High School

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Given the expression:

\[ \frac{15}{20} \cdot \frac{24}{30} \cdot \frac{18}{27} - \frac{3}{5} \cdot \frac{27}{18} = \frac{22}{30} = \frac{9}{10} \]

Which of the following is correct?

B) \( \frac{9}{10} \)
C) \( \frac{1}{10} \)
D) \( \frac{1}{9} \)

Answer :

To solve the expression given and compare the results to the multiple-choice options provided, we need to simplify each part step-by-step.

  1. Simplify each fraction:

    • [tex]\frac{15}{20} \ = \ \frac{3}{4}[/tex]
    • [tex]\frac{24}{30} \ = \ \frac{4}{5}[/tex]
    • [tex]\frac{18}{27} \ = \ \frac{2}{3}[/tex]
    • [tex]\frac{3}{5}[/tex] (already simplified)
    • [tex]\frac{27}{18} \ = \ \frac{3}{2}[/tex]

  2. Multiply the fractions in the first part of the expression:
    (
    \frac{3}{4} \cdot \frac{4}{5} \cdot \frac{2}{3} \ = \ \frac{3 \cdot 4 \cdot 2}{4 \cdot 5 \cdot 3} = \frac{24}{60} \ = \ \frac{2}{5}
    )

  3. Multiply the fractions in the subtraction part of the expression:
    (
    \frac{3}{5} \cdot \frac{3}{2} \ = \ \frac{3 \cdot 3}{5 \cdot 2} = \frac{9}{10}
    )

  4. Subtract the results of the two calculations:
    (
    \frac{2}{5} - \frac{9}{10}
    )

    To subtract these fractions, find a common denominator, which is 10:
    (
    \frac{2}{5} \ = \ \frac{4}{10}
    )

    Subtract the fractions:
    (
    \frac{4}{10} - \frac{9}{10} \ = \ -\frac{5}{10} \ = \ -\frac{1}{2}
    )

  5. Compare to the given options:
    None of the positive fractions [tex]\frac{9}{10}[/tex], [tex]\frac{1}{10}[/tex], or [tex]\frac{1}{9}[/tex] matches, and we noted the calculation resulted in a negative fraction [tex]-\frac{1}{2}[/tex], indicating an error in matching choices. Typically, such evaluations might require re-evaluation of initial conditions or choices given. As none are exact matches, [tex]\mathbf{None \ of \ the \ options \ provided \ match \ the \ exact \ simplified \ result.}[/tex]

The correct evaluation does not align with the positive options provided. Please verify the exact match or potential errors in the option provided context.

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Rewritten by : Barada