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Answer :
To solve the given expression [tex]\(\left(\frac{5}{6}+\frac{7}{12}\right) \times\left(\frac{8}{15} \div \frac{4}{9}\right)\)[/tex], let's break it down step-by-step:
1. Calculate the sum of the fractions:
We start by adding [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{7}{12}\)[/tex]. To do this, we need a common denominator.
The least common denominator (LCD) of 6 and 12 is 12.
- Convert [tex]\(\frac{5}{6}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\][/tex]
- Now add the fractions:
[tex]\[
\frac{10}{12} + \frac{7}{12} = \frac{17}{12}
\][/tex]
2. Calculate the division of fractions:
We need to divide [tex]\(\frac{8}{15}\)[/tex] by [tex]\(\frac{4}{9}\)[/tex].
Dividing by a fraction is the same as multiplying by its reciprocal. So, invert [tex]\(\frac{4}{9}\)[/tex] to get [tex]\(\frac{9}{4}\)[/tex], and multiply:
[tex]\[
\frac{8}{15} \times \frac{9}{4}
\][/tex]
- Multiply the numerators and the denominators:
[tex]\[
\frac{8 \times 9}{15 \times 4} = \frac{72}{60}
\][/tex]
- Simplify [tex]\(\frac{72}{60}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 12:
[tex]\[
\frac{72 \div 12}{60 \div 12} = \frac{6}{5}
\][/tex]
3. Multiply the two results:
Now, we multiply the result from the sum [tex]\(\frac{17}{12}\)[/tex] by the result from the division [tex]\(\frac{6}{5}\)[/tex]:
[tex]\[
\frac{17}{12} \times \frac{6}{5}
\][/tex]
- Multiply the numerators and the denominators:
[tex]\[
\frac{17 \times 6}{12 \times 5} = \frac{102}{60}
\][/tex]
- Simplify [tex]\(\frac{102}{60}\)[/tex] by dividing both by their greatest common divisor, which is 6:
[tex]\[
\frac{102 \div 6}{60 \div 6} = \frac{17}{10}
\][/tex]
Therefore, the simplified result of the expression is [tex]\(\frac{17}{10}\)[/tex].
Since this fraction does not match any of the multiple-choice answers exactly, it indicates that the choices might consider the decimals. Among the provided options, the correct answer choice is not available because [tex]\(\frac{17}{10}\)[/tex] simplifies to [tex]\(1.7\)[/tex], which is distinct from the given options.
1. Calculate the sum of the fractions:
We start by adding [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{7}{12}\)[/tex]. To do this, we need a common denominator.
The least common denominator (LCD) of 6 and 12 is 12.
- Convert [tex]\(\frac{5}{6}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\][/tex]
- Now add the fractions:
[tex]\[
\frac{10}{12} + \frac{7}{12} = \frac{17}{12}
\][/tex]
2. Calculate the division of fractions:
We need to divide [tex]\(\frac{8}{15}\)[/tex] by [tex]\(\frac{4}{9}\)[/tex].
Dividing by a fraction is the same as multiplying by its reciprocal. So, invert [tex]\(\frac{4}{9}\)[/tex] to get [tex]\(\frac{9}{4}\)[/tex], and multiply:
[tex]\[
\frac{8}{15} \times \frac{9}{4}
\][/tex]
- Multiply the numerators and the denominators:
[tex]\[
\frac{8 \times 9}{15 \times 4} = \frac{72}{60}
\][/tex]
- Simplify [tex]\(\frac{72}{60}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 12:
[tex]\[
\frac{72 \div 12}{60 \div 12} = \frac{6}{5}
\][/tex]
3. Multiply the two results:
Now, we multiply the result from the sum [tex]\(\frac{17}{12}\)[/tex] by the result from the division [tex]\(\frac{6}{5}\)[/tex]:
[tex]\[
\frac{17}{12} \times \frac{6}{5}
\][/tex]
- Multiply the numerators and the denominators:
[tex]\[
\frac{17 \times 6}{12 \times 5} = \frac{102}{60}
\][/tex]
- Simplify [tex]\(\frac{102}{60}\)[/tex] by dividing both by their greatest common divisor, which is 6:
[tex]\[
\frac{102 \div 6}{60 \div 6} = \frac{17}{10}
\][/tex]
Therefore, the simplified result of the expression is [tex]\(\frac{17}{10}\)[/tex].
Since this fraction does not match any of the multiple-choice answers exactly, it indicates that the choices might consider the decimals. Among the provided options, the correct answer choice is not available because [tex]\(\frac{17}{10}\)[/tex] simplifies to [tex]\(1.7\)[/tex], which is distinct from the given options.
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