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 Chukwuemeka



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.) Benedict and Benedicta went to dine at La Casa restaurant.
They left a $12\%$ tip of their bill of $34.99$
How much was their tip?


$ Tip = 12\% \:\:of\:\: 34.99 \\[3ex] = \dfrac{12}{100} * 34.99 \\[3ex] = 4.1988 $
Tip = $$4.20$
(2.) The cost of a Blackberry Z10 phone at Emmanuel's Phones is $$220.98$.
This cost excludes a $7\%$ sales tax.
How much is the sales tax?


$ Sales\:\: tax\:\: = 7\% \:\:of\:\: 220.98 \\[3ex] = \dfrac{7}{100} * 220.98 \\[3ex] = 15.4686 $
Sales tax = $$15.47$
(3.) The cost of a Blackberry Z10 phone at Emmanuel's Phones is $$220.98$.
This cost includes a $7\%$ sales tax.
How much is the sales tax?


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 = $$14.46$
(4.) A gaming laptop at Ifa's Electronics is on sale for $9\%$ off.
The initial price of the laptop is $$960$
Calculate the discount and the sale price.


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 = $$873.60$
(5.) Expecting new arrivals, Ifa's Electronics laptops were marked down for $20\%$ off the initial price.
A customer purchases a laptop for $$588.00$. The cost excludes a $5\%$ sales tax.
Determine the initial price of the laptop.


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 = $$735.00$
(6.) Expecting new arrivals, Ifa's Electronics laptops were marked down for $20\%$ off the initial price.
A customer purchases a laptop for $$588.00$. The cost includes a $5\%$ sales tax.
Determine the initial price of the laptop.


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.) JAMB A car dealer bought a second-hand car for ₦250,000.00 and spent ₦70,000.00 refurbishing it. He then sold the car for ₦400,000.00. What is the percentage gain?

$ A.\:\: 60\% \\[3ex] B.\:\: 32\% \\[3ex] C.\:\: 25\% \\[3ex] D.\:\: 20\% \\[3ex] $

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\% $
(8.)ACT If the length of a rectangle is increased by $25\%$ and the width is decreased by $10\%$, the area of the resulting rectangle is larger than the area of the original rectangle by what percent?


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 $12.5\%$

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?
(9.) ACT A coat originally priced at $\$80$ is discounted to $\$60$.
What is the percent of discount on this coat?

$ A.\:\: 13\% \\[3ex] B.\:\: 20\% \\[3ex] C.\:\: 25\% \\[3ex] D.\:\: 30\% \\[3ex] E.\:\: 33\dfrac{1}{3}\% \\[5ex] $

$ 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\% $
(10.)


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 $12.5\%$

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?