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

# Solved Examples - Numbers

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.

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

(1.) ACT If $a$ and $b$ are real numbers such that $a \gt 0$ and $b \lt 0$, then which of the following is equivalent to $|a| - |b|$?

$F.\:\: |a - b| \\[3ex] G.\:\: |a + b| \\[3ex] H.\:\: |a| + |b| \\[3ex] J.\:\: a - b \\[3ex] K.\:\: a + b \\[3ex]$

We shall solve this question using two methods.
If you really do not know how to begin to solve this question, use the first method.
For faster solution because the ACT is a timed test, use the second method.

First Method - Arithmetic method (Test Numbers)

$a\:\: and\:\: b\:\: are\:\: real\:\: numbers \\[3ex] a \gt 0, b \lt 0 \\[3ex] Let\:\: a = 1, b = -1 \\[3ex] |a| - |b| = |1| - |-1| = 1 - 1 = 0 \\[3ex] Test \\[3ex] |a - b| = |1 - (-1)| = |1 + 1| = |2| = 2 \:\: NO \\[3ex] |a + b| = |1 + (-1)| = |1 - 1| = |0| = 0 \:\: Maybe \\[3ex] |a| + |b| = |1| + |-1| = 1 + 1 = 2 \:\: NO \\[3ex] a - b = 1 - (-1) = 1 + 1 = 2 \:\: NO \\[3ex] a + b = 1 + (-1) = 1 - 1 = 0 \:\: Maybe \\[3ex]$ So, we are left with two options: $|a + b|$ and $a + b$
Specific to the question, $a$ and $b$ are real numbers.
Their absolute values are also real numbers.
The subtraction of their absolute values should give us real numbers, rather than the absolute value of real numbers.
In that sense, $a + b$ is a better answer.

Second Method - Algebraic method

$a\:\: and\:\: b\:\: are\:\: real\:\: numbers \\[3ex] a \gt 0, b \lt 0 \\[3ex]$ Be definition
$|a|$ means that $a$ could be positive, zero, or negative.
However, from the question; $a \gt 0 \implies a$ is positive
$|b|$ means that $b$ could be positive, zero, or negative.
However, from the question; $b \lt 0 \implies b$ is negative
This means:
$|a| - |b| \\[3ex] = (+a) - (-b) \\[3ex] = a - (-b) \\[3ex] = a + b$
(2.) ACT If $x \lt y$ and $y \lt 4$, then what is the greatest possible integer value of $x + y$

$A.\:\: 0 \\[3ex] B.\:\: 3 \\[3ex] C.\:\: 4 \\[3ex] D.\:\: 7 \\[3ex] E.\:\: 8 \\[3ex]$

We shall solve this question using the Arithmetic Method by testing numbers.
We shall use maximum numbers to test.

Student: Why do we have to use "maximum numbers" to test?
Teacher: We shall use "maximum numbers" to test because the question is asking for the greatest possible integer value
Keep in mind that $x$ and $y$ do not have to be integer values.
But, we are interested in the integer value of their sum

$\underline{Arithmetic\:\:Method:\:\:Test\:\:Numbers} \\[3ex] y \lt 4 \\[3ex] Let\:\: y = 3.9 \\[3ex] x \lt y \\[3ex] Let\:\: x = 3.8 \\[3ex] x + y = 3.9 + 3.8 = 7.7 \\[3ex]$ Greatest possible integer value = $7$

Student: Why is the greatest possible integer value not equal to $8$?
$Teacher:\:\: 7.7 \ne 8 \\[3ex] 7.7 \gt 7 \\[3ex] 7.7 \lt 8$
(3.) ACT What is the smallest integer greater than $\sqrt{85}$

$A.\:\: 5 \\[3ex] B.\:\: 9 \\[3ex] C.\:\: 10 \\[3ex] D.\:\: 12 \\[3ex] E.\:\: 43 \\[3ex]$

$\sqrt{85} \approx 9.21954 \\[3ex]$ Next integer $\gt 9.21954$ is $10$
(4.) Evaluate $||-2| - |-10||$

