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Mathematics LibreTexts

1.E: Review Exercises and Sample Exam

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    24448
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    Review Exercises

    Exercise \(\PageIndex{1}\) Real Numbers and the Number Line

    Choose an appropriate scale and graph the following sets of real numbers on a number line.

    1. \(\{−4, 0, 4\}\)
    2. \(\{−30, 10, 40\}\)
    3. \(\{−12, −3, 9\}\)
    4. \(\{−10, 8, 10\}\)
    Answer

    1. 

    Screenshot (977).png

    Figure 1.E.1

    3. 

    Screenshot (979).png

    Figure 1.E.2

    Exercise \(\PageIndex{2}\) Real Numbers and the Number Line

    Fill in the blank with \(<, =,\text{ or }>\).

    1. \(0\underline{\quad}-9\)
    2. \(-75\underline{\quad}-5\)
    3. \(-12\underline{\quad}-(-3)\)
    4. \(-(-23)\underline{\quad}23\)
    5. \(|-20|\underline{\quad}-|-30|\)
    6. \(-|6|\underline{\quad}-|-(-8)|\)
    Answer

    1. \(>\)

    3. \(<\)

    5. \(>\) 

    Exercise \(\PageIndex{3}\) Real Numbers and the Number Line

    Determine the unknown.

    1. \(|?|=2\)
    2. \(|?|=1\)
    3. \(|?|=-7\)
    4. \(|?|=0\)
    Answer

    1. \(\pm 2\)

    3. \(Ø\), No solution

    Exercise \(\PageIndex{4}\) Real Numbers and the Number Line

    Translate the following into a mathematical statement.

    1. Negative eight is less than or equal to zero.
    2. Seventy-eight is not equal to twelve.
    3. Negative nine is greater than negative ten.
    4. Zero is equal to zero.
    Answer

    1. \(-8\leq 0\)

    3. \(-9>-10\)

    Exercise \(\PageIndex{5}\) Adding and Subtracting Integers

    Simplify.

    1. \(12+(−7)\)
    2. \(20+(−32)\)
    3. \(−23−(−7)\)
    4. \(−8−(−8)\)
    5. \(−3−(−13)+(−1)\)
    6. \(9+(−15)−(−8)\)
    7. \((7−10)−3\)
    8. \((−19+6)−2\)
    Answer

    1. \(5\)

    3. \(-16\)

    5. \(9\)

    7. \(-6\)

    Exercise \(\PageIndex{6}\) Adding and Subtracting Integers

    Find the distance between the given numbers on a number line.

    1. \(−8\) and \(14\)
    2. \(−35\) and \(−6\)
    3. What is \(2\) less than \(17\)?
    4. What is \(3\) less than \(−20\)?
    5. Subtract \(30\) from the sum of \(8\) and \(12\).
    6. Subtract \(7\) from the difference of \(−5\) and \(7\).
    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?
    8. 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

    Exercise \(\PageIndex{7}\) Multiplying and Dividing Integers

    Simplify.

    1. \(10÷5⋅2\)
    2. \(36÷6⋅2\)
    3. \(−6(4)÷2(−3)\)
    4. \(120÷(−5)(−3)(−2)\)
    5. \(−8(−5)÷0\)
    6. \(−24(0)÷8\)
    7. Find the product of \(−6\) and \(9\).
    8. Find the quotient of \(−54\) and \(−3\).
    9. James was able to drive \(234\) miles on \(9\) gallons of gasoline. How many miles per gallon did he get?
    10. 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

    Exercise \(\PageIndex{8}\) Fractions

    Reduce each fraction to lowest terms.

    1. \(\frac{180}{300}\)
    2. \(\frac{252}{324}\)
    3. Convert to a mixed number: \(\frac{23}{8}\).
    4. Convert to an improper fraction: \(3\frac{5}{9}\).
    Answer

    1. \(\frac{3}{5}\)

    3. \(2\frac{7}{8}\)

    Exercise \(\PageIndex{9}\) Fractions

    Simplify.

