Answer :

To evaluate the expression [tex]\left( 5 \frac{2}{5} \times 3 - 1 \frac{13}{15} \div 4 \frac{13}{15} + 3 \frac{5}{6} \right)[/tex], follow these steps:

First, it is helpful to convert all mixed numbers to improper fractions for easier calculation.


  1. Convert Mixed Numbers to Improper Fractions:


    • For [tex]5 \frac{2}{5}[/tex]:
      [tex]5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{27}{5}[/tex]

    • For [tex]1 \frac{13}{15}[/tex]:
      [tex]1 \frac{13}{15} = \frac{1 \times 15 + 13}{15} = \frac{28}{15}[/tex]

    • For [tex]4 \frac{13}{15}[/tex]:
      [tex]4 \frac{13}{15} = \frac{4 \times 15 + 13}{15} = \frac{73}{15}[/tex]

    • For [tex]3 \frac{5}{6}[/tex]:
      [tex]3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{23}{6}[/tex]



  2. Substitute back into the expression:
    [tex]\left( \frac{27}{5} \times 3 - \frac{28}{15} \div \frac{73}{15} + \frac{23}{6} \right)[/tex]


  3. Multiply [tex]\frac{27}{5}[/tex] by [tex]3[/tex]:


    • Convert [tex]3[/tex] to a fraction: [tex]\frac{3}{1}[/tex]

    • Multiply:
      [tex]\frac{27}{5} \times \frac{3}{1} = \frac{81}{5}[/tex]



  4. Divide [tex]\frac{28}{15}[/tex] by [tex]\frac{73}{15}[/tex]:


    • When dividing by a fraction, multiply by its reciprocal:
      [tex]\frac{28}{15} \div \frac{73}{15} = \frac{28}{15} \times \frac{15}{73} = \frac{28}{73}[/tex]



  5. Simplify the expression:
    [tex]\left( \frac{81}{5} - \frac{28}{73} + \frac{23}{6} \right)[/tex]


    • To subtract and add fractions, find a common denominator. The least common multiple of 5, 73, and 6 is 2190.



  6. Convert each fraction to have a common denominator:
    [tex]\frac{81}{5} = \frac{81 \times 438}{5 \times 438} = \frac{35478}{2190}[/tex]
    [tex]\frac{28}{73} = \frac{28 \times 30}{73 \times 30} = \frac{840}{2190}[/tex]
    [tex]\frac{23}{6} = \frac{23 \times 365}{6 \times 365} = \frac{8395}{2190}[/tex]


  7. Perform the operations with the common denominator:
    [tex]\frac{35478}{2190} - \frac{840}{2190} + \frac{8395}{2190} = \frac{35478 - 840 + 8395}{2190} = \frac{43033}{2190}[/tex]


    • Simplify [tex]\frac{43033}{2190}[/tex] if necessary.




Therefore, the evaluated expression is [tex]\frac{43033}{2190}[/tex] or simplified further to approximately [tex]19.64[/tex].

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