$||-2| - |-10|| \\[3ex] |-2| = 2 \\[3ex] |-10| = 10 \\[3ex] \rightarrow |2 - 10| \\[3ex] = |-8| \\[3ex] = 8$
(5.) ACT For every pair of real numbers $x$ and $y$ such that $xy = 0$ and $\dfrac{x}{y} = 0$, which of the following statements is true?
$A.\:\: x = 0 \:\:and\:\: y = 0 \\[3ex] B.\:\: x \ne 0 \:\:and\:\: y = 0 \\[3ex] C.\:\: x = 0 \:\:and\:\: y \ne 0 \\[3ex] D.\:\: x \ne 0 \:\:and\:\: y \ne 0 \\[3ex]$ $E.\:\:$ None of the statements is true for every such pair of real numbers $x$ and $y$

$xy = 0$ means that:
Either $x = 0$ OR $y = 0$ OR both $x = 0$ AND $y = 0$
This is known as the Zero Product Property

$\dfrac{x}{y} = 0$ means that $x = 0$
$0$ divided by anything is $0$
However, $y$ cannot be $0$.
Dividing anything by $0$ is undefined.

Combining the two conditions:
$x = 0 \:\:and\:\: y \ne 0$
(6.) CMAT $\sqrt{188 + \sqrt{51 + \sqrt{169}}} = ?$

$1.\:\: 16.4 \\[3ex] 2.\:\: 14.4 \\[3ex] 3.\:\: 16 \\[3ex] 4.\:\: 14 \\[3ex]$

$\sqrt{188 + \sqrt{51 + \sqrt{169}}} \\[3ex] \sqrt{169} = 13 \\[3ex] 51 + 13 = 64 \\[3ex] \sqrt{64} = 8 \\[3ex] 188 + 8 = 196 \\[3ex] \sqrt{196} = 14$
(7.) JAMB Simplify $(\sqrt{0.7} + \sqrt{70})^2$

$A.\:\:70.7 \\[3ex] B.\:\: 84.7 \\[3ex] C.\:\: 217.7 \\[3ex] D.\:\: 168.7 \\[3ex]$

$(\sqrt{0.7} + \sqrt{70})^2 \\[3ex] = (\sqrt{0.7} + \sqrt{70})(\sqrt{0.7} + \sqrt{70}) \\[3ex] \sqrt{0.7} * \sqrt{0.7} = (\sqrt{0.7})^2 = 0.7 \\[3ex] \sqrt{0.7} * \sqrt{70} = \sqrt{0.7 * 70} = \sqrt{0.7 * 7 * 10} = \sqrt{7 * 7} = \sqrt{49} = 7 \\[3ex] \sqrt{70} * \sqrt{0.7} = \sqrt{70 * 0.7} = \sqrt{7 * 10 * 0.7} = \sqrt{7 * 7} = \sqrt{49} = 7 \\[3ex] \sqrt{70} * \sqrt{70} = (\sqrt{70})^2 = 70 \\[3ex] = 0.7 + 7 + 7 + 70 \\[3ex] = 84.7$
(8.) ACT $|9(-6) + 5(4)| = ?$

$A.\:\: -34 \\[3ex] B.\:\: 12 \\[3ex] C.\:\: 23 \\[3ex] D.\:\: 34 \\[3ex] E.\:\: 74 \\[3ex]$

$|9(-6) + 5(4)| \\[3ex] = |-54 + 20| \\[3ex] = |-34| \\[3ex] = 34$
(9.) Simplify the expression using the order of operations

$\dfrac{5 * 2 - 3^2}{[2^2 - (-7)]^2} \\[5ex]$

$\dfrac{5 * 2 - 3^2}{[2^2 - (-7)]^2} \\[5ex] \underline{Numerator} \\[3ex] 5 * 2 - 3^2 \\[3ex] = 5 * 2 - 9 \\[3ex] = 10 - 9 \\[3ex] = 1 \\[3ex] \underline{Denominator} \\[3ex] [2^2 - (-7)]^2 \\[3ex] = [4 - (-7)]^2 \\[3ex] = [4 + 7]^2 \\[3ex] = 11^2 \\[3ex] = 121 \\[3ex] \therefore \dfrac{5 * 2 - 3^2}{[2^2 - (-7)]^2} = \dfrac{1}{121}$
(10.) Use the order of operations to simplify the expression

