1.E: Review Exercises and Sample Exam
- Page ID
- 24448
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Review Exercises
Choose an appropriate scale and graph the following sets of real numbers on a number line.
- \(\{−4, 0, 4\}\)
- \(\{−30, 10, 40\}\)
- \(\{−12, −3, 9\}\)
- \(\{−10, 8, 10\}\)
- Answer
-
1.
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Figure 1.E.1
3.
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Figure 1.E.2
Fill in the blank with \(<, =,\text{ or }>\).
- \(0\underline{\quad}-9\)
- \(-75\underline{\quad}-5\)
- \(-12\underline{\quad}-(-3)\)
- \(-(-23)\underline{\quad}23\)
- \(|-20|\underline{\quad}-|-30|\)
- \(-|6|\underline{\quad}-|-(-8)|\)
- Answer
-
1. \(>\)
3. \(<\)
5. \(>\)
Determine the unknown.
- \(|?|=2\)
- \(|?|=1\)
- \(|?|=-7\)
- \(|?|=0\)
- Answer
-
1. \(\pm 2\)
3. \(Ø\), No solution
Translate the following into a mathematical statement.
- Negative eight is less than or equal to zero.
- Seventy-eight is not equal to twelve.
- Negative nine is greater than negative ten.
- Zero is equal to zero.
- Answer
-
1. \(-8\leq 0\)
3. \(-9>-10\)
Simplify.
- \(12+(−7)\)
- \(20+(−32)\)
- \(−23−(−7)\)
- \(−8−(−8)\)
- \(−3−(−13)+(−1)\)
- \(9+(−15)−(−8)\)
- \((7−10)−3\)
- \((−19+6)−2\)
- Answer
-
1. \(5\)
3. \(-16\)
5. \(9\)
7. \(-6\)
Find the distance between the given numbers on a number line.
- \(−8\) and \(14\)
- \(−35\) and \(−6\)
- What is \(2\) less than \(17\)?
- What is \(3\) less than \(−20\)?
- Subtract \(30\) from the sum of \(8\) and \(12\).
- Subtract \(7\) from the difference of \(−5\) and \(7\).
- An airplane flying at \(22,000\) feet descended \(8,500\) feet and then ascended \(5,000\) feet. What is the new altitude of the airplane?
- The width of a rectangle is \(5\) inches less than its length. If the length measures \(22\) inches, then determine the width.
- Answer
-
1. \(22\) units
3. \(15\)
5. \(-10\)
7. \(18,500\) feet
Simplify.
- \(10÷5⋅2\)
- \(36÷6⋅2\)
- \(−6(4)÷2(−3)\)
- \(120÷(−5)(−3)(−2)\)
- \(−8(−5)÷0\)
- \(−24(0)÷8\)
- Find the product of \(−6\) and \(9\).
- Find the quotient of \(−54\) and \(−3\).
- James was able to drive \(234\) miles on \(9\) gallons of gasoline. How many miles per gallon did he get?
- If a bus travels at an average speed of \(54\) miles per hour for \(3\) hours, then how far does the bus travel?
- Answer
-
1. \(4\)
3. \(36\)
5. Undefined
7. \(-54\)
9. \(26\) miles per gallon
Reduce each fraction to lowest terms.
- \(\frac{180}{300}\)
- \(\frac{252}{324}\)
- Convert to a mixed number: \(\frac{23}{8}\).
- Convert to an improper fraction: \(3\frac{5}{9}\).
- Answer
-
1. \(\frac{3}{5}\)
3. \(2\frac{7}{8}\)
Simplify.
