ex] 10\%\;\;increase \\[3ex] = 9583.2 + 958.32 \\[3ex] = 10541.52 \\[3ex] Salary = \$10541.52 \\[3ex] \underline{5th\;\;Year} \\[3ex] 10\%\;\;of\;\;10541.52 = 0.1(10541.52) = 1054.152 \\[3ex] 10\%\;\;increase \\[3ex] = 10541.52 + 1054.152 \\[3ex] = 11595.672 \\[3ex] Salary \approx \$11595.67 \\[5ex] \underline{2nd\;\;Approach:\;\;Financial\;\;Mathematics} \\[3ex] Compound\;\;Interest\;\;Formula \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] P = \$7200 \\[3ex] r = 10\% = 0.1 \\[3ex] Compounded\;\;yearly \implies m = 1 \\[3ex] t = 5\;years \\[3ex] A = ? \\[3ex] \implies \\[3ex] A = 7200 * \left(1 + \dfrac{0.1}{1}\right)^{1 * 5} \\[5ex] = 7200(1 + 0.1)^5 \\[3ex] = 7200(1.1)^5 \\[3ex] = 7200(1.61051) \\[3ex] = 11595.672 \\[3ex] \approx \$11595.67 \\[3ex] $ In 5 years, you would expect to earn ≈ $11595.67 per year.

$ \$11595.672\;\;per\;\;year \\[3ex] = \dfrac{11595.672}{12} \;\;per\;\;month \\[5ex] = 966.306 \\[3ex] \approx \$966.31 \\[3ex] $ In 5 years, you would expect to earn ≈ $966.31 per month.

Let that minimum score be: $p$
Let us represent this information in a table
You may do it any way you prefer.

Assessment	Weight (%)	Score	Weighted Score
Test 1	20	78	1560
Test 2	20	86	1720
Test 3	20	82	1640
Final Exam	40	$p$	$40p$
$\Sigma Weights = 100$		$\Sigma Weighted\;\;Scores = 4920 + 40p$

$ \dfrac{\Sigma Weighted\;\;Scores}{\Sigma Weights} = Course\;\;Grade \\[5ex] At\;\;least\;\;86 \;\;means\;\; \ge 86 \\[3ex] \implies \\[3ex] \dfrac{4920 + 40p}{100} \ge 86 \\[5ex] 4920 + 40p \ge 86(100) \\[3ex] 4920 + 40p \ge 8600 \\[3ex] 40p \ge 8600 - 4920 \\[3ex] 40p \ge 3680 \\[3ex] p \ge \dfrac{3680}{40} \\[5ex] p \ge 92 \\[3ex] $ Sani will have to earn 92% on the final exam in order to receive a course grade of at least 86.

$ (a.) \\[3ex] \Sigma students = 16 + 22 = 38 \\[3ex] n(men) = 16 \\[3ex] \%(men) = \dfrac{n(men)}{\Sigma students} * 100 \\[5ex] = \dfrac{16}{38} * 100 \\[5ex] = 42.10526316 \\[3ex] \approx 42.1\% \\[5ex] (b.) \\[3ex] n(men) = \%(men) * \Sigma students \\[3ex] n(men) = 54.9\% * 235 \\[3ex] = \dfrac{54.9}{100} * 235 \\[5ex] = 129.015 \\[3ex] \approx 130\;men \\[5ex] (c.) \\[3ex] \%(women) * \Sigma students = n(women) \\[3ex] 66\% * \Sigma students = 16 \\[3ex] \dfrac{66}{100} * \Sigma students = 16 \\[3ex] 0.66 * \Sigma students = 16 \\[3ex] \Sigma students = \dfrac{16}{0.66} \\[5ex] = 24.24242424 \\[3ex] \approx 25\;students $