$6^2 - 64 \div 4^2 * 7 - 5 \\[3ex]$

$6^2 - 64 \div 4^2 * 7 - 5 \\[3ex] = 36 - 64 \div 16 * 7 - 5 \\[3ex] = 36 - 4 * 7 - 5 \\[3ex] = 36 - 28 - 5 \\[3ex] = 8 - 5 \\[3ex] = 3$
(11.) Use the order of operations to simplify the expression

$4 - 6[-4(5 - 7) - 7(5 - 4)] \\[3ex]$

$4 - 6[-4(5 - 7) - 7(5 - 4)] \\[3ex] = 4 - 6[-4(-2) - 7(1)] \\[3ex] = 4 - 6[8 - 7] \\[3ex] = 4 - 6(1) \\[3ex] = 4 - 6 \\[3ex] = -2$
(12.) CSEC Using a calculator, or otherwise, calculate the EXACT value of

$(12.8)^2 - (30 \div 0.375) \\[3ex]$

$(12.8)^2 - (30 \div 0.375) \\[3ex] = 163.84 - 80 \\[3ex] = 83.84$
(13.) CMAT $2\:3\:7\:4\:3\:2\:1\:5\:7\:3\:2\:7\:1\:0\:9\:8\:7\:5\:4\:7\:5\:4\:7\:2\:3$
Find the number of $7$ in the given series that are followed by an even number but are not preceded by a prime number?

$1.\:\: 1 \\[3ex] 2.\:\: 2 \\[3ex] 3.\:\: 3\\[3ex] 4.\:\: 4 \\[3ex]$

$1st\:\:7 \\[3ex] Preceed\:\:3 = prime\:\:number \\[3ex] Succeed\:\:4 = even\:\:number \\[3ex] NO \\[3ex] 2nd\:\:7 \\[3ex] Preceed\:\:5 = prime\:\:number \\[3ex] Succeed\:\:3 = odd\:\:number \\[3ex] NO \\[3ex] 3rd\:\:7 \\[3ex] Preceed\:\:2 = prime\:\:number \\[3ex] Succeed\:\:1 = odd\:\:number \\[3ex] NO \\[3ex] 4th\:\:7 \\[3ex] Preceed\:\:8 = not\:\:prime\:\:number \\[3ex] Succeed\:\:5 = odd\:\:number \\[3ex] NO \\[3ex] 5th\:\:7 \\[3ex] Preceed\:\:4 = not\:\:prime\:\:number \\[3ex] Succeed\:\:5 = odd\:\:number \\[3ex] NO \\[3ex] 6th\:\:7 \\[3ex] Preceed\:\:4 = not\:\:prime\:\:number \\[3ex] Succeed\:\:2 = even\:\:number \\[3ex] YES \\[3ex] There\:\:is\:\:only\:\:one\:\:7$
(14.) ACT The square root of a certain number is approximately $9.2371$
The certain number is between what $2$ integers?

$F.\:\: 3\:\:and\:\:4 \\[3ex] G.\:\: 4\:\:and\:\:5 \\[3ex] H.\:\: 9\:\:and\:\:10 \\[3ex] J.\:\: 18\:\:and\:\:19 \\[3ex] K.\:\: 81\:\:and\:\:99 \\[3ex]$

$Let\:\:the\:\:number = x \\[3ex] \sqrt{x} = 9.2371 \\[3ex] Square\:\:both\:\:sides \\[3ex] (\sqrt{x})^2 = 9.2371^2 \\[3ex] x \gt 9^2 \\[3ex] x \lt 10^2 \\[3ex] 9^2 \lt x \lt 10^2 \\[3ex] 81 \lt x \lt 100$
(15.) Simplify the expression using the order of operations

