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Evaluate the expression: \(\frac{\frac{1}{2} + \frac{2}{3}}{\frac{14}{15}}\)

Answer :

To solve the expression [tex](\frac{1}{2} + \frac{2}{3}) / (\frac{14}{15})[/tex], follow these steps:

  1. Simplify the Numerator: Start by adding the fractions in the numerator, [tex]\frac{1}{2} + \frac{2}{3}[/tex]. To do this, first find a common denominator:

    • The denominators are 2 and 3. The least common denominator (LCD) is 6.

    • Convert [tex]\frac{1}{2}[/tex] to [tex]\frac{3}{6}[/tex] and [tex]\frac{2}{3}[/tex] to [tex]\frac{4}{6}[/tex].

    • Now, add the fractions: [tex]\frac{3}{6} + \frac{4}{6} = \frac{7}{6}[/tex].

  2. Divide by the Denominator: Now divide the result by [tex]\frac{14}{15}[/tex]:

    • This is equivalent to multiplying by the reciprocal of [tex]\frac{14}{15}[/tex], which is [tex]\frac{15}{14}[/tex].

    • So, calculate [tex]\frac{7}{6} \times \frac{15}{14}[/tex].

  3. Multiply the Fractions: To multiply fractions, multiply the numerators and then the denominators:

    • Numerators: [tex]7 \times 15 = 105[/tex].

    • Denominators: [tex]6 \times 14 = 84[/tex].

    • Therefore, [tex]\frac{7}{6} \times \frac{15}{14} = \frac{105}{84}[/tex].

  4. Simplify the Result: Simplify [tex]\frac{105}{84}[/tex] to its lowest terms:

    • Find the greatest common divisor (GCD) of 105 and 84, which is 21.

    • Divide the numerator and denominator by 21.

    • [tex]\frac{105}{84} = \frac{105 \div 21}{84 \div 21} = \frac{5}{4}[/tex].

    So, the simplified answer is [tex]\frac{5}{4}[/tex] or [tex]1.25[/tex] when expressed as a decimal.

In conclusion, the result of the expression [tex](\frac{1}{2} + \frac{2}{3}) / (\frac{14}{15})[/tex] is [tex]\frac{5}{4}[/tex].

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