$ (a.) \\[3ex] n(male\;\;nurses) = \%(male\;\;nurses) * \Sigma nurses \\[3ex] = 33\% * 336 \\[3ex] = \dfrac{33}{100} * 336 \\[5ex] = 110.88 \\[3ex] \approx 111\;male\;\;nurses \\[5ex] (b.) \\[3ex] n(female\;\;engineers) + n(male\;\;engineers) = \Sigma engineers \\[3ex] n(female\;\;engineers) + 106 = 175 \\[3ex] n(female\;\;engineers) = 175 - 106 \\[3ex] n(female\;\;engineers) = 69 \\[3ex] \%\;female\;\;engineers = \dfrac{n(female\;\;engineers)}{\Sigma engineers} * 100 \\[5ex] = \dfrac{69}{175} * 100 \\[5ex] = 39.42857143 \\[3ex] \approx 39.4\% \\[5ex] (c.) \\[3ex] \%(male\;\;lawyers) * \Sigma lawyers = n(male\;\;lawyers) \\[3ex] 60\% * \Sigma lawyers = 174 \\[3ex] \dfrac{60}{100} * \Sigma lawyers = 174 \\[3ex] 0.6 * \Sigma lawyers = 174 \\[3ex] \Sigma lawyers = \dfrac{174}{0.6} \\[5ex] = 290\;\;lawyers $

C. The house decreased in value over the past five years because the relative change is negative.

The percent of adults from the sample that said they spend an average of less than $100 each week on food are:
(1.) the percent of adults from the sample that spend no less than $50 and no more than $90
plus
(2.) the percent of adults from the sample that spend less than $50

This is: 17% + 8% = 25%
Therefore, the approximate number of adults from the sample that said they spend an average of less than $100 each week on food
= 25% of 1014
= 0.25(1014)

At least $150 means $150 or more
The percent of adults from the sample responded that they spend, on average, at least $150 each week on food are:
(1.) the percent of adults from the sample that spend no less than $150 and no more than $199
plus
(2.) the percent of adults from the sample that spend no less than $200 and no more than $299
plus
(3.) the percent of adults from the sample that spend $300 or more

This is = 15% + 21% + 10%
= 46%

$ n(Men) = 8 + 13 + 7 + 2 = 30 \\[3ex] n(Women) = 3 + 3 + 7 + 7 = 20 \\[3ex] n(People) = n(Men) + n(Women) \\[3ex] = 30 + 20 \\[3ex] = 50 \\[3ex] (a.) \\[3ex] n(Men\;\;with\;\;persuasion) = 8 + 7 = 15 \\[3ex] n(Women\;\;with\;\;persuasion) = 3 + 7 = 10 \\[3ex] n(People\;\;with\;\;persuasion) = 15 + 10 = 25 \\[3ex] n(Men\;\;with\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 8 \\[3ex] n(Women\;\;with\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 3 \\[3ex] n(People\;\;with\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 8 + 3 = 11 \\[3ex] \%\;\;of\;\;People\;\;with\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment \\[3ex] = \dfrac{n(People\;\;with\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment)}{n(People\;\;with\;\;persuasion)} * 100 \\[5ex] = \dfrac{11}{25} * 100 \\[5ex] = 44\% \\[3ex] (b.) \\[3ex] n(Men\;\;without\;\;persuasion) = 13 + 2 = 15 \\[3ex] n(Women\;\;without\;\;persuasion) = 3 + 7 = 10 \\[3ex] n(People\;\;without\;\;persuasion) = 15 + 10 = 25 \\[3ex] n(Men\;\;without\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 13 \\[3ex] n(Women\;\;without\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 3 \\[3ex] n(People\;\;without\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment) = 13 + 3 = 16 \\[3ex] \%\;\;of\;\;People\;\;without\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment \\[3ex] = \dfrac{n(People\;\;without\;\;persuasion\;\;who\;\;support\;\;capital\;\;punishment)}{n(People\;\;without\;\;persuasion)} * 100 \\[5ex] = \dfrac{16}{25} * 100 \\[5ex] = 64\% \\[3ex] $ (c.)
A. The student spoke against capital​ punishment, and fewer who heard her statements against it supported capital​ punishment, compared to those who did not hear the​ student's persuasion.