$\dfrac{(6 - 7)^2 - 3|4 - 9|}{128 - 2 * 6^2} \\[5ex]$

$\dfrac{(6 - 7)^2 - 3|4 - 9|}{128 - 2 * 6^2} \\[5ex] \underline{Numerator} \\[3ex] (6 - 7)^2 - 3|4 - 9| \\[3ex] = (-1)^2 - 3|-5| \\[3ex] = (-1)(-1) - 3(5) \\[3ex] = 1 - 15 \\[3ex] = -14 \\[3ex] \underline{Denominator} \\[3ex] 128 - 2 * 6^2 \\[3ex] = 128 - 2 * 36 \\[3ex] = 128 - 72 \\[3ex] = 56 \\[3ex] \therefore \dfrac{(6 - 7)^2 - 3|4 - 9|}{128 - 2 * 6^2} = \dfrac{-14}{56} \\[5ex] = -\dfrac{1}{4}$
(16.) ACT For all nonzero values of $a$ and $b$, the value of which of the following expressions is always negative?

$F.\:\: a - b \\[3ex] G.\:\: -a - b \\[3ex] H.\:\: |a| + |b| \\[3ex] J.\:\: |a| - |b| \\[3ex] K.\:\: -|a| - |b| \\[3ex]$

Let us analyze each of the options.
$a$ is any nonzero value
This means that $a$ could be positive or negative

$b$ is any nonzero value
This means that $b$ could be positive or negative

Neither $a$ nor $b$ is zero

$Option\:\:F \\[3ex] a - b \\[3ex] Assume\:\: a = 5, b = 2 \\[3ex] 5 - 2 = 3 \\[3ex] 3\:\:is\:\:not\:\:negative \\[3ex] Incorrect \\[3ex] Option\:\:G \\[3ex] -a - b \\[3ex] Assume\:\: a = -5, b = 2 \\[3ex] -(-5) - 2 = 5 - 2 = 3 \\[3ex] 3\:\:is\:\:not\:\:negative \\[3ex] Incorrect \\[3ex] Option\:\:H \\[3ex] |a| + |b| \\[3ex] Assume\:\: a = 5, b = 2 \\[3ex] |5| + |2| = 5 + 2 = 7 \\[3ex] 7\:\:is\:\:not\:\:negative \\[3ex] Incorrect \\[3ex] Option\:\:J \\[3ex] |a| - |b| \\[3ex] Assume\:\: a = 5, b = 2 \\[3ex] |5| - |2| = 5 - 2 = 3 \\[3ex] 3\:\:is\:\:not\:\:negative \\[3ex] Incorrect \\[3ex] Option\:\:K \\[3ex] -|a| - |b| \\[3ex] Assume\:\: a = 5, b = 2 \\[3ex] -|5| - |2| = -5 - 2 = -7...okay \\[3ex] Assume\:\: a = -5, b = 2 \\[3ex] -|-5| - |2| = -5 - 2 = -7...okay \\[3ex] Assume\:\: a = 5, b = -2 \\[3ex] -|5| - |-2| = -5 - 2 = -7...okay \\[3ex] Assume\:\: a = -5, b = -2 \\[3ex] -|-5| - |-2| = -5 - 2 = -7...okay \\[3ex] Correct!$
(17.) JAMB Without using tables, evaluate $(343)^{\dfrac{1}{3}} * (0.14)^{-1} * (25)^{-\dfrac{1}{2}}$

$A.\:\: 12 \\[3ex] B.\:\: 10 \\[3ex] C.\:\: 8 \\[3ex] D.\:\: 7 \\[3ex]$

$(343)^{\dfrac{1}{3}} * (0.14)^{-1} * (25)^{-\dfrac{1}{2}} \\[7ex] (343)^{\dfrac{1}{3}} = \sqrt[3]{343} = 7 \\[7ex] (0.14)^{-1} = \dfrac{1}{(0.14)^1} = \dfrac{1}{0.14} \\[5ex] (25)^{-\dfrac{1}{2}} = \dfrac{1}{(25)^{\dfrac{1}{2}}} \\[7ex] (25)^{\dfrac{1}{2}} = \sqrt{25} = 5 \\[3ex] \rightarrow (25)^{-\dfrac{1}{2}} = \dfrac{1}{5} \\[5ex] = 7 * \dfrac{1}{0.14} * \dfrac{1}{5} \\[5ex] = \dfrac{7 * 1 * 1}{0.14 * 5} \\[5ex] = \dfrac{7 * 100}{0.14 * 100 * 5} \\[5ex] = \dfrac{7 * 100}{14 * 5} \\[5ex] = \dfrac{1 * 20}{2 * 1} \\[5ex] = \dfrac{20}{2} \\[5ex] = 10$
(18.) ACT What is $|5 - x|$ when $x = 9?$

