Question:
Find the value of $\frac{12 + 3\frac{2}{3} \div \left[1\frac{3}{4} \div \left\{5\frac{1}{4} \times \left(3\frac{3}{5} \div 4\frac{1}{2}\right)\right\}\right]}{1\frac{1}{3} \times 5\frac{1}{5}}$.
$\frac{12 + 3\frac{2}{3} \div \left[1\frac{3}{4} \div \left\{5\frac{1}{4} \times \left(3\frac{3}{5} \div 4\frac{1}{2}\right)\right\}\right]}{1\frac{1}{3} \times 5\frac{1}{5}}$ का मान ज्ञात कीजिए।
Options:
- a) 1
- b) 3
- c) 5
- d) 2
- a) 1
- b) 3
- c) 5
- d) 2
correct answer : b)
Explanation:
First, convert all mixed fractions to improper fractions:
– $3\frac{2}{3} = \frac{11}{3}$
– $1\frac{3}{4} = \frac{7}{4}$
– $5\frac{1}{4} = \frac{21}{4}$
– $3\frac{3}{5} = \frac{18}{5}$
– $4\frac{1}{2} = \frac{9}{2}$
– $1\frac{1}{3} = \frac{4}{3}$
– $5\frac{1}{5} = \frac{26}{5}$
Now, simplify step-by-step from the inside:
1. Innermost brackets: $\frac{18}{5} \div \frac{9}{2} = \frac{18}{5} \times \frac{2}{9} = \frac{4}{5}$.
2. Braces: $\frac{21}{4} \times \frac{4}{5} = \frac{21}{5}$.
3. Square brackets: $\frac{7}{4} \div \frac{21}{5} = \frac{7}{4} \times \frac{5}{21} = \frac{5}{12}$.
4. Divide by 11/3: $\frac{11}{3} \div \frac{5}{12} = \frac{11}{3} \times \frac{12}{5} = \frac{44}{5}$.
5. Numerator becomes: $12 + \frac{44}{5} = \frac{60 + 44}{5} = \frac{104}{5}$.
6. Denominator is: $\frac{4}{3} \times \frac{26}{5} = \frac{104}{15}$