(I.) The statement makes sense.
A 200​% increase in price corresponds to a price that is 300​% of the original price.
Let's verify
Let the price of tuition when John's parents went to school = $100

$ price = \$100 \\[3ex] 200\% = \dfrac{200}{100} = 2 \\[5ex] 200\%\;\;of\;\;100 = 2(100) = 200 \\[3ex] 200\%\;\;increase\;\;in\;\;price = 100 + 200 = \$300 \\[3ex] = \\[3ex] Triple\;\;the\;\;price = 3(100) = \$300 \\[3ex] $ (II.) Let's verify
Let the amount that Mary earns = $100

$ \underline{John's\;\;earnings} \\[3ex] Mary's\;\;earnings = \$100 \\[3ex] 20\% = \dfrac{20}{100} = 0.2 \\[5ex] 20\%\;\;of\;\;100 = 0.2(100) = 20 \\[3ex] 20\%\;\;more = 100 + 20 = \$120 \\[5ex] \underline{Mary's\;\;earnings} \\[3ex] John's\;\;earnings = \$120 \\[3ex] 20\% = \dfrac{20}{100} = 0.2 \\[5ex] 20\%\;\;of\;\;120 = 0.2(120) = 24 \\[3ex] 20\%\;\;less = 120 - 24 = \$96 \\[5ex] \$96 \ne \$100 \\[3ex] $ This statement is not true.
But, what if we want to find how percent less than John's earnings is earned by Mary?
We have to: first: reword the problem as:
What percent of John's earnings is Mary's earnings?
What percent of 120 is 100?

$ \dfrac{is}{of} = \dfrac{what\%}{100} \\[5ex] \dfrac{100}{120} = \dfrac{what\%}{100} \\[5ex] 120 * what = 100 * 100 \\[3ex] what = \dfrac{10000}{120} \\[5ex] what = 83.33333333\% \\[3ex] $ Second: How many percent less of John's earnings is Mary's earnings?

$ 100\% - 83.33333333\% \\[3ex] = 16.66666667\% \\[3ex] \approx 16.7\% \\[3ex] $ The statement does not make sense because if John earns​ 20% more than​ Mary, then Mary must earn approximately​ 16.7% less than John does.

(III.) The statement makes sense.
A percentage change in population can be negative if the data reflects that.

(IV.) The statement does not make sense.
His new caloric intake would be −18​% of his previous​ intake, and a positive caloric intake is necessary for survival.

(V.) Let's verify

$ initial = 1.8\% \\[3ex] new = 3.6\% \\[3ex] \%\;\;increase = \dfrac{new - initial}{initial} * 100 \\[5ex] = \dfrac{3.6 - 1.8}{1.8} * 100 \\[5ex] = 100\% \\[3ex] $ The statement makes sense because if the interest rate increased by​ 100%, to 3.6​%, then the previous rate was 1.8​%, which makes sense.

(VI.) Let's verify
Let the number of cancers found in the general population = 1

$ number = 1 \\[3ex] 700\% = \dfrac{700}{100} = 7 \\[5ex] 700\%\;\;of\;\;1 = 7(1) = 7 \\[3ex] 700\%\;\;more = 1 + 7 = 8 \\[3ex] $ The statement makes sense because the number of rare cancers found in children living near the toxic landfill is 8 times the number of rare cancers found in the general​ population, which is possible.

(VII.) Let's verify
Verify by using the 2nd Approach: Compound Interest Formula.
However, ask students to verify by using the 1st Approach: Quantitative Reasoning.
Let the income = $100

$ \underline{2nd\;\;Approach:\;\;Financial\;\;Mathematics} \\[3ex] Compound\;\;Interest\;\;Formula \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] P = \$100 \\[3ex] r = 10\% = 0.1 \\[3ex] Compounded\;\;yearly \implies m = 1 \\[3ex] t = 10\;years \\[3ex] A = ? \\[3ex] \implies \\[3ex] A = 100 * \left(1 + \dfrac{0.1}{1}\right)^{1 * 10} \\[5ex] = 100(1 + 0.1)^{10} \\[3ex] = 100(1.1)^{10} \\[3ex] = 100(2.59374246) \\[3ex] = 259.374246 \\[3ex] \approx \$259.37 \\[3ex] $ What percent of 100 is 259.374246?