    1. \(\frac{3}{5}(−\frac{2}{7})\)
    2. \(−\frac{5}{8}(−\frac{1}{3})\)
    3. \(−\frac{3}{4}÷\frac{6}{7}\)
    4. \(\frac{4}{15}÷\frac{28}{3}\)
    5. \(4\frac{4}{5}÷6\)
    6. \(5÷8\frac{1}{3}\)
    7. \(\frac{5}{4}÷\frac{15}{2}⋅6\)
    8. \(\frac{5}{24}÷\frac{3}{2}÷\frac{5}{12}\)
    9. \(\frac{1}{12}−\frac{1}{4}\)
    10. \(\frac{5}{6}−\frac{3}{14}\)
    11. \(\frac{3}{4}+\frac{2}{3}−\frac{1}{12}\)
    12. \(\frac{3}{10}+\frac{5}{12}−\frac{1}{6}\)
    13. Subtract \(\frac{2}{3}\) from the sum of \(−\frac{1}{2}\) and \(\frac{2}{9}\).
    14. Subtract \(\frac{5}{6}\) from the difference of \(\frac{1}{3}\) and \(\frac{7}{2}\).
    15. 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?
    16. 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 

    Exercise \(\PageIndex{10}\) Decimals and Percents

    1. Write as a mixed number: \(5.32\).
    2. Write as a decimal: \(7\frac{3}{25}\)
    Answer

    1. \(5\frac{8}{25}\)

    Exercise \(\PageIndex{11}\) Decimals and Percents

    Perform the operations.

    1. \(6.032+2.19\)
    2. \(12.106−9.21\)
    3. \(4.23×5.13\)
    4. \(9.246÷4.02\)
    Answer

    1. \(8.222\)

    3. \(21.6999\)

    Exercise \(\PageIndex{12}\) Decimals and Percents

    Convert to a decimal.

    1. \(7.2\)%
    2. \(5\frac{3}{8}\)%
    3. \(147\)%
    4. \(27\frac{1}{2}\)%
    Answer

    1. \(0.072\)

    3. \(1.47\)

    Exercise \(\PageIndex{13}\) Decimals and Percents

    Convert to a percent.

    1. \(0.055\)
    2. \(1.75\)
    3. \(\frac{9}{10}\)
    4. \(\frac{5}{6}\)
    5. 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?
    6. A retail outlet is offering \(12\)% off the original $\(39.99\) price of tennis shoes. What is the price after the discount?
    7. If an item costs $\(129.99\), then what is the total after adding \(7\frac{1}{4}\)% sales tax?
    8. 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. $\(139.41\)

    Exercise \(\PageIndex{14}\) Exponents and Square Roots

    Simplify.

    1. \(8^{2}\)
    2. \((−5)^{2}\)
    3. \(−4^{2}\)
    4. \(−(−3)^{2}\)
    5. \((\frac{2}{9})^{2}\)
    6. \((1\frac{2}{3})^{2}\)
    7. \(3^{3}\)
    8. \((−4)^{3}\)
    9. \((\frac{2}{5})^{3}\)
    10. \((−\frac{1}{6})^{3}\)
    11. \(−(−2)^{4}\)
    12. \(−(−1)^{5}\)
    13. \(\sqrt{49}\)
    14. \(\sqrt{225}\)
    15. \(2\sqrt{25}\)
    16. \(−\sqrt{121}\)
    17. \(3\sqrt{50}\)
    18. \(−4\sqrt{12}\)
    19. \(4\sqrt{9}\)
    20. \(8\sqrt{25}\)
    21. Calculate the area of a square with sides measuring \(3\) centimeters. \((A=s^{2})\)
    22. Calculate the volume of a cube with sides measuring \(3\) centimeters. \((V=s^{3})\)
    23. Determine the length of the diagonal of a square with sides measuring \(3\) centimeters.
    24. 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

    Exercise \(\PageIndex{15}\) Order of Operations

    Simplify.