$A.\:\: -14 \\[3ex] B.\:\: -4 \\[3ex] C.\:\: 4 \\[3ex] D.\:\: 9 \\[3ex] E.\:\: 14 \\[3ex]$

$|5 - x| \\[3ex] x = 9 \\[3ex] = |5 - 9| \\[3ex] = |-4| \\[3ex] = 4$
(19.) Evaluate the expression: $3(-9)(2 - 10 - 2(10))$

$3(-9)(2 - 10 - 2(10)) \\[3ex] = -27(2 - 10 - 20) \\[3ex] = -27(-8 - 20) \\[3ex] = -27(-28) \\[3ex] = 756$
(20.) Simplify $5[2 + 2(3 * 9 - 19)]$

$5[2 + 2(3 * 9 - 19)] \\[3ex] = 5[2 + 2(27 - 19)] \\[3ex] = 5[2 + 2(8)] \\[3ex] = 5[2 + 16] \\[3ex] = 5[18] \\[3ex] = 90$

(21.) ACT An integer is abundant if its positive integer factors, excluding the integer itself, have a sum that is greater than the integer. How many of the integers $6, 8, 10, \:\:and\:\: 12$ are abundant?

$A.\:\: 0 \\[3ex] B.\:\: 1 \\[3ex] C.\:\: 2 \\[3ex] D.\:\: 3 \\[3ex] E.\:\: 4 \\[3ex]$

Positive integer factors (exclude $6$) of $6 = 1, 2, 3$

$1 + 2 + 3 = 6 \\[3ex] 6 \ngtr 6 \\[3ex]$ Positive integer factors (exclude $8$) of $8 = 1, 2, 4$

$1 + 2 + 4 = 7 \\[3ex] 7 \ngtr 8 \\[3ex]$ Positive integer factors (exclude $10$) of $10 = 1, 2, 5$

$1 + 2 + 5 = 8 \\[3ex] 8 \ngtr 10 \\[3ex]$ Positive integer factors (exclude $12$) of $12 = 1, 2, 3, 4, 6$

$1 + 2 + 3 + 4 + 6 = 16 \\[3ex] 16 \gt 12 \\[3ex]$ Only one of those integers, $12$ is abundant.
The answer is Option $B$
(22.) ACT Which of the following numbers has the greatest value?

$A.\:\: 0.\bar{3} \\[3ex] B.\:\: 0.3 \\[3ex] C.\:\: 0.33 \\[3ex] D.\:\: 0.333 \\[3ex] E.\:\: 0.3333 \\[3ex]$

$0.\bar{3} = 0.33333\bar{3} \\[3ex] 0.33333\bar{3} \gt 0.3333 \gt 0.333 \gt 0.33 \gt 0.3 \\[3ex] \therefore Greatest\:\:value = 0.\bar{3}$
(23.) ACT What is the value of the expression

$\dfrac{|-3 - 2|^2 + (-1)^3}{16 \div 4 * 2 - 5}? \\[5ex] F.\:\: -8 \\[3ex] G.\:\: -\dfrac{2}{3} \\[5ex] H.\:\: \dfrac{2}{3} \\[5ex] J.\:\: \dfrac{26}{3} \\[5ex] K.\:\: 8 \\[3ex]$

$\dfrac{|-3 - 2|^2 + (-1)^3}{16 \div 4 * 2 - 5} \\[5ex] \underline{Numerator} \\[3ex] |-3 - 2|^2 \\[3ex] = |-5|^2 \\[3ex] = 5^2 \\[3ex] = 25 \\[3ex] (-1)^3 \\[3ex] = (-1)(-1)(-1) \\[3ex] = -1 \\[3ex] \therefore |-3 - 2|^2 + (-1)^3 \\[3ex] = 25 + -1 \\[3ex] = 25 - 1 \\[3ex] = 24 \\[3ex] \underline{Denominator} \\[3ex] 16 \div 4 * 2 - 5 \\[3ex] = 4 * 2 - 5 \\[3ex] = 8 - 5 \\[3ex] = 3 \\[3ex] \underline{Entire\:\:Question} \\[3ex] = \dfrac{24}{3} \\[5ex] = 8$
(24.) ACT The difference $\dfrac{3}{5} - \dfrac{-1}{3}$ lies in which of the following intervals graphed on the real number line?