$ \dfrac{is}{of} = \dfrac{what\%}{100} \\[5ex] \dfrac{259.374246}{100} = \dfrac{what\%}{100} \\[5ex] what = 259.374246\% \\[3ex] 259.374246\% \ne 100\% \\[3ex] $ The statement does not make sense because each year there is a different reference value.

(VIII.) Let's verify
Let the initial value = p%

$ \%\;increase = 50\% \\[3ex] new\;\;value = 15\% \\[3ex] initial\;\;value = p\% \\[3ex] \%\;increase = \dfrac{new - initial}{initial} * 100 \\[5ex] 50 = \dfrac{15 - p}{p} * 100 \\[5ex] \dfrac{50}{100} = \dfrac{15 - p}{p} \\[5ex] \dfrac{15 - p}{p} = \dfrac{1}{2} \\[5ex] p(1) = 2(15 - p) \\[3ex] p = 30 - 2p \\[3ex] p + 2p = 30 \\[3ex] 3p = 30 \\[3ex] p = \dfrac{30}{3} \\[5ex] p = 10\% \\[3ex] $ The statement makes sense because if the rate of return on our fund increased by​ 50%, to​ 15%, then the previous rate is​ 10%, which makes sense.

$ 30\% = \dfrac{30}{100} = 0.3 \\[5ex] 40\% = \dfrac{40}{100} = 0.4 \\[5ex] \underline{Female\;\;students} \\[3ex] Number = 50 \\[3ex] Live\;\;in\;\major\;\;city = 30\%\;\;of\;\;50 \\[3ex] = 0.3(50) \\[3ex] = 15 \\[5ex] \underline{Male\;\;students} \\[3ex] Number = 50 \\[3ex] Live\;\;in\;\major\;\;city = 30\%\;\;of\;\;50 \\[3ex] = 0.4(30) \\[3ex] = 12 \\[5ex] \underline{Both\;\;Female\;\;students\;\;and\;\;Male\;\;students} \\[3ex] Number = 50 + 30 = 80 \\[3ex] Live\;\;in\;\major\;\;city = 15 + 12 = 27 \\[3ex] What\;\;percent\;\;live\;\;in\;\;major\;\;city? \\[3ex] What\;\;\%\;\;of\;\;80\;\;is\;\;27? \\[3ex] \dfrac{is}{of} = \dfrac{what\%}{100} \\[5ex] \dfrac{27}{80} = \dfrac{what\%}{100} \\[5ex] what = \dfrac{27(100)}{80} \\[5ex] what = 33.75\% \\[3ex] $ Suppose that​ 30% of the female students and​ 40% of the male students in a college history class are from a major city.
If there are 50 female students and 30 male students in the​ class, then 15 of the female students and 12 of the male student are from a major city.
Of the 80 students in the class in​ total, 33.75% are from a major city.
The percentage of students from a major city is not equal to the average of​ 30% and​ 40%, which is​ 35%, because the number of female and male students in the class differs.

As a general​ rule, percentages should never be averaged.
The only case in which the total percentage of students in the class from a major city would be​ 35% is if there were the same number of female and male students in the class.

$ 22\dfrac{1}{2} = \dfrac{2 * 22 + 1}{2} = \dfrac{44 + 1}{2} = \dfrac{45}{2} \\[5ex] 22\dfrac{1}{2}\% = \dfrac{22\dfrac{1}{2}\%}{100} \\[5ex] = \dfrac{\dfrac{45}{2}}{100} \\[5ex] = \dfrac{45}{2} \div \dfrac{100}{1} \\[5ex] = \dfrac{45}{2} * \dfrac{1}{100} \\[5ex] = \dfrac{9}{2} * \dfrac{1}{20} \\[5ex] = \dfrac{9 * 1}{2 * 20} \\[5ex] = \dfrac{9}{40} \\[5ex] 17\dfrac{1}{10} = \dfrac{10 * 17 + 1}{10} = \dfrac{170 + 1}{10} = \dfrac{171}{10} \\[5ex] 17\dfrac{1}{10}\% = \dfrac{17\dfrac{1}{10}\%}{100} \\[5ex] = \dfrac{\dfrac{171}{10}}{100} \\[5ex] = \dfrac{171}{10} \div \dfrac{100}{1} \\[5ex] = \dfrac{171}{10} * \dfrac{1}{100} \\[5ex] = \dfrac{171 * 1}{10 * 100} \\[5ex] = \dfrac{171}{1000} \\[5ex] \underline{Proportional\:\:Reasoning\:\:Method} \\[3ex] $

