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.

For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

Solve all questions.
Use at least two methods (two or more methods) whenever applicable.
Check your work whenever applicable.
Show all work.

(2.) Write 12.5% as a simplified fraction.

$ 12.5\% \\[3ex] = \dfrac{12.5}{100} \\[5ex] = \dfrac{12.5 * 10}{100 * 10} \\[5ex] = \dfrac{125}{1000} \\[5ex] = \dfrac{125 \div 25}{1000 \div 25} \\[5ex] = \dfrac{5}{40} \\[5ex] = \dfrac{5 \div 5}{40 \div 5} \\[5ex] = \dfrac{1}{8} \\[5ex] \underline{Check} \\[3ex] \dfrac{12.5}{100} = 0.125 \\[5ex] \dfrac{1}{8} = 0.125 $

(3.) 160% is equivalent to what fraction?

$ 160\% \\[3ex] = \dfrac{160}{100} \\[5ex] = \dfrac{160 \div 10}{100 \div 10} \\[5ex] = \dfrac{16}{10} \\[5ex] = \dfrac{16 \div 2}{10 \div 2} \\[5ex] = \dfrac{8}{5} $

(4.) Convert $6\dfrac{2}{5}$ to a percent

$ 6\dfrac{2}{5} \\[5ex] = \dfrac{5 * 6 + 2}{5} \\[5ex] = \dfrac{30 + 2}{5} \\[5ex] = \dfrac{32}{5} \\[5ex] to\:\:percent \\[3ex] = \dfrac{32}{5} * 100 \\[5ex] = 32 * 20 \\[3ex] 640\% $

(5.) ACT If 115% of a number is 460, what is 75% of the number?

$ F.\:\: 280 \\[3ex] G.\:\: 300 \\[3ex] H.\:\: 320 \\[3ex] J.\:\: 345 \\[3ex] K.\:\: 400 \\[3ex] $

Let us use two different methods to solve each part of the question:
Let us use Percent-Equation to find the first part
Then, we use Percent-Proportion to find the second part.
Use whichever method you prefer for any of them

$ First\:\:Part: \\[3ex] 115\%\:\:of\:\:a\:\:number\:\:is\:\:460 \\[3ex] Let\:\:the\:\:number = c \\[3ex] \underline{Percent-Equation} \\[3ex] \dfrac{115}{100} * c = 460 \\[3ex] Multiply\:\;both\:\:sides\:\:by\:\:\dfrac{100}{115} \\[5ex] \dfrac{100}{115} * \dfrac{115}{100} * c = \dfrac{100}{115} * 460 \\[5ex] c = \dfrac{100 * 115}{460} \\[5ex] c = 400 \\[3ex] Second\:\:Part: \\[3ex] What\:\:is\:\:75\%\:\:of\:\:400? \\[3ex] \underline{Percent-Proportion} \\[3ex] \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] Let\:\:the\:\:what(variable) = d \\[3ex] \dfrac{d}{400} = \dfrac{75}{100} \\[5ex] Multiply\:\;both\:\:sides\:\:by\:\:400 \\[3ex] 400 * \dfrac{d}{400} = 400 * \dfrac{75}{100} \\[5ex] d = 4 * 75 \\[3ex] d = 300 \\[3ex] $ The number is $400$
The actual answer is $300$

$ \underline{Check} \\[3ex] 115\%\:\:of\:\:400 \\[3ex] = \dfrac{115}{100} * 400 \\[5ex] = 115 * 4 \\[3ex] = 460 \\[3ex] 75\%\:\:of\:\:400 \\[3ex] = \dfrac{75}{100} * 400 \\[5ex] = 75 * 4 \\[3ex] = 300 $

(6.) ACT The sum of 2 and 200% of 1 has the same value as which of the following calculations?

$ A.\:\: 100\%\:\:of\:\:2 \\[3ex] B.\:\: 150\%\:\:of\:\:2 \\[3ex] C.\:\: 300\%\:\:of\:\:2 \\[3ex] D.\:\: 300\%\:\:of\:\:1 \\[3ex] E.\:\: 400\%\:\:of\:\:1 \\[3ex] $

$ 200\%\:\:of\:\:1 \\[3ex] = \dfrac{200}{100} * 1 \\[5ex] = 2 * 1 \\[3ex] = 2 \\[3ex] Sum\:\:of\:\:2\:\:and\:\:200\%\:\:of\:\:1 \\[3ex] = 2 + 2 \\[3ex] = 4 \\[3ex] 400\%\:\:of\:\:1 \\[3ex] = \dfrac{400}{100} * 1 \\[5ex] = 4 * 1 \\[3ex] = 4 \\[3ex] Option\:\:E $

(7.) ACT What is 10% of 10% of 1?