$\dfrac{3}{5} - \dfrac{-1}{3} \\[5ex] = \dfrac{3}{5} - -\dfrac{1}{3} \\[5ex] = \dfrac{3}{5} + \dfrac{1}{3} \\[5ex] LCD = 15 \\[3ex] = \dfrac{9}{15} + \dfrac{5}{15} \\[5ex] = \dfrac{9 + 5}{15} \\[5ex] = \dfrac{14}{15} \\[5ex] = 0.933333333 \\[3ex] \dfrac{4}{5} = 0.8 \\[3ex]$ Looking at the options, $\dfrac{14}{15}$ is between $\dfrac{4}{5}$ and $1$
(25.) ACT How many integers, but not including, $20$ and $30$ have a prime factorization with exactly $3$ factors that are NOT necessarily unique?
(Note: $1$ is NOT a prime number.)

$F.\:\: 1 \\[3ex] G.\:\: 2 \\[3ex] H.\:\: 3 \\[3ex] J.\:\: 4 \\[3ex] K.\:\: 5 \\[3ex]$

$21 = 3 * 7 \\[3ex] 22 = 2 * 11 \\[3ex] 23 = 23 \\[3ex] 24 = 2 * 2 * 2 * 3 \\[3ex] 25 = 5 * 5 \\[3ex] 26 = 2 * 13 \\[3ex] 27 = 3 * 3 * 3...three\:\:factors \\[3ex] 28 = 2 * 2 * 7 ...three\:\:factors \\[3ex] 29 = 29 \\[3ex]$ Two numbers, $27$ and $28$ have exactly three factors.
(26.) ACT What is the smallest positive integer having exactly $5$ different positive integer divisors?

$A.\:\: 5 \\[3ex] B.\:\: 6 \\[3ex] C.\:\: 12 \\[3ex] D.\:\: 16 \\[3ex] E.\:\: 18 \\[3ex]$

Positive integer divisors of $5 = 1, 5$
Number of positive integer divisors = $2$

Positive integer divisors of $6 = 1, 2, 3, 6$
Number of positive integer divisors = $4$

Positive integer divisors of $12 = 1, 2, 3, 4, 6, 12$
Number of positive integer divisors = $6$

Positive integer divisors of $16 = 1, 2, 4, 8, 16$
Number of positive integer divisors = $5$

Positive integer divisors of $18 = 1, 2, 3, 6, 9, 18$
Number of positive integer divisors = $6$

The correct option is $D.$
(27.) ACT Which of the following arranges the numbers

$\dfrac{9}{5}, 1.\overline{8}, 1.08$, and $1.\overline{08}$ into ascending order? (Note: The overbar notation shows that the digits under the bar will repeat. For example, $1.\overline{73} = 1.737373...$)

$F.\:\: \dfrac{9}{5} \lt 1.\overline{08} \lt 1.08 \lt 1.\overline{8} \\[5ex] G.\:\: \dfrac{9}{5} \lt 1.08 \lt 1.\overline{08} \lt 1.\overline{8} \\[5ex] H.\:\: 1.\overline{08} \lt 1.08 \lt \dfrac{9}{5} \lt 1.\overline{8} \\[5ex] J.\:\: 1.08 \lt 1.\overline{08} \lt 1.\overline{8} \lt \dfrac{9}{5} \\[5ex] K.\:\: 1.08 \lt 1.\overline{08} \lt \dfrac{9}{5} \lt 1.\overline{8} \\[5ex]$

