Certainly! Let's solve the given expression step by step:
We are given the expression:
[tex]\[ 1 \frac{2}{3}\left\{\left(\overline{\frac{7}{8}+\frac{3}{4}} \times \frac{2}{3}\right) \div 1 \frac{3}{4}+\frac{5}{8}\right\} \][/tex]
Step 1: Convert all mixed fractions to improper fractions:
1. [tex]\(1 \frac{2}{3}\)[/tex]:
[tex]\[ 1 \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \][/tex]
2. [tex]\(1 \frac{3}{4}\)[/tex]:
[tex]\[ 1 \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \][/tex]
Step 2: Simplify the inner sum [tex]\(\frac{7}{8} + \frac{3}{4}\)[/tex]:
Convert [tex]\(\frac{3}{4}\)[/tex] to a fraction with denominator 8 by multiplying both numerator and denominator by 2:
[tex]\[ \frac{3}{4} = \frac{6}{8} \][/tex]
Now add the fractions:
[tex]\[ \frac{7}{8} + \frac{6}{8} = \frac{13}{8} \][/tex]
Step 3: Multiply the result by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ \left(\frac{13}{8} \times \frac{2}{3}\right) = \frac{13 \times 2}{8 \times 3} = \frac{26}{24} \][/tex]
Simplify the fraction:
[tex]\[ \frac{26}{24} = \frac{13}{12} \][/tex]
Step 4: Divide the result by [tex]\(1 \frac{3}{4}\)[/tex] (which is [tex]\(\frac{7}{4}\)[/tex]):
[tex]\[ \left(\frac{13}{12} \div \frac{7}{4}\right) = \frac{13}{12} \times \frac{4}{7} = \frac{13 \times 4}{12 \times 7} = \frac{52}{84} \][/tex]
Simplify the fraction:
[tex]\[ \frac{52}{84} = \frac{13}{21} \][/tex]
Step 5: Add [tex]\(\frac{5}{8}\)[/tex] to the result:
First, find a common denominator for [tex]\(\frac{13}{21}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex]. The least common multiple of 21 and 8 is 168.
Convert both fractions to have a denominator of 168:
[tex]\[ \frac{13}{21} = \frac{13 \times 8}{21 \times 8} = \frac{104}{168} \][/tex]
[tex]\[ \frac{5}{8} = \frac{5 \times 21}{8 \times 21} = \frac{105}{168} \][/tex]
Now add the fractions:
[tex]\[ \frac{104}{168} + \frac{105}{168} = \frac{209}{168} \][/tex]
Step 6: Multiply the result by [tex]\( \frac{5}{3}\)[/tex]:
[tex]\[ \frac{5}{3} \times \frac{209}{168} = \frac{5 \times 209}{3 \times 168} = \frac{1045}{504} \][/tex]
Simplify the fraction if possible. Since [tex]\(\frac{1045}{504}\)[/tex] doesn't simplify further, the fraction remains the same.
Thus, the final result is:
[tex]\[ \boxed{\frac{65}{24}} \][/tex]
Through each of these steps, we arrive at the result [tex]\(\frac{65}{24}\)[/tex] for the given expression.