$M$	$₦$
$\dfrac{171}{1000}$	$\dfrac{9}{40}$
$1$	$p$

$ \dfrac{\dfrac{171}{1000}}{1} = \dfrac{\dfrac{9}{40}}{p} \\[7ex] Cross\:\:Multiply \\[3ex] \dfrac{171p}{1000} = 1\left(\dfrac{9}{40}\right) \\[5ex] \dfrac{171p}{1000} = \dfrac{9}{40} \\[5ex] \dfrac{1000}{171} * \dfrac{171p}{1000} = \dfrac{1000}{171} * \dfrac{9}{40} \\[5ex] p = \dfrac{1000}{171} * \dfrac{9}{40} \\[5ex] p = \dfrac{75}{57} \\[5ex] p = \dfrac{25}{19} \\[5ex] p = 1\dfrac{6}{19} \\[5ex] \therefore 1M = 1\dfrac{6}{19}₦ \\[5ex] Option\:\:C \\[3ex] 1\dfrac{18}{57} = 1\dfrac{6}{19} $

(a.)
Men that support capital punishment = Men who support capital punishment with persuasion + Men who support capital punishment without persuasion
= 4 + 12
= 16

Men against capital punishment = Men against capital punishment with persuasion + Men against capital punishment without persuasion
= 11 + 3
= 14

Women that support capital punishment = Women who support capital punishment with persuasion + Women who support capital punishment without persuasion
= 4 + 5
= 9

Women against capital punishment = Women against capital punishment with persuasion + Women against capital punishment without persuasion
= 6 + 5
= 11

	Men	Women
For capital punishment	16	9
Against capital punishment	14	11

$ (b.) \\[3ex] n(Men) = 16 + 14 = 30 \\[3ex] n(Men\;\;who\;\;support\;\;capital\;\;punishment) = 16 \\[3ex] \%(Men\;\;who\;\;support\;\;capital\;\;punishment) \\[3ex] = \dfrac{n(Men\;\;who\;\;support\;\;capital\;\;punishment)}{n(Men)} * 100 \\[5ex] = \dfrac{16}{30} * 100 \\[5ex] = 53.33333333 \\[3ex] \approx 53.3\% \\[3ex] (c.) \\[3ex] n(Women) = 9 + 11 = 20 \\[3ex] n(Women\;\;who\;\;support\;\;capital\;\;punishment) = 9 \\[3ex] \%(Women\;\;who\;\;support\;\;capital\;\;punishment) \\[3ex] = \dfrac{n(Women\;\;who\;\;support\;\;capital\;\;punishment)}{n(Women)} * 100 \\[5ex] = \dfrac{9}{20} * 100 \\[5ex] = 45\% \\[3ex] $ (d.) Based on these results, for any jury for murder trial, a jury of women would be preferable because a higher percentage of them (100 - 45% = 55%) are not in favor of capital punishment.

$ \% Error = \dfrac{Measured - Actual}{Actual} * 100 \\[5ex] \% Error = \dfrac{16.8 - 15}{15} * 100 \\[5ex] = \dfrac{1.8}{15} * 100 \\[5ex] = \dfrac{180}{15} \\[5ex] = \dfrac{60}{5} \\[5ex] = 12\% $

$ Let\:\:the\:\:number\:\:of\:\:goats = g \\[3ex] Cost\:\:Price = 4000 * g = 4000g \\[3ex] Selling\:\:Price = 180000 \\[3ex] \%Loss = 25\% = \dfrac{25}{100} = \dfrac{1}{4} \\[5ex] \%Loss = \dfrac{Cost\:\:Price - Selling\:\:Price}{Cost\:\:Price} \\[5ex] \rightarrow \dfrac{1}{4} = \dfrac{4000g - 180000}{4000g} \\[5ex] Cross\:\:Multiply \\[3ex] 4000g = 4(4000g - 180000) \\[3ex] 4000g = 16000g - 720000 \\[3ex] 720000 = 16000g - 4000g \\[3ex] 720000 = 12000g \\[3ex] 12000g = 720000 \\[3ex] g = \dfrac{720000}{12000} \\[5ex] g = 60 \\[3ex] $ He bought $60$ goats.