    1. \(−5(2)−7^{2}\)
    2. \(1−4^{2}+2(−3)^{2}\)
    3. \(2+3(6−2⋅4)^{3}\)
    4. \(5−3(8−3⋅4)^{2}\)
    5. \(−2^{3}+6(3^{2}−4)+(−3)^{2}\)
    6. \(5^{2}−40÷5(−2)^{2}−(−4)\)
    7. \(\frac{3}{4}[\frac{2}{9}(−3)^{2}−4]^{2}\)
    8. \((\frac{1}{2})^{2}−\frac{3}{4}÷\frac{9}{16}−\frac{1}{3}\)
    9. \(\frac{2−3(6−3^{2})^{2}}{4⋅5−5^{2}}\)
    10. \(\frac{(2⋅8−6^{2})^{2}−10}{273−(2(−5)^{3}−7) }\)
    11. \(8−5|3⋅4−(−2)^{4}|\)
    12. \(|14|-|−3−52|\)
    Answer

    1. \(-59\)

    3. \(-22\)

    5. \(31\)

    7. \(3\)

    9. \(5\)

    11. \(-12\)

    Exercise \(\PageIndex{16}\) Order of Operations

    Find the distance between the given numbers on a number line.

    1. \(−14\) and \(22\)
    2. \(−42\) and \(−2\)
    3. \(\frac{7}{8}\) and \(−\frac{1}{5}\)
    4. \(−5\frac{1}{2}\) and \(−1\frac{1}{4}\)
    Answer

    1. \(36\) units

    3. \(\frac{43}{40}\) units

    Sample Exam

    Exercise \(\PageIndex{17}\)

    1. List three integers greater than \(−10\).
    2. Determine the unknown(s): \(| ? |=13\).
    3. Fill in the blank with \(<,=,\text{ or }\) : \(-|-100|\underline{\:\:}9^{2}\). 
    4. Convert to a fraction: \(33\frac{1}{3}\)%
    5. Convert to a percent: \(2\frac{3}{4}\).
    6. Reduce: \(\frac{75}{225}\). 
    Answer

    1. \(\{−5, 0, 5\}\) (answers may vary)

    3. \(<\)

    5. \(275\)%

    Exercise \(\PageIndex{18}\)

    Calculate the following.

    1. a. \((−7)^{2}\); b. \(−(−7)^{2}\); c. \(−7^{2}\)
    2. a. \((−3)^{3}\); b. \(−(−3)^{3}\); c. \(−3^{3}\)
    3. a. \(|10|\); b. \(−|−10|\); c. \(−|10|\)
    Answer

    1. a. \(49\); b. \(−49\); c. \(−49\)

    3. a. \(10\); b. \(−10\); c. \(−10\)

    Exercise \(\PageIndex{19}\)

    Simplify.

    1. \(−(−(−1))\)
    2. \(\frac{2}{3}+\frac{1}{5}−\frac{3}{10}\)
    3. \(10−(−12)+(−8)−20\)
    4. \(−8(4)(−3)÷2\)
    5. \(\frac{1}{2}⋅(−\frac{4}{5})÷\frac{14}{15}\)
    6. \(\frac{3}{5}⋅\frac{1}{2}−\frac{2}{3}\)
    7. \(4⋅5−20÷5⋅2\)
    8. \(10−7(3−8)−5^{2}\)
    9. \(3+2|−2^{2}−(−1)|+(−2)^{2}\)
    10. \(\frac{1}{3}[5^{2}−(7−|−2|)+15⋅2÷3]\)
    11. \(\sqrt{116}\)
    12. \(3\sqrt{72}\)
    Answer

    2. \(\frac{17}{30}\)

    4. \(48\)

    6. \(-\frac{11}{30}\)

    8. \(20\)

    10. \(10\)

    12. \(18\sqrt{2}\)

    Exercise \(\PageIndex{20}\)

    1. Subtract \(2\) from the sum of \(8\) and \(−10\).
    2. Subtract \(10\) from the product of \(8\) and \(−10\).
    3. A student earns \(9, 8, 10, 7,\) and \(8\) points on the first \(5\) chemistry quizzes. What is her quiz average?
    4. 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