- \(\frac{3}{5}(−\frac{2}{7})\)
- \(−\frac{5}{8}(−\frac{1}{3})\)
- \(−\frac{3}{4}÷\frac{6}{7}\)
- \(\frac{4}{15}÷\frac{28}{3}\)
- \(4\frac{4}{5}÷6\)
- \(5÷8\frac{1}{3}\)
- \(\frac{5}{4}÷\frac{15}{2}⋅6\)
- \(\frac{5}{24}÷\frac{3}{2}÷\frac{5}{12}\)
- \(\frac{1}{12}−\frac{1}{4}\)
- \(\frac{5}{6}−\frac{3}{14}\)
- \(\frac{3}{4}+\frac{2}{3}−\frac{1}{12}\)
- \(\frac{3}{10}+\frac{5}{12}−\frac{1}{6}\)
- Subtract \(\frac{2}{3}\) from the sum of \(−\frac{1}{2}\) and \(\frac{2}{9}\).
- Subtract \(\frac{5}{6}\) from the difference of \(\frac{1}{3}\) and \(\frac{7}{2}\).
- If a bus travels at an average speed of \(54\) miles per hour for \(2\frac{1}{3}\) hours, then how far does the bus travel?
- Determine the length of fencing needed to enclose a rectangular pen with dimensions \(12\frac{1}{2}\) feet by \(8\frac{3}{4}\) feet.
- Answer
-
1. \(-\frac{6}{35}\)
3. \(-\frac{7}{8}\)
5. \(\frac{4}{5}\)
7. \(1\)
9. \(-\frac{1}{6}\)
11. \(\frac{4}{3}\)
13. \(-\frac{17}{18}\)
15. \(126\) miles
- Write as a mixed number: \(5.32\).
- Write as a decimal: \(7\frac{3}{25}\)
- Answer
-
1. \(5\frac{8}{25}\)
Perform the operations.
- \(6.032+2.19\)
- \(12.106−9.21\)
- \(4.23×5.13\)
- \(9.246÷4.02\)
- Answer
-
1. \(8.222\)
3. \(21.6999\)
Convert to a decimal.
- \(7.2\)%
- \(5\frac{3}{8}\)%
- \(147\)%
- \(27\frac{1}{2}\)%
- Answer
-
1. \(0.072\)
3. \(1.47\)
Convert to a percent.
- \(0.055\)
- \(1.75\)
- \(\frac{9}{10}\)
- \(\frac{5}{6}\)
- Mary purchased \(3\) boxes of t-shirts for a total of $\(126\). If each box contains \(24\) t-shirts, then what is the cost of each t-shirt?
- A retail outlet is offering \(12\)% off the original $\(39.99\) price of tennis shoes. What is the price after the discount?
- If an item costs $\(129.99\), then what is the total after adding \(7\frac{1}{4}\)% sales tax?
- It is estimated that \(8.3\)% of the total student population carpools to campus each day. If there are \(13,000\) students, then estimate the number of students that carpool to campus.
- Answer
-
1. \(5.5\)%
3. \(90\)%
5. $\(1.75\)
7. $\(139.41\)
Simplify.
- \(8^{2}\)
- \((−5)^{2}\)
- \(−4^{2}\)
- \(−(−3)^{2}\)
- \((\frac{2}{9})^{2}\)
- \((1\frac{2}{3})^{2}\)
- \(3^{3}\)
- \((−4)^{3}\)
- \((\frac{2}{5})^{3}\)
- \((−\frac{1}{6})^{3}\)
- \(−(−2)^{4}\)
- \(−(−1)^{5}\)
- \(\sqrt{49}\)
- \(\sqrt{225}\)
- \(2\sqrt{25}\)
- \(−\sqrt{121}\)
- \(3\sqrt{50}\)
- \(−4\sqrt{12}\)
- \(4\sqrt{9}\)
- \(8\sqrt{25}\)
- Calculate the area of a square with sides measuring \(3\) centimeters. \((A=s^{2})\)
- Calculate the volume of a cube with sides measuring \(3\) centimeters. \((V=s^{3})\)
- Determine the length of the diagonal of a square with sides measuring \(3\) centimeters.
- Determine the length of the diagonal of a rectangle with dimensions \(2\) inches by \(4\) inches.
- Answer
-
1. \(64\)
3. \(−16\)
5. \(\frac{4}{81}\)
7. \(27\)
9. \(\frac{81}{25}\)
11. \(−16\)
13. \(7\)
15. \(10\)
17. \(15\sqrt{2}\)
19. \(\frac{2}{3}\)
21. \(9\) square centimeters
23. \(3\sqrt{2}\) centimeters
Simplify.
