Samuel Dominic Chukwuemeka (SamDom For Peace)

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka



Fractions and Percents

For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

Solve all questions.
Show all work.

(1.) ACT The ratio of Alani's height to Baahir's height is $5:7$
The ratio of Baahir's height to Connor's height is $4:3$
What is the ratio of Alani's height to Connor's height?


Let Alani's height = $A$
Let Baahir's height = $B$
Let Connor's height = $C$
$ \dfrac{A}{B} = \dfrac{5}{7} \\[5ex] \dfrac{B}{C} = \dfrac{4}{3} \\[5ex] \dfrac{A}{C} = \dfrac{A}{B} * \dfrac{B}{C} \\[5ex] = \dfrac{5}{7} * \dfrac{4}{3} \\[5ex] = \dfrac{20}{21} $
Student: Why did you multiply?
Teacher: The question asked for the ratio of Alani's height to Connor's height
That is $\dfrac{A}{C}$

$ \dfrac{A}{C} = \dfrac{A}{B} * \dfrac{B}{C} $
The $B$ cancels out.
(2.)


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


Let the initial cost of the phone = $p$
$ 7\%\:\: sales\:\: tax\:\: of\:\: p = 7\% * p \\[3ex] = \dfrac{7}{100} * p \\[3ex] = 0.07 * p \\[3ex] = 0.07p \\[3ex] $
The initial price of the phone and the tax equals $$220.98$
$ p + 0.07p = 220.98 \\[3ex] 1.07p = 220.98 \\[3ex] p = \dfrac{220.98}{1.07} \\[5ex] p = 206.5233645 \\[3ex] 7\%\:\: sales\:\: tax = 0.07p \\[3ex] = 0.07 * 206.5233645 \\[3ex] = 14.45663551 $
The initial cost of the phone = $$206.52$
The sales tax = $$14.46$
(4.)


The initial price of the laptop = $960$

$ Discount = 9\%\:\:of\:\: 960 \\[3ex] = 9\% * 960 \\[3ex] = 0.09 * 960 \\[3ex] = 86.4 $
The discount = $$86.40$

Sale price = Initial price - Discount

$ Sale\:\: price = 9\%\:\:off\:\: 960 \\[3ex] = 960 - (9\%\:\:of\:\: 960) \\[3ex] = 960 - 86.4 \\[3ex] = 873.60 $
The sale price = $$873.60$
(5.)


Selling price = $$588.00$
Let the initial price = $p$

$ Discount = 20\%\:\:of\:\: p \\[3ex] = 20\% * p \\[3ex] = 0.2 * p \\[3ex] = 0.2p $

Sale price = Initial price - Discount

$ Sale\:\: price = p - 0.2p \\[3ex] = 0.8p $

Tax is excluded.
Sale price = Selling price.

$ \therefore 0.8p = 588 \\[3ex] p = \dfrac{588}{0.8} \\[5ex] p = 735.00 $
The initial price = $$735.00$
(6.)


Selling price = $$588.00$
Let the initial price = $p$

$ Discount = 20\%\:\:of\:\: p \\[3ex] = 20\% * p \\[3ex] = 0.2 * p \\[3ex] = 0.2p $

Sale price = Initial price - Discount

$ Sale\:\: price = p - 0.2p \\[3ex] = 0.8p $

Tax is included.

$ 5\%\:\: sales\:\: tax\:\: of\:\: 0.8p \\[3ex] = 0.05 * 0.8p \\[3ex] = 0.04p \\[3ex] $

Sale price + Tax = Selling price

$ \therefore 0.8p + 0.04p = 588 \\[3ex] 0.84p = 588 \\[3ex] p = \dfrac{588}{0.84} \\[5ex] p = 700.00 $
The initial price = $$700.00$
(7.)


$s = ut + \dfrac{1}{2}at^2$
Newton's Laws of Motion - Physics/Mechanics

$s = ut + \dfrac{1}{2}at^2$

Swap. Let the LHS = RHS, and the RHS = LHS

$ ut + \dfrac{1}{2}at^2 = s \\[5ex] LCD = 2 $
Multiply both sides by $2$

$ 2 * \left(ut + \dfrac{1}{2}at^2\right) = 2 * s \\[5ex] 2 * ut + 2 * \dfrac{1}{2}at^2 = 2s \\[5ex] 2ut + at^2 = 2s $
Subtract $at^2$ from both sides

$2ut = 2s - at^2$

Divide both sides by $2t$

$ u = \dfrac{2s - at^2}{2t} $
(8.)


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 = $1.125LW - LW = 0.125LW$
$0.125LW = 12.5\%LW$
Area of resulting rectangle is larger than the Area of original rectangle by $12.5\%$