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Mathematics

Simplify the following:

(i) 123+212+341\dfrac{2}{3} + 2\dfrac{1}{2} + \dfrac{3}{4}

(ii) 329+213+27123\dfrac{2}{9} + 2\dfrac{1}{3} + 2\dfrac{7}{12}

(iii) 712+89−56\dfrac{7}{12} + \dfrac{8}{9} - \dfrac{5}{6}

(iv) 1325+720−251\dfrac{3}{25} + \dfrac{7}{20} - \dfrac{2}{5}

(v) 11314−256+1671\dfrac{13}{14} - 2\dfrac{5}{6} + 1\dfrac{6}{7}

(vi) 3−116−7153 - 1\dfrac{1}{6} - \dfrac{7}{15}

Fractions

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Answer

(i) 123+212+341\dfrac{2}{3} + 2\dfrac{1}{2} + \dfrac{3}{4}

LCM of 3, 2 and 4 = 12.

⇒53+52+34⇒5×43×4+5×62×6+3×34×3⇒2012+3012+912⇒20+30+912⇒5912⇒41112\Rightarrow \dfrac{5}{3} + \dfrac{5}{2} + \dfrac{3}{4}\\[1em] \Rightarrow \dfrac{5 \times 4}{3 \times 4} + \dfrac{5 \times 6}{2 \times 6} + \dfrac{3 \times 3}{4 \times 3}\\[1em] \Rightarrow \dfrac{20}{12} + \dfrac{30}{12} + \dfrac{9}{12}\\[1em] \Rightarrow \dfrac{20 + 30 + 9}{12}\\[1em] \Rightarrow \dfrac{59}{12}\\[1em] \Rightarrow 4\dfrac{11}{12}

Hence, 123+212+34=411121\dfrac{2}{3} + 2\dfrac{1}{2} + \dfrac{3}{4} = 4\dfrac{11}{12}.

(ii) 329+213+27123\dfrac{2}{9} + 2\dfrac{1}{3} + 2\dfrac{7}{12}

LCM of 9, 3 and 12 = 36.

⇒299+73+3112⇒29×49×4+7×123×12+31×312×3⇒11636+8436+9336⇒116+84+9336⇒29336⇒8536\Rightarrow \dfrac{29}{9} + \dfrac{7}{3} + \dfrac{31}{12}\\[1em] \Rightarrow \dfrac{29 \times 4}{9 \times 4} + \dfrac{7 \times 12}{3 \times 12} + \dfrac{31 \times 3}{12 \times 3}\\[1em] \Rightarrow \dfrac{116}{36} + \dfrac{84}{36} + \dfrac{93}{36}\\[1em] \Rightarrow \dfrac{116 + 84 + 93}{36}\\[1em] \Rightarrow \dfrac{293}{36}\\[1em] \Rightarrow 8\dfrac{5}{36}

Hence, 329+213+2712=85363\dfrac{2}{9} + 2\dfrac{1}{3} + 2\dfrac{7}{12} = 8\dfrac{5}{36}.

(iii) 712+89−56\dfrac{7}{12} + \dfrac{8}{9} - \dfrac{5}{6}

LCM of 12, 9 and 6 = 36.

⇒7×312×3+8×49×4−5×66×6⇒2136+3236−3036⇒21+32−3036⇒2336\Rightarrow \dfrac{7 \times 3}{12 \times 3} + \dfrac{8 \times 4}{9 \times 4} - \dfrac{5 \times 6}{6 \times 6}\\[1em] \Rightarrow \dfrac{21}{36} + \dfrac{32}{36} - \dfrac{30}{36}\\[1em] \Rightarrow \dfrac{21 + 32 - 30}{36}\\[1em] \Rightarrow \dfrac{23}{36}

Hence, 712+89−56=2336\dfrac{7}{12} + \dfrac{8}{9} - \dfrac{5}{6} = \dfrac{23}{36}.

(iv) 1325+720−251\dfrac{3}{25} + \dfrac{7}{20} - \dfrac{2}{5}

LCM of 25, 20 and 5 = 100.

⇒2825+720−25⇒28×425×4+7×520×5−2×205×20⇒112100+35100−40100⇒112+35−40100⇒107100⇒17100\Rightarrow \dfrac{28}{25} + \dfrac{7}{20} - \dfrac{2}{5}\\[1em] \Rightarrow \dfrac{28 \times 4}{25 \times 4} + \dfrac{7 \times 5}{20 \times 5} - \dfrac{2 \times 20}{5 \times 20}\\[1em] \Rightarrow \dfrac{112}{100} + \dfrac{35}{100} - \dfrac{40}{100}\\[1em] \Rightarrow \dfrac{112 + 35 - 40}{100}\\[1em] \Rightarrow \dfrac{107}{100}\\[1em] \Rightarrow 1\dfrac{7}{100}

Hence, 1325+720−25=171001\dfrac{3}{25} + \dfrac{7}{20} - \dfrac{2}{5} = 1\dfrac{7}{100}.

(v) 11314−256+1671\dfrac{13}{14} - 2\dfrac{5}{6} + 1\dfrac{6}{7}

LCM of 14, 6 and 7 = 42.

⇒2714−176+137⇒27×314×3−17×76×7+13×67×6⇒8142−11942+7842⇒81−119+7842⇒4042⇒2021\Rightarrow \dfrac{27}{14} - \dfrac{17}{6} + \dfrac{13}{7}\\[1em] \Rightarrow \dfrac{27 \times 3}{14 \times 3} - \dfrac{17 \times 7}{6 \times 7} + \dfrac{13 \times 6}{7 \times 6}\\[1em] \Rightarrow \dfrac{81}{42} - \dfrac{119}{42} + \dfrac{78}{42}\\[1em] \Rightarrow \dfrac{81 - 119 + 78}{42}\\[1em] \Rightarrow \dfrac{40}{42}\\[1em] \Rightarrow \dfrac{20}{21}

Hence, 11314−256+167=20211\dfrac{13}{14} - 2\dfrac{5}{6} + 1\dfrac{6}{7} = \dfrac{20}{21}.

(vi) 3−116−7153 - 1\dfrac{1}{6} - \dfrac{7}{15}

LCM of 1, 6 and 15 = 30.

⇒31−76−715⇒3×301×30−7×56×5−7×215×2⇒9030−3530−1430⇒90−35−1430⇒4130⇒11130\Rightarrow \dfrac{3}{1} - \dfrac{7}{6} - \dfrac{7}{15}\\[1em] \Rightarrow \dfrac{3 \times 30}{1 \times 30} - \dfrac{7 \times 5}{6 \times 5} - \dfrac{7 \times 2}{15 \times 2}\\[1em] \Rightarrow \dfrac{90}{30} - \dfrac{35}{30} - \dfrac{14}{30}\\[1em] \Rightarrow \dfrac{90 - 35 - 14}{30}\\[1em] \Rightarrow \dfrac{41}{30}\\[1em] \Rightarrow 1\dfrac{11}{30}

Hence, 3−116−715=111303 - 1\dfrac{1}{6} - \dfrac{7}{15} = 1\dfrac{11}{30}.

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