$ A.\:\: 0.001 \\[3ex] B.\:\: 0.01 \\[3ex] C.\:\: 0.1 \\[3ex] D.\:\: 1.0 \\[3ex] E.\:\: 10.0 \\[3ex] $

Work backwards...one-by-one

$ 10\%\:\:of\:\:10\%\:\:1 \\[3ex] 10\%\:\:of\:\:1 \\[3ex] = \dfrac{10}{100} * 1 \\[3ex] = 0.1(1) \\[3ex] = 0.1 \\[3ex] 10\%\:\:of\:\:0.1 \\[3ex] = \dfrac{10}{100} * 0.1 \\[3ex] = 0.1(0.1) \\[3ex] = 0.01 $

(8.) ACT What is 4% of $1.36 * 10^4 ?$

$ F.\:\: 340 \\[3ex] G.\:\: 544 \\[3ex] H.\:\: 3,400 \\[3ex] J.\:\: 5,440 \\[3ex] K.\:\: 54,400 \\[3ex] $

$ 4\% \:\:of\:\: 1.36 * 10^4 \\[3ex] = \dfrac{4}{100} * 1.36 * 10000\\[5ex] = 4 * 1.36 * 100 \\[3ex] = 4 * 136 \\[3ex] = 544 $

(9.) ACT A calculator has a regular price of $59.95 before taxes.
It goes on sale at 20% below the regular price.
Before taxes are added, what is the sale price of the calculator?

$ A.\:\: \$11.99 \\[3ex] B.\:\: \$29.98 \\[3ex] C.\:\: \$39.95 \\[3ex] D.\:\: \$47.96 \\[3ex] E.\:\: \$54.95 \\[3ex] $

$ 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 $

(10.) ACT What is 3% of $5.13 * 10^5?$

$ A.\:\: 15,390 \\[3ex] B.\:\: 17,100 \\[3ex] C.\:\: 171,000 \\[3ex] D.\:\: 153,900 \\[3ex] E.\:\: 1,539,000 \\[3ex] $

$ 3\%\:\:of\:\:5.13 * 10^5 \\[3ex] = \dfrac{3}{100} * 5.13 * 100000 \\[5ex] = 3 * 5.13 * 1000 \\[3ex] = 3 * 5130 \\[3ex] = 15390 $

(11.) ACT A shirt has a sale price of $30.40, which is 20% off the original price.
How much less than the original price is the sale price?

$ A.\:\: \$0.38 \\[3ex] B.\:\: \$1.52 \\[3ex] C.\:\: \$6.08 \\[3ex] D.\:\: \$7.60 \\[3ex] E.\:\: \$10.40 \\[3ex] $

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

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

(12.) WASSCE A trader made a profit of 15% by selling an article for Le 345.00.
Calculate the actual profit.

$ A.\:\: Le\:300.00 \\[3ex] B.\:\: Le\:117.00 \\[3ex] C.\:\: Le\:51.75 \\[3ex] D.\:\: Le\:45.00 \\[3ex] $

$ \%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 $

(13.) JAMB A trader bought goats for $₦$4,0000 each.
He sold them for $₦$180,000 at a loss of 25%.
How many goats did he buy?

$ A.\:\: 50 \\[3ex] B.\:\: 60 \\[3ex] C.\:\: 36 \\[3ex] D.\:\: 45 \\[3ex] $

$ 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.

(14.) ACT 40% of 250 is equal to 60% of what number?

$ F.\;\; 150 \\[3ex] G.\;\; 160 \\[3ex] H.\;\; 166\dfrac{2}{3} \\[5ex] J.\;\; 270 \\[3ex] K.\;\; 375 \\[3ex] $

$ 40\%\;\;of\;\;250 \\[3ex] = \dfrac{40}{100} * 250 \\[5ex] = 0.4 * 250 \\[3ex] = 100 \\[3ex] 100 \;\;is\;\; 60\%\;\;of\;\;what\;\;number \\[3ex] 100 = \dfrac{60}{100} * number \\[5ex] 100 = \dfrac{6}{10} * number \\[5ex] 100 * 10 = 6 * number \\[3ex] 6 * number = 100 * 10 \\[3ex] number = \dfrac{100 * 10}{6} \\[5ex] number = \dfrac{100 * 5}{3} \\[5ex] number = \dfrac{500}{3} \\[5ex] number = 166\dfrac{2}{3} $

(15.) ACT Widely considered one of the greatest film directors, Alfred Hitchcock directed over 60 films.
The table below gives some information about Hitchcock's last 12 films.