Ascending order means from least to greatest

$\dfrac{9}{5}, 1.\overline{8}, 1.08, 1.\overline{08} \\[5ex] \dfrac{9}{5} = 1.8 \\[5ex] 1.\overline{8} = 1.888888888888... \\[3ex] 1.08 \\[3ex] 1.\overline{08} = 1.0808080808080808... \\[3ex] 1.88 \gt 1.8 \\[3ex] 1.8 \gt 1.0808 \\[3ex] 1.0808 \gt 1.08 \\[3ex] 1.08 \lt 1.0808 \lt 1.8 \lt 1.88 \\[3ex] \therefore 1.08 \lt 1.\overline{08} \lt \dfrac{9}{5} \lt 1.\overline{8}$
(28.) ACT Walter recently vacationed in Paris.
While there, he visited the Louvre, a famous art museum.
Afterward, he took a $3.7-kilometer$ cab ride from the Louvre to the Eiffel Tower.
A tour guide named Amélie informed him that $2.5$ million rivets were used to build the tower, which stands $320$ meters tall.

When written in scientific notation, the number of rivets used to build the Eiffel Tower is equal to which of the following expressions?

$A.\:\: 2.5 * 10^6 \\[3ex] B.\:\: 2.5 * 10^7 \\[3ex] C.\:\: 2.5 * 10^8 \\[3ex] D.\:\: 25 * 10^6 \\[3ex] E.\:\: 25 * 10^7 \\[3ex]$

$2.5$ million is two million, five hundred thousand

$2.5\:\:million \\[3ex] = 2,500,000 \\[3ex] = 2.5 * 10^6$
(29.) ACT Let $a$ and $b$ be real numbers.
If $(a + b)^2 = a^2 + b^2$, it must be true that:

$A.$ either $a$ or $b$ is zero
$B.$ both $a$ and $b$ are zero.
$C.$ both $a$ and $b$ are positive.
$D.$ $a$ is positive and $b$ is negative.
$E.$ $a$ is negative and $b$ is positive.

(30.) ACT What is the value of $\left(9^{\dfrac{1}{2}} + 16^{\dfrac{1}{2}}\right)^2 ?$

$A.\:\: 7 \\[3ex] B.\:\: 25 \\[3ex] C.\:\: 49 \\[3ex] D.\:\: 337 \\[3ex] E.\:\: 625 \\[3ex]$

$\left(9^{\dfrac{1}{2}} + 16^{\dfrac{1}{2}}\right)^2 \\[5ex] = \left(\sqrt{9} + \sqrt{16}\right)^2 \\[3ex] = (3 + 4)^2 \\[3ex] = 7^2 \\[3ex] = 49$
(31.) ACT

Ascending order means from least to greatest

$\dfrac{9}{5}, 1.\overline{8}, 1.08, 1.\overline{08} \\[5ex] \dfrac{9}{5} = 1.8 \\[5ex] 1.\overline{8} = 1.888888888888... \\[3ex] 1.08 \\[3ex] 1.\overline{08} = 1.0808080808080808... \\[3ex] 1.88 \gt 1.8 \\[3ex] 1.8 \gt 1.0808 \\[3ex] 1.0808 \gt 1.08 \\[3ex] 1.08 \lt 1.0808 \lt 1.8 \lt 1.88 \\[3ex] \therefore 1.08 \lt 1.\overline{08} \lt \dfrac{9}{5} \lt 1.\overline{8}$
(32.) ACT What is the $358th$ digit after the decimal point in the repeating decimal $0.\overline{3178}?$

$F.\:\: 0 \\[3ex] G.\:\: 3 \\[3ex] H.\:\: 1 \\[3ex] J.\:\: 7 \\[3ex] K.\:\: 8 \\[3ex]$