$ \underline{Check} \\[3ex] Cost\:\:price\:\:of\:\:one\:\:goats = ₦4000 \\[3ex] Cost\:\:price\:\:of\:\:all\:\:goats = 60 * 4000 = ₦240000 \\[3ex] 25\%\:\:of\:\:240000 \\[3ex] = \dfrac{25}{100} * 240000 \\[5ex] = 25 * 2400 \\[3ex] = 60000 \\[3ex] 25\%\:\:Loss \\[3ex] = 240000 - 60000 \\[3ex] = 180000 \\[3ex] Selling\:\:price = ₦180,000.00 $

$ \%Profit = \dfrac{Profit}{Cost\:\:Price} * 100 \\[5ex] \rightarrow \%Profit = \dfrac{Selling\:\:Price - Cost\:\:Price}{Cost\:\:Price} * 100 \\[5ex] Let\:\:Cost\:\:Price = C \\[3ex] Selling\:\:Price = 345.00 = 345 \\[3ex] \%Profit = 15\% = \dfrac{15}{100} \\[5ex] \rightarrow \dfrac{15}{100} = \dfrac{345 - C}{C} \\[5ex] Cross\:\:Multiply \\[3ex] 15C = 100(345 - C) \\[3ex] 15C = 34500 - 100C \\[3ex] 15C + 100C = 34500 \\[3ex] 115C = 34500 \\[3ex] C = \dfrac{34500}{115} \\[5ex] C = 300 \\[3ex] Profit = Selling\:\:Price - Cost\:\:Price \\[3ex] Profit = 345 - 300 \\[3ex] Profit = Le\:45.00 $

$ (a.) \\[3ex] \underline{1st\;\;Half} \\[3ex] Micah:\;\; \dfrac{30}{93} * 100 = 32.25806452\% \\[5ex] Nahum:\;\; \dfrac{5}{25} * 100 = 20\% \\[5ex] \underline{2nd\;\;Half} \\[3ex] Micah:\;\; \dfrac{19}{33} * 100 = 57.57575758\% \\[5ex] Nahum:\;\; \dfrac{44}{94} * 100 = 46.80851064\% \\[5ex] $ (b.) Micah had the higher free throw percentage in the first half of the season.

(c.) Micah had the higher free throw percentage in the second half of the season.

$ (d.) \\[3ex] \underline{Complete\;\;Season} \\[3ex] Micah: \\[3ex] Free\;\;Throws = 30 + 19 = 49 \\[3ex] Attempts = 93 + 33 = 126 \\[3ex] \dfrac{49}{126} * 100 = 38.88888889\% \\[5ex] Nahum \\[3ex] Free\;\;Throws = 5 + 44 = 49 \\[3ex] Attempts = 25 + 94 = 119 \\[3ex] \dfrac{49}{119} * 100 = 41.17647059\% \\[5ex] $ (e.) Nahum had the higher free throw percentage for the complete season.

(f.) Micah performed better in the first and second halves of the​ season, and yet Nahum outperformed Micah over the course of the entire season.
While both players took a similar number of shots​ overall, they took a different number of shots in each half of the season.

