Answer :

To solve this problem, we will evaluate the expression [tex]4 \frac{2}{7} \cdot \frac{13}{15} - 2 \cdot \frac{1}{13} \cdot \frac{4}{9}[/tex] step-by-step.

First, let's convert the mixed number into an improper fraction:


  1. Convert [tex]4 \frac{2}{7}[/tex] to an improper fraction:

    • [tex]4 \frac{2}{7} = \frac{4 \cdot 7 + 2}{7} = \frac{28 + 2}{7} = \frac{30}{7}[/tex]




Now the expression becomes:

[tex]\frac{30}{7} \cdot \frac{13}{15} - 2 \cdot \frac{1}{13} \cdot \frac{4}{9}[/tex]

Let's solve this expression by evaluating each part:


  1. Multiply [tex]\frac{30}{7} \cdot \frac{13}{15}[/tex]:


    • [tex]\frac{30}{7} \cdot \frac{13}{15} = \frac{30 \cdot 13}{7 \cdot 15} = \frac{390}{105}[/tex]

    • Simplify [tex]\frac{390}{105}[/tex] by finding the greatest common divisor (GCD) of 390 and 105, which is 15:

      • [tex]390 \div 15 = 26[/tex]

      • [tex]105 \div 15 = 7[/tex]

      • This gives us [tex]\frac{26}{7}[/tex]





  2. Multiply [tex]2 \cdot \frac{1}{13} \cdot \frac{4}{9}[/tex]:


    • [tex]2 \cdot \frac{1}{13} = \frac{2}{13}[/tex]

    • [tex]\frac{2}{13} \cdot \frac{4}{9} = \frac{2 \cdot 4}{13 \cdot 9} = \frac{8}{117}[/tex]



  3. Subtract [tex]\frac{26}{7} - \frac{8}{117}[/tex]:


    • To subtract these fractions, find a common denominator. The least common multiple (LCM) of 7 and 117 is 819.

    • Convert [tex]\frac{26}{7}[/tex] to the new denominator:

      • [tex]\frac{26}{7} = \frac{26 \cdot 117}{819} = \frac{3042}{819}[/tex]



    • Convert [tex]\frac{8}{117}[/tex] to the new denominator:

      • [tex]\frac{8}{117} = \frac{8 \cdot 7}{819} = \frac{56}{819}[/tex]



    • Now subtract:

      • [tex]\frac{3042}{819} - \frac{56}{819} = \frac{2986}{819}[/tex]



    • This fraction can be simplified by finding the GCD of 2986 and 819, which is 1, so it remains [tex]\frac{2986}{819}[/tex].




Therefore, the solution to the expression is [tex]\frac{2986}{819}[/tex].

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