$0.\overline{3178} \\[3ex] means\:\: 0.\overline{3178317831783178....} \\[3ex] The\:\:digits\:\: 3178 \:\:repeats \\[3ex] Every\:\:4th\:\:digit = 8 \\[3ex] Every\:\:8th\:\:digit = 8 \\[3ex] Every\:\:12th\:\:,16th\:\:,20th\:\:...digits = 8 \\[3ex] Every\:\:multiple\:\:of\:\:4\:\:-th\:\:digit = 8 \\[3ex] For\:\:358th\:\:digit, \\[3ex] \dfrac{358}{4} = 89.5 \\[3ex] Integer\:\:part = 89 \\[3ex] 89 * 4 = 356 \\[3ex] 356th\:\:digit = 8 \\[3ex] Back-to-repeating \\[3ex] 357th\:\:digit = 3 \\[3ex] 358th\:\:digit = 1 \\[3ex] OR \\[3ex] \dfrac{358}{4} = 89.5 \\[3ex] Decimal\:\:part = 0.5 \\[3ex] 0.5 * 4 = 2 \\[3ex] Count\:\:the\:\:2nd\:\:digit\:\:in\:\:the\:\:sequence \\[3ex] The\:\:2nd\:\:digit = 358th\:\:digit \\[3ex] 2nd\:\:digit = 1 \\[3ex] \therefore the\:\: 358th\:\:digit = 1$
(33.) ACT For real numbers $a$, $b$, and $c$ such that $a \gt b \gt c$ and $b \gt 0$, which of the statements below is(are) always true?

$I.\:\: |a| \gt |b| \\[3ex] II.\:\: |a| \gt |c| \\[3ex] III.\:\: |b| \gt |c| \\[3ex] A.\:\: I \:\:only \\[3ex] B.\:\: II \:\:only \\[3ex] C.\:\: I \:\:and\:\: II \:\:only \\[3ex] D.\:\: II \:\:and\:\: III \:\:only \\[3ex] E.\:\: I, \:\:II,\:\:and\:\: III \\[3ex]$

$\underline{First\:\:Method:\:\:Arithmetically} \\[3ex] Try\:\:several\:\:numbers \\[3ex] Real\:\:Numbers\:\:includes\:\:positive\:\:numbers,\:\:negative\:\:numbers,\:\:and\:\:zero \\[3ex] Use\:\:both\:\:positive\:\:and\:\:negative\:\:real\:\:numbers\:\:as\:\:applicable \\[3ex] b \gt 0 \\[3ex] Let\:\: b = 1 \\[3ex] a \gt b \\[3ex] Let\:\: a = 2 \\[3ex] b \gt c \\[3ex] Let\:\: c = -1 \\[3ex] a \gt b \gt c \implies 2 \gt 1 \gt -1 \\[3ex] I.\:\: |a| \gt |b| \\[3ex] |2| \gt |1| ? \\[3ex] 2 \gt 1 ...correct...YES \\[3ex] I\:\:works \\[3ex] II.\:\: |a| \gt |c| \\[3ex] |2| \gt |-1| ? \\[3ex] 2 \gt 1 ...correct \\[3ex] But\:\:what\:\:if\:\:c = -2 \\[3ex] c\:\:can\:\:be\:\:-2\:\:because\:\: b \gt c \implies 1 \gt -2 \\[3ex] Try\:\: c = -2 \\[3ex] |2| \gt |-2| ? \\[3ex] 2 \gt 2 ...NO \\[3ex] II\:\:will\:\:not\:\:work \\[3ex] III.\:\: |b| \gt |c| \\[3ex] |1| \gt |-1| ? \\[3ex] 1 \gt -1...NO \\[3ex] III\:\:will\:\:not\:\:work \\[3ex] Correct\:\:Option = I\:\:only = A$
(34.) ACT Which of the following inequalities orders the numbers $0.2$, $0.03$, and $\dfrac{1}{4}$ from least to greatest?

$F.\:\: 0.2 \lt 0.03 \lt \dfrac{1}{4} \\[5ex] G.\:\: 0.03 \lt 0.2 \lt \dfrac{1}{4} \\[5ex] H.\:\: 0.03 \lt \dfrac{1}{4} \lt 0.2 \\[5ex] J.\:\: \dfrac{1}{4} \lt 0.03 \lt 0.2 \\[5ex] K.\:\: \dfrac{1}{4} \lt 0.2 \lt 0.03 \\[5ex]$

$\dfrac{1}{4} = 0.25 \\[5ex] 0.03 \lt 0.2 \lt 0.25 \\[3ex] \therefore 0.03 \lt 0.2 \lt \dfrac{1}{4}$