$ Let\:\:L = length\:\:of\:\:the\:\:Square \\[3ex] Let\:\:W = width\:\:of\:\:the\:\:Square \\[3ex] Area\:\:of\:\:the\:\:Square = L * W = LW \\[3ex] 20\% = \dfrac{20}{100} = 0.2 \\[3ex] 20\%\:\:increase\:\:in\:\:length = L + 0.2L = 1.2L \\[3ex] 20\%\:\:decrease\:\:in\:\:width = W - 0.2W = 0.8W \\[3ex] Area\:\:of\:\:the\:\:Rectangle = 1.2L * 0.8W = 0.96LW \\[3ex] Ratio = \dfrac{Area\:\:of\:\:the\:\:Rectangle}{Area\:\:of\:\:the\:\:Square} \\[5ex] Ratio = \dfrac{0.96LW}{LW} \\[5ex] = \dfrac{0.96}{1} \\[5ex] = 0.96 $

We can solve this question in two ways
Use any method you like

$ Let\:\:the\:\:amount\:\:of\:\:initial\:\:ice\:\:cream = C \\[3ex] \underline{First\:\:Method:\:\:Quantitative\:\:Reasoning - More\:\:Work} \\[3ex] 25\% = \dfrac{25}{100} = 0.25 \\[5ex] First\:\:Day:\:\:Eats\:\:25\%\:\:of\:\:C = 0.25 * C = 0.25C \\[3ex] Remaining:\:\: C - 0.25C = 0.75C \\[3ex] Second\:\:Day:\:\:Eats\:\:25\%\:\:of\:\:0.75C = 0.25 * 0.75C = 0.1875C \\[3ex] Remaining:\:\: 0.75C - 0.1875C = 0.5625C \\[3ex] Third\:\:Day:\:\:Eats\:\:25\%\:\:of\:\:0.5625C = 0.25 * 0.5625C = 0.140625C \\[3ex] Remaining:\:\: 0.5625C - 0.140625C = 0.421875C \\[3ex] Fourth\:\:Day:\:\:Eats\:\:25\%\:\:of\:\:0.421875C = 0.25 * 0.421875C = 0.10546875C \\[3ex] Remaining:\:\: 0.421875C - 0.10546875C = 0.31640625C \\[3ex] Fifth\:\:Day:\:\:Eats\:\:25\%\:\:of\:\:0.31640625C = 0.25 * 0.31640625C = 0.0791015625C \\[3ex] Remaining:\:\: 0.31640625C - 0.0791015625C = 0.237304688C \\[3ex] Remaining\:\:in\:\:\% = 0.237304688C * 100 = 23.7304688\%C \\[3ex] \approx 24\%C \\[4ex] \underline{Second\:\:Method:\:\:Declining\:\:Balance\:\:Method:\:\:Formula - Less\:\:Work} \\[3ex] V = N(1 - r)^t \\[4ex] N = 1 \\[3ex] r = 25\% = \dfrac{25}{100} = 0.25 \\[5ex] t = 5\:days \\[3ex] V = 1(1 - 0.25)^5 \\[4ex] V = 1(0.75)^5 \\[4ex] V = 0.237304688 \\[3ex] to\:\:\%: V = 0.237304688 * 100 \\[3ex] V = 23.7304688\% \\[3ex] V \approx 24\% $

$ Let\:\:his\:\:income = x \\[3ex] Tax-free = \dfrac{1}{8}th\:\:of\:\:x \\[5ex] = \dfrac{1}{8} * x \\[5ex] = \dfrac{1}{8}x \\[5ex] = \dfrac{1 * x}{8} \\[5ex] = \dfrac{x}{8} \\[5ex] Remainder = x - \dfrac{x}{8} \\[5ex] = \dfrac{8x}{8} - \dfrac{x}{8} \\[5ex] = \dfrac{8x - x}{8} \\[5ex] = \dfrac{7x}{8} \\[5ex] 20\%\:\:tax\:\:on\:\:Remainder \\[5ex] = \dfrac{20}{100} * \dfrac{7x}{8} \\[5ex] = \dfrac{1}{5} * \dfrac{7x}{8} \\[5ex] = \dfrac{1 * 7x}{5 * 8} \\[5ex] = \dfrac{7x}{40} \\[5ex] This\:\:is\:\:equal\:\:to\:\; 490 \\[3ex] \rightarrow \dfrac{7x}{40} = 490 \\[5ex] \dfrac{7x}{40} = \dfrac{490}{1} \\[5ex] Cross\:\:Multiply \\[3ex] 7x(1) = 40(490) \\[3ex] 7x = 40 * 490 \\[3ex] x = \dfrac{40 * 490}{7} \\[5ex] x = 40 * 70 \\[3ex] x = 2800 \\[3ex] x = ₦2,800.00 $

