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Simplify the following expressions: a) 3 rac14 - 2 rac25 + 1 rac110 and d) 5 - 3 rac17 + 2 rac314.

Mathematics

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2 parts
a

119201\frac{19}{20}

d

41144\frac{1}{14}

Step-by-step solution

12 steps

Okay, Sheela, let's simplify these expressions.

Part a· 6 steps

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

  1. Convert 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}

  2. Rewrite the expression.

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

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

    The LCM is 20.

  4. 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}

  5. 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}

  6. 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}}.

Part d

41144\frac{1}{14}

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

  1. Convert mixed numbers to improper fractions.

    5=515 = \frac{5}{1} 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}

  2. Rewrite the expression.

    51−227+3114\frac{5}{1} - \frac{22}{7} + \frac{31}{14}

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

    The LCM is 14.

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

    51=5×141×14=7014\frac{5}{1} = \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}

  5. 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}

  6. 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}}.

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