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Convert the following fractions and simplify the expression: a) 85/105, b) -48/144. Simplify 3 1/4 - 2 2/5 + 1 1/10.

Mathematics

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Convert the following fractions:

a) To convert 85105\frac{85}{105} to its simplest form, find the greatest common divisor (GCD) of the numerator and the denominator.

  1. Find the prime factors of 85 and 105.

    85=5×1785 = 5 \times 17 105=3×5×7105 = 3 \times 5 \times 7

  2. The GCD of 85 and 105 is 5.

  3. Divide both the numerator and the denominator by 5.

    85÷5105÷5=1721\frac{85 \div 5}{105 \div 5} = \frac{17}{21}

    The simplified fraction is 1721\boxed{\frac{17}{21}}.

    b) To convert −48144\frac{-48}{144} to its simplest form, find the greatest common divisor (GCD) of the numerator and the denominator.

  4. Find the prime factors of 48 and 144.

    48=24×348 = 2^4 \times 3 144=24×32=3×48144 = 2^4 \times 3^2 = 3 \times 48

  5. The GCD of 48 and 144 is 48.

  6. Divide both the numerator and the denominator by 48.

    −48÷48144÷48=−13\frac{-48 \div 48}{144 \div 48} = \frac{-1}{3}

    The simplified fraction is −13\boxed{\frac{-1}{3}}.

    Simplify:

    a) Simplify 314−225+11103\frac{1}{4} - 2\frac{2}{5} + 1\frac{1}{10}.

  7. Convert the mixed numbers to improper fractions.

    314=(3×4)+14=1343\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4} 225=(2×5)+25=1252\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{12}{5} 1110=(1×10)+110=11101\frac{1}{10} = \frac{(1 \times 10) + 1}{10} = \frac{11}{10}

  8. Rewrite the expression with improper fractions.

    134−125+1110\frac{13}{4} - \frac{12}{5} + \frac{11}{10}

  9. Find the least common multiple (LCM) of the denominators 4, 5, and 10.

    The LCM is 20.

  10. Convert each fraction to an equivalent fraction with a denominator of 20.

    134=13×54×5=6520\frac{13}{4} = \frac{13 \times 5}{4 \times 5} = \frac{65}{20} 125=12×45×4=4820\frac{12}{5} = \frac{12 \times 4}{5 \times 4} = \frac{48}{20} 1110=11×210×2=2220\frac{11}{10} = \frac{11 \times 2}{10 \times 2} = \frac{22}{20}

  11. Perform the addition and subtraction.

    6520−4820+2220=65−48+2220=17+2220=3920\frac{65}{20} - \frac{48}{20} + \frac{22}{20} = \frac{65 - 48 + 22}{20} = \frac{17 + 22}{20} = \frac{39}{20}

  12. Convert the improper fraction back to a mixed number.

    3920=11920\frac{39}{20} = 1\frac{19}{20}

    The simplified expression is 11920\boxed{1\frac{19}{20}}.

    d) Simplify 5−317+23145 - 3\frac{1}{7} + 2\frac{3}{14}.

  13. Convert the mixed numbers to improper fractions.

    317=(3×7)+17=2273\frac{1}{7} = \frac{(3 \times 7) + 1}{7} = \frac{22}{7} 2314=(2×14)+314=31142\frac{3}{14} = \frac{(2 \times 14) + 3}{14} = \frac{31}{14}

  14. Rewrite the expression with improper fractions.

    5−227+31145 - \frac{22}{7} + \frac{31}{14}

  15. Find the least common multiple (LCM) of the denominators 1, 7, and 14.

    The LCM is 14.

  16. Convert each term to an equivalent fraction with a denominator of 14.

    5=5×141×14=70145 = \frac{5 \times 14}{1 \times 14} = \frac{70}{14} 227=22×27×2=4414\frac{22}{7} = \frac{22 \times 2}{7 \times 2} = \frac{44}{14}

  17. Perform the addition and subtraction.

    7014−4414+3114=70−44+3114=26+3114=5714\frac{70}{14} - \frac{44}{14} + \frac{31}{14} = \frac{70 - 44 + 31}{14} = \frac{26 + 31}{14} = \frac{57}{14}

  18. Convert the improper fraction back to a mixed number.

    5714=4114\frac{57}{14} = 4\frac{1}{14}

    The simplified expression is 4114\boxed{4\frac{1}{14}}.

    Compare the fractions:

    a) Compare 58\frac{5}{8} and 47\frac{4}{7}.

  19. Find a common denominator for 8 and 7.

    The LCM is 8×7=568 \times 7 = 56.

  20. Convert both fractions to equivalent fractions with a denominator of 56.

    58=5×78×7=3556\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56} 47=4×87×8=3256\frac{4}{7} = \frac{4 \times 8}{7 \times 8} = \frac{32}{56}

  21. Compare the numerators.

    Since 35>3235 > 32, it follows that 3556>3256\frac{35}{56} > \frac{32}{56}.
    Therefore, 58>47\boxed{\frac{5}{8} > \frac{4}{7}}.

    e) Compare −419\frac{-4}{19} and −317\frac{-3}{17}.

  22. Find a common denominator for 19 and 17.

    The LCM is 19×17=32319 \times 17 = 323.

  23. Convert both fractions to equivalent fractions with a denominator of 323.

    −419=−4×1719×17=−68323\frac{-4}{19} = \frac{-4 \times 17}{19 \times 17} = \frac{-68}{323} −317=−3×1917×19=−57323\frac{-3}{17} = \frac{-3 \times 19}{17 \times 19} = \frac{-57}{323}

  24. Compare the numerators.

    For negative numbers, the number closer to zero is greater.
    Since −68<−57-68 < -57, it follows that −68323<−57323\frac{-68}{323} < \frac{-57}{323}.
    Therefore, −419<−317\boxed{\frac{-4}{19} < \frac{-3}{17}}.

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