$ (a.) \\[3ex] Sticker\:\:Price = 900 \\[3ex] 30\%\:\:of\:\:Sticker\:\:Price = \dfrac{30}{100} * 900 = 30 * 9 = 270 \\[5ex] 30\%\:\:Increase = 900 + 270 = 1170 \\[3ex] New\:\:Sticker\:\:Price = \$1,170.00 \\[3ex] (b.) \\[3ex] 15\%\:\:of\:\:Sticker\:\:Price = \dfrac{15}{100} * 1170 = 175.5 \\[5ex] Discount = \$175.50 \\[3ex] (c.) \\[3ex] 15\%\:\:off = 1170 - 175.5 = 994.5 \\[3ex] New\:\:Sticker\:\:Price = \$994.50 \\[3ex] (d.) \\[3ex] Profit = Selling\:\:Price - Cost\:\:Price \\[3ex] = 994.50 - 900 \\[3ex] = 94.5 \\[3ex] Profit = \$94.50 \\[3ex] (e.) \\[3ex] \% Profit = \dfrac{Profit}{Cost\:\:Price} * 100 \\[5ex] = \dfrac{94.5}{900} * 100 \\[5ex] = \dfrac{94.5}{9} \\[5ex] = 10.5\% \\[3ex] Profit = 10.5\% $

We can do this question in two ways.
Use any method you prefer.

$ \underline{First\:\:Method:\:\:Arithmetically - Long\:\:Way} \\[3ex] \underline{2003} \\[3ex] Population = 3000 \\[3ex] \underline{Beginning\:\:of\:\:2003} \\[3ex] GH¢15.00\:\:per\:\:head = 3000(15) = GH¢45000 \\[3ex] 20\%\:\:increase\:\:during\:\:2003 \\[3ex] = 20\%\:\:of\:\:3000 \\[3ex] = \dfrac{20}{100} * 3000 \\[5ex] = 20(30) \\[3ex] = 600 \\[3ex] Population = 3000 + 600 = 3600 \\[3ex] \underline{Beginning\:\:of\:\:2004} \\[3ex] GH¢15.00\:\:per\:\:head = 3600(15) = GH¢54000 \\[3ex] 20\%\:\:increase\:\:during\:\:2004 \\[3ex] = 20\%\:\:of\:\:3600 \\[3ex] = \dfrac{20}{100} * 3600 \\[5ex] = 20(36) \\[3ex] = 720 \\[3ex] Population = 3600 + 720 = 4320 \\[3ex] \underline{Beginning\:\:of\:\:2005} \\[3ex] GH¢15.00\:\:per\:\:head = 4320(15) = GH¢64800 \\[3ex] 20\%\:\:increase\:\:during\:\:2005 \\[3ex] = 20\%\:\:of\:\:4320 \\[3ex] = \dfrac{20}{100} * 4320 \\[5ex] = 2(432) \\[3ex] = 864 \\[3ex] Population = 4320 + 864 = 5184 \\[3ex] \underline{Beginning\:\:of\:\:2006} \\[3ex] GH¢15.00\:\:per\:\:head = 5184(15) = GH¢77760 \\[3ex] 20\%\:\:increase\:\:during\:\:2006 \\[3ex] = 20\%\:\:of\:\:5184 \\[3ex] = \dfrac{20}{100} * 5184 \\[5ex] = 0.2(5184) \\[3ex] = 1036.8 \\[3ex] Population = 5184 + 1036.8 = 6220.8 \approx 6221...no\:\:decimal\:\:human\:\:being \\[3ex] \underline{Beginning\:\:of\:\:2007} \\[3ex] GH¢15.00\:\:per\:\:head = 6221(15) = GH¢93315 \\[3ex] Total\:\:Grant\:\:from\:\:2003\:\:to\:\:2007 \\[3ex] = 45000 + 54000 + 64800 + 77760 + 93315 \\[3ex] = GH¢334,875 \\[3ex] $