45 Practice Answer Key: Pre Algebra (1.3 – 1.6) — Intermediate Algebra

1.3 Order of Operations

  1. \begin{array}{l} \\ \\ \\ -6 \cdot -\dfrac{4}{1}\\ \\ -6 \cdot -4 \\ 24 \end{array}
  2. (-1)^3=-1
  3. \begin{array}{l} \\ \\ 3+8 \div 4 \\ 3+2 \\ 5 \end{array}
  4. \begin{array}{l} \\ \\ 5(1)\cdot 36 \\ 5\cdot 36 \\ 180 \end{array}
  5. \begin{array}{l} \\ \\ 8 \div 4\cdot 2 \\ 2 \cdot 2 \\ 4 \end{array}
  6. 2+6=8
  7. \begin{array}{l} \\ \\ \\ \left[-9-(2-5)\right] \div -6 \\ \left[-9-(-3)\right]\div -6 \\ \left[-6\right] \div -6 \\ 1 \end{array}
  8. \begin{array}{l} \\ \\ (-2 \cdot 8\cdot 2) \div (-4) \\ -32 \div -4 \\ 8 \end{array}
  9. \begin{array}{l} \\ \\ \\ -6 + (-6)^2\div 3 \\ -6 + 36 \div 3 \\ -6 + 12 \\ 6 \end{array}
  10. \begin{array}{l} \\ \\ -12 \div [-2-2+6] \\ -12 \div 2 \\ -6 \end{array}
  11. \begin{array}{l} \\ \\ \\ 4-2|9-16| \\ 4-2(7) \\ 4-14 \\ -10 \end{array}
  12. \begin{array}{l} \\ \\ -16 \div 4 -5 \\ -4-5 \\ -9 \end{array}
  13. \begin{array}{l} \\ \\ (-1+5)(5) \\ 4(5) \\ 20 \end{array}
  14. \begin{array}{l} \\ \\ \\ \\ -3-\{3-[-3(6)+2]\} \\ -3-\{3-[-18+2]\} \\ -3-\{3-[-16]\} \\ -3-\{19\} \\ -22 \end{array}
  15. \begin{array}{l} \\ \\ \\ \\ \left[2+4|7+4| \right] \div \left[8+15\right] \\ \left[2+4(11)\right] \div 23 \\ 2+44 \div 23 \\ 46\div 23 \\ 2 \end{array}
  16. \begin{array}{l} \\ \\ -4-\left[2-24-4-22-10\right] \\ -4-\left[-58\right] \\ 54 \end{array}
  17. \begin{array}{l} \\ \\ \\ \left[12+2+6\right](-5+ |-3|) \\ (20)(-5+3) \\ (20)(-2) \\ -40 \end{array}
  18. \begin{array}{l} \\ \\ \\ \\ -6+3-6\left[-2-(-4)\right] \\ -3-6\left[-2+4\right] \\ -3-6\left[2\right] \\ -3-12 \\ -15 \end{array}
  19. \begin{array}{l} \\ \\ \\ \\ \\ \dfrac{-15}{2-1-6-[-1+3]} \\ \\ -15 \div (-3-[2]) \\ -15 \div (-5) \\ 3 \end{array}
  20. \begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \dfrac{25+25}{|16-32| - 6} \\ \\ \dfrac{50}{|-16|-6} \\ \\ \dfrac{50}{16-6} \\ \\ \dfrac{50}{10} \\ \\ 5 \end{array}
  21. \begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \dfrac{-48-4-4-[-4+3]}{(16+9)\div 5} \\ \\ \dfrac{-56-[-1]}{25\div 5} \\ \\ \dfrac{-55}{5} \\ \\ -11 \end{array}
  22. \begin{array}{l} \\ \\ \\ \\ \\ \dfrac{-18-(-3)}{1-(-1)+3} \\ \\ -\dfrac{15}{5} \\ \\ -3 \end{array}
  23. \begin{array}{l} \\ \\ \\ \\ \\ \dfrac{8+4}{-24-4-[-25]} \\ \\ -\dfrac{12}{3} \\ \\ -4 \end{array}
  24. \begin{array}{l} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \dfrac{13+9-12+1-[-10+6]}{\{9 \div [16-9(1)-8]\}+12} \\ \\ \dfrac{11-[-4]}{\{9\div [-1]\}+12} \\ \\ \dfrac{15}{-9+12} \\ \\ \dfrac{15}{3} \\ \\ 5 \end{array}

1.4 Properties of Algebra

  1. r+1
  2. -4x-2
  3. 2n
  4. 11b+7
  5. 15v
  6. 7x
  7. -9x
  8. -7a-1
  9. k+5
  10. -3p
  11. -5x-9
  12. -10n-9
  13. -m
  14. -r-5
  15. -8x+32
  16. 24r+27
  17. 8n^2+72n
  18. -9a+5
  19. -7k^2+42k
  20. 20x^2+10x
  21. -36x-6
  22. -2n-2
  23. -8m^2+40m
  24. -18p^2+2p
  25. 9x^2-36x
  26. 32n-8
  27. -9b^2+90b
  28. -28r-4
  29. 9b+90+5b
    14b+90
  30. 4v-7+56v
    60v-7
  31. -3x+12x^2-4x^2
    8x^2-3x
  32. -8x-81x+81
    -89x+81
  33. -4k^2-64k^2-8k
    -68k^2-8k
  34. -9-10-90a
    -90a-19
  35. 1-35-49p
    -49p-34
  36. -10x+20-3
    -10x+17
  37. -10-4n+20
    -4n+10
  38. -30+6m+3m
    9m-30
  39. 4x+28+8x+32
    12x+60
  40. -2r-8r^2-8r^2+32r
    -16r^2+30r
  41. -8n-48-8n^2-64n
    -8n^2-72n-48
  42. 54b+45-4b^2-12b
    -4b^2+42b+45
  43. 49+21v+30-100v
    -79v+79
  44. -28x+42+20x-20
    -8x+22
  45. -20n^2+10n-42+70n
    -20n^2+80n-42
  46. -12-3a+54a^2+60a
    54a^2+57a-12
  47. 5-30k+10k-80
    -20k-75
  48. -28x-21-100x-100
    -128x-121
  49. 8n^2-3n-5-4n^2
    4n^2-3n-5
  50. 7x^2-3-5x^2-6x
    2x^2-6x-3
  51. 5p-6+1-p
    4p-5
  52. 3x^2-x-7+8x
    3x^2+7x-7
  53. 2-4v^2+3v^2+2v
    -v^2+2v+2
  54. 2b-8+b-7b^2
    -7b^2+3b-8
  55. 4-2k^2+8-2k^2
    -4k^2+12
  56. 7a^2+7a-6a^2-4a
    a^2+3a
  57. x^2-8+2x^2-7
    3x^2-15
  58. 3-7n^2+6n^2+3
    -n^2+6

1.6 Unit Conversion Word Problems

  1. \begin{array}{rrl} \\ \\ 7\text{ mi}&=&7\text{ \cancel{mi}}\times \dfrac{1760\text{ yd}}{\text{ \cancel{mi}}} \\ \\ &=&12,320\text{ yd} \end{array}
  2. \begin{array}{rrl} \\ \\ 234\text{ oz}&=&234\text{ \cancel{oz}}\times \dfrac{1\text{ \cancel{lb}}}{16\text{ \cancel{oz}}}\times \dfrac{1\text{ ton}}{2000\text{ \cancel{lb}}} \\ \\ &=&0.0073\text{ tons} \end{array}
  3. \begin{array}{rrl} \\ \\ 11.2\text{ mg}&=&11.2\text{ \cancel{mg}}\times \dfrac{1\text{ g}}{1000\text{ \cancel{mg}}} \\ \\ &=&0.0112\text{ g} \end{array}
  4. \begin{array}{rrl} \\ \\ 1.35\text{ km}&=&1.35\text{ \cancel{km}}\times \dfrac{1000\text{ \cancel{m}}}{1\text{ \cancel{km}}}\times \dfrac{100\text{ cm}}{1\text{ \cancel{m}}} \\ \\ &=&135,000\text{ cm} \end{array}
  5. \begin{array}{rrl} \\ \\ 9,800,000\text{ mm}&=&9,800,000\text{ \cancel{mm}}\times \dfrac{1\text{ \cancel{m}}}{1000\text{ \cancel{mm}}}\times \dfrac{1\text{ \cancel{km}}}{1000\text{ \cancel{m}}}\times \dfrac{1\text{ mi}}{1.61\text{ \cancel{km}}} \\ \\ &=&6.09\text{ mi (rounded)} \end{array}
  6. \begin{array}{rrl} \\ \\ 4.5\text{ ft}^2&=&4.5\cancel{\text{ ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ \cancel{ft}})^2} \\ \\ &=&0.5\text{ yd}^2 \end{array}
  7. \begin{array}{rrl} \\ \\ 435,000\text{ m}^2&=&435,000\cancel{\text{ m}^2}\times \dfrac{(1\text{ km}^2)}{(1000\text{ \cancel{m}})^2} \\ \\ &=&0.435\text{ km}^2 \end{array}
  8. \begin{array}{rrl} \\ \\ 8\text{ km}^2&=&8\cancel{\text{ km}^2}\times \dfrac{1\cancel{\text{ mi}^2}}{(1.61\text{ \cancel{km}})^2}\times \dfrac{(5280 \text{ ft})^2}{(1\text{ \cancel{mi}})^2} \\ \\ &=&86,000,000\text{ ft}^2\text{ (rounded)} \end{array}
  9. \begin{array}{rrl} \\ \\ 0.0065\text{ km}^3&=&0.0065 \cancel{\text{ km}^3}\times \dfrac{(1000\text{ m})^3}{(1\text{ \cancel{km}})^3} \\ \\ &=&6,500,000\text{ m}^3 \end{array}
  10. \begin{array}{rrl} \\ \\ 14.62\text{ in}^2&=&14.62\cancel{\text{ in}^2}\times \dfrac{(2.54\text{ cm})^2}{(1\text{ \cancel{in}})^2} \\ \\ &=&94.3\text{ cm}^2 \end{array}
  11. \begin{array}{rrl} \\ \\ 5500\text{ cm}^3&=&5500\cancel{\text{ cm}^3}\times \dfrac{(1\text{ \cancel{in}})^3}{(2.54\text{ \cancel{cm}})^3}\times \dfrac{(1\text{ yd})^3}{(36\text{ \cancel{in}})^3} \\ \\ &=&0.0072\text{ yd}^3 \end{array}
  12. \begin{array}{rrl} \\ \\ 3.5\text{ mi/h}&=&\dfrac{3.5\text{ \cancel{mi}}}{\cancel{\text{h}}}\times \dfrac{5280\text{ ft}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{h}}}{3600\text{ s}} \\ \\ &=&5.13\text{ ft/s} \end{array}
  13. \begin{array}{rrl} \\ \\ 185\text{ yd/min.}&=&\dfrac{185 \text{ \cancel{yd}}}{\text{\cancel{min.}}}\times \dfrac{1\text{ mi}}{1760\text{ \cancel{yd}}}\times \dfrac{60\text{ \cancel{min.}}}{1\text{ h}} \\ \\ &=&6.31\text{ mi/h} \end{array}
  14. \begin{array}{rrl} \\ \\ 153\text{ ft/s}&=&\dfrac{153\text{ \cancel{ft}}}{\text{s}}\times \dfrac{1\text{ mi}}{5280\text{ \cancel{ft}}}\times \dfrac{3600\text{ \cancel{s}}}{1\text{ h}} \\ \\ &=&104.3\text{ mi/h} \end{array}
  15. \begin{array}{rrl} \\ \\ 248\text{ mi/h}&=&\dfrac{248\text{ \cancel{mi}}}{\cancel{\text{h}}}\times \dfrac{1.61\text{ \cancel{km}}}{1\text{ \cancel{mi}}}\times \dfrac{1000\text{ m}}{1\text{ \cancel{km}}}\times \dfrac{1\text{ \cancel{h}}}{3600\text{ s}} \\ \\ &=&111\text{ m/s} \end{array}
  16. \begin{array}{rrl} \\ \\ 186,000\text{ mi/h}&=&\dfrac{186,000\text{ \cancel{mi}}}{\cancel{\text{h}}}\times \dfrac{1.61\text{ km}}{1\text{ \cancel{mi}}}\times \dfrac{24\text{ \cancel{h}}}{1\text{ \cancel{day}}}\times \dfrac{365\text{ \cancel{days}}}{1\text{ yr}} \\ \\ &=&2,620,000,000\text{ km/yr (rounded)} \end{array}
  17. \begin{array}{rrl} \\ \\ 7.50\text{ t/yd}^2&=&\dfrac{7.50\text{ \cancel{t}}}{\cancel{\text{yd}^2}}\times \dfrac{2000\text{ lb}}{1\text{ \cancel{t}}}\times \dfrac{(1\text{ \cancel{yd}})^2}{(36\text{ in})^2} \\ \\ &=&11.57\text{ lb/in}^2 \end{array}
  18. \begin{array}{rrl} \\ \\ 16\text{ ft/s}^2&=&\dfrac{16\text{ \cancel{ft}}}{\cancel{\text{s}^2}}\times \dfrac{1\text{ \cancel{mi}}}{5280\text{ \cancel{ft}}}\times \dfrac{1.61\text{ km}}{1\text{ \cancel{mi}}}\times \dfrac{(3600 \text{ \cancel{s}})^2}{(1\text{ h})^2} \\ \\ &=&63,200\text{ km/h}^2\text{ (rounded)} \end{array}
  19. \phantom{1}
    \text{mpg: }\dfrac{260\text{ mi}}{8\text{ gal}}\Rightarrow 32.5\text{ mpg} \\
    \begin{array}{rrl} 32.5\text{ mi/gal}&=&\dfrac{32.5\text{ \cancel{mi}}}{\text{\cancel{gal}}}\times \dfrac{1.602\text{ km}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{gal}}}{3.785\text{ litres}} \\ \\ &=&13.8\text{ km/litre} \\ \\ 32.5\text{ mpg}&=&13.8\text{ km/l} \end{array}
  20. \begin{array}{ccccccl} \\ \\ \\ 12\text{ pg/min}&=&\dfrac{12\text{ pg}}{\cancel{\text{min}}}&\times&\dfrac{60\text{ \cancel{min}}}{1\text{ hr}}&\Rightarrow &720\text{ pg/hr} \\ \\ 720\text{ pg/hr}&=&\dfrac{720\text{ pg}}{\cancel{\text{hr}}}&\times &\dfrac{24\text{ \cancel{hr}}}{1\text{ d}}&\Rightarrow &17,280\text{ pg/d} \end{array}
  21. \begin{array}{rrl} \\ \\ \\ \\ 60\text{ beats/min}&=&\dfrac{60\text{ beats}}{\cancel{\text{min}}}\times \dfrac{60\text{ \cancel{min}}}{1\text{ \cancel{hr}}}\times \dfrac{24\text{ \cancel{hr}}}{1\text{ \cancel{day}}}\times \dfrac{365.24\text{ \cancel{days}}}{\cancel{\text{yr}}}\times \dfrac{86\text{ \cancel{yrs}}}{\text{life}} \\ \\ 60\text{ beats/min}&=&2,713,879,296\text{ beats/life} \\ \\ &=&2.71\text{ million beats} \end{array}
  22. \begin{array}{rrl} \\ \\ 128\text{ mg/dL}&=&\dfrac{128\text{ \cancel{mg}}}{\cancel{\text{dL}}}\times \dfrac{1\text{ g}}{1000\text{ \cancel{mg}}}\times \dfrac{10\text{ \cancel{dL}}}{\text{L}} \\ \\ 128\text{ mg/dL}&=&1.28\text{ g/L} \end{array}
  23. \begin{array}{ccccccl} \\ \\ \\ \\ \\ \\ \\ \\ 38\text{ ft}&=&38\text{ \cancel{ft}}&\times&\dfrac{1\text{ yd}}{3\text{ \cancel{ft}}}&\Rightarrow &12\dfrac{2}{3}\text{ yd} \\ \\ 40\text{ ft}&=&40\text{ \cancel{ft}}&\times &\dfrac{1\text{ yd}}{3\text{ \cancel{ft}}}&\Rightarrow &13\dfrac{1}{3}\text{ yd} \\ \\ \text{Area}&=&12\dfrac{2}{3}\text{ yd}&\times &13\dfrac{1}{3}\text{ yd}&\Rightarrow &168\dfrac{8}{9}\text{ yd}^2 \\ \\ \text{Cost}&=&\$18\text{/yd}^2&\times &168\dfrac{8}{9}\text{ yd}^2&\Rightarrow &\$3040 \end{array}
  24. \begin{array}{ccccl} \\ \\ \\ \\ \\ \\ \text{Volume}&=&50\text{ ft}\times 10\text{ ft}\times 8\text{ ft}&\Rightarrow&4000\text{ ft}^3 \\ \\ 4000\text{ ft}^3&=&4000\cancel{\text{ ft}^3}\times \dfrac{1\text{ yd}^3}{(3\text{ \cancel{ft}})^3}&\Rightarrow &148\text{ yd}^3 \\ \\ 148\text{ yd}^3&=&148\cancel{\text{ yd}^3}\times \dfrac{(0.9144\text{ m})^3}{1\cancel{\text{ yd}^3}}&\Rightarrow &113\text{ m}^3 \end{array}
  25. \begin{array}{rcccl} \\ \\ \\ \\ \\ \\ \text{Area of lot}&\Rightarrow&\dfrac{1}{3}\left(43,560\text{ ft}^2\right)&\Rightarrow &14,520\text{ ft}^2 \\ \\ \text{Now }\dfrac{1}{4}\text{ of this is}&\Rightarrow &\dfrac{1}{4}\left(14,520\text{ ft}^2\right)&\Rightarrow &3630\text{ ft}^2 \\ \\ \text{Convert to square metres}&\Rightarrow&3630\cancel{\text{ ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ \cancel{ft}})^2}&\Rightarrow &403\dfrac{1}{3}\text{ yd}^2 \end{array}
  26. \begin{array}{rcccl} \\ \\ \text{Car speed}&\Rightarrow &\dfrac{23\text{ km}}{15\text{ \cancel{min}}}\times \dfrac{60\text{ \cancel{min}}}{1\text{ h}}&\Rightarrow& \dfrac{92\text{ km}}{\text{h}} \\ \\ \text{Convert to m/s}&\Rightarrow &\dfrac{92\text{ \cancel{km}}}{\text{ \cancel{h}}}\times \dfrac{1000\text{ m}}{1\text{ \cancel{km}}}\times \dfrac{1\text{ \cancel{h}}}{3600\text{ s}}&\Rightarrow &25.6\text{ m/s} \end{array}
  27. \begin{array}{rcccl} \\ \\ \\ \\ \\ \\ 3106\text{ carats}&=&3106\text{ \cancel{carats}}\times \dfrac{0.20\text{ g}}{1\text{ \cancel{carat}}}&\Rightarrow &621.2\text{ g} \\ \\ 621.2\text{ g}&=&621.2\text{ \cancel{g}}\times \dfrac{1000\text{ mg}}{1\text{ \cancel{g}}}&\Rightarrow &621,200\text{ mg} \\ \\ 621.2\text{ g}&=&621.2\text{ \cancel{g}}\times \dfrac{0.0022005\text{ lbs}}{1\text{ \cancel{g}}}&\Rightarrow &1.37\text{ lbs} \end{array}

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