Title	Year of release	Length(minutes)
The Trouble with Harry	$1955$	$99$
The Man Who Knew Too Much	$1956$	$120$
The Wrong Man	$1956$	$105$
Vertigo	$1958$	$128$
North by Northwest	$1959$	$136$
Psycho	$1960$	$109$
The Birds	$1963$	$119$
Marnie	$1964$	$130$
Torn Curtin	$1966$	$128$
Topaz	$1969$	$143$
Frenzy	$1972$	$?$
Family Plot	$1976$	$?$

Recently, a director made a new version of Vertigo.
The new version is $20\%$ shorter in length than Hitchcock's version.
Which of the following values is closest to the length, in minutes, of the new version?

$ F.\:\: 64 \\[3ex] G.\:\: 102 \\[3ex] H.\:\: 105 \\[3ex] J.\:\: 108 \\[3ex] K.\:\: 125 \\[3ex] $

$ 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.

(16.) JAMB P sold his bicycle to Q at a profit of 10%.
Q sold it to R for $₦$209 at a loss of 5%.
How much did the bicycle cost P?

$ A.\:\: ₦200 \\[3ex] B.\:\: ₦196 \\[3ex] C.\:\: ₦180 \\[3ex] D.\:\: ₦205 \\[3ex] E.\:\: ₦150 \\[3ex] $

$ \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 $₦200.00$

(17.) ACT The changes in a city's population from one decade to the next decade for 3 consecutive decades were a 20\% increase, a 30% increase, and a 20% decrease.
About what percent was the increase in the city's population over the 3 decades?

$ A.\:\: 10\% \\[3ex] B.\:\: 20\% \\[3ex] C.\:\: 25\% \\[3ex] D.\:\: 30\% \\[3ex] E.\:\: 70\% \\[3ex] $

Please be careful of this common mistake!

$ 20\% + 30\% - 20\% = 20\% \\[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] 30\%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\% $

(18.) JAMB $22\dfrac{1}{2}\%$ of the Nigerian Naira, $₦$ is equal to $17\dfrac{1}{10}\%$ of a foreign curreny M.
What is the conversion rate of the M to the Naira?

$ A.\:\: 1M = \dfrac{15}{57}₦ \\[5ex] B.\:\: 1M = 2\dfrac{11}{57}₦ \\[5ex] C.\:\: 1M = 1\dfrac{18}{57}₦ \\[5ex] D.\:\: 1M = 38\dfrac{1}{4}₦ \\[5ex] E.\:\: 1M = 384\dfrac{3}{4}₦ \\[5ex] $

$ 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} $

(20.) ACT One number is 25% of a second number, and the second number is 70% of a third number.
The first number is what percent of the third number?

$ A.\;\; 17.5\% \\[3ex] B.\;\; 42.5\% \\[3ex] C.\;\; 45\% \\[3ex] D.\;\; 87.5\% \\[3ex] E.\;\; 95\% \\[3ex] $

$ Let: \\[3ex] first\;\;number = p \\[3ex] second\;\;number = m \\[3ex] third\;\;number = k \\[3ex] \implies \\[3ex] p = 25\%\;\;of\;\;m \\[3ex] p = \dfrac{25}{100} * m \\[5ex] p = 0.25m ... eqn.(1) \\[3ex] m = 70\%\;\;of\;\;k \\[3ex] m = \dfrac{70}{100} * k \\[5ex] m = 0.7k ... eqn.(2) \\[3ex] Substitute\;\;eqn.(2)\;\;into\;\;eqn.(1) \\[3ex] p = 0.25m \\[3ex] p = 0.25(0.7k) \\[3ex] p = 0.175k \\[3ex] p = \dfrac{175}{1000} * k \\[5ex] p = \dfrac{17.5}{100} * k \\[3ex] p = 17.5\% * k \\[3ex] $ The first number is 17.5% of the third number

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