- \(−5(2)−7^{2}\)
- \(1−4^{2}+2(−3)^{2}\)
- \(2+3(6−2⋅4)^{3}\)
- \(5−3(8−3⋅4)^{2}\)
- \(−2^{3}+6(3^{2}−4)+(−3)^{2}\)
- \(5^{2}−40÷5(−2)^{2}−(−4)\)
- \(\frac{3}{4}[\frac{2}{9}(−3)^{2}−4]^{2}\)
- \((\frac{1}{2})^{2}−\frac{3}{4}÷\frac{9}{16}−\frac{1}{3}\)
- \(\frac{2−3(6−3^{2})^{2}}{4⋅5−5^{2}}\)
- \(\frac{(2⋅8−6^{2})^{2}−10}{273−(2(−5)^{3}−7) }\)
- \(8−5|3⋅4−(−2)^{4}|\)
- \(|14|-|−3−52|\)
- Answer
-
1. \(-59\)
3. \(-22\)
5. \(31\)
7. \(3\)
9. \(5\)
11. \(-12\)
Find the distance between the given numbers on a number line.
- \(−14\) and \(22\)
- \(−42\) and \(−2\)
- \(\frac{7}{8}\) and \(−\frac{1}{5}\)
- \(−5\frac{1}{2}\) and \(−1\frac{1}{4}\)
- Answer
-
1. \(36\) units
3. \(\frac{43}{40}\) units
Sample Exam
- List three integers greater than \(−10\).
- Determine the unknown(s): \(| ? |=13\).
- Fill in the blank with \(<,=,\text{ or }\) : \(-|-100|\underline{\:\:}9^{2}\).
- Convert to a fraction: \(33\frac{1}{3}\)%
- Convert to a percent: \(2\frac{3}{4}\).
- Reduce: \(\frac{75}{225}\).
- Answer
-
1. \(\{−5, 0, 5\}\) (answers may vary)
3. \(<\)
5. \(275\)%
Calculate the following.
- a. \((−7)^{2}\); b. \(−(−7)^{2}\); c. \(−7^{2}\)
- a. \((−3)^{3}\); b. \(−(−3)^{3}\); c. \(−3^{3}\)
- a. \(|10|\); b. \(−|−10|\); c. \(−|10|\)
- Answer
-
1. a. \(49\); b. \(−49\); c. \(−49\)
3. a. \(10\); b. \(−10\); c. \(−10\)
Simplify.
- \(−(−(−1))\)
- \(\frac{2}{3}+\frac{1}{5}−\frac{3}{10}\)
- \(10−(−12)+(−8)−20\)
- \(−8(4)(−3)÷2\)
- \(\frac{1}{2}⋅(−\frac{4}{5})÷\frac{14}{15}\)
- \(\frac{3}{5}⋅\frac{1}{2}−\frac{2}{3}\)
- \(4⋅5−20÷5⋅2\)
- \(10−7(3−8)−5^{2}\)
- \(3+2|−2^{2}−(−1)|+(−2)^{2}\)
- \(\frac{1}{3}[5^{2}−(7−|−2|)+15⋅2÷3]\)
- \(\sqrt{116}\)
- \(3\sqrt{72}\)
- Answer
-
2. \(\frac{17}{30}\)
4. \(48\)
6. \(-\frac{11}{30}\)
8. \(20\)
10. \(10\)
12. \(18\sqrt{2}\)
- Subtract \(2\) from the sum of \(8\) and \(−10\).
- Subtract \(10\) from the product of \(8\) and \(−10\).
- A student earns \(9, 8, 10, 7,\) and \(8\) points on the first \(5\) chemistry quizzes. What is her quiz average?
- An \(8\frac{3}{4}\) foot plank is cut into \(5\) pieces of equal length. What is the length of each piece?
- Answer
-
2. \(-90\)
4. \(1\frac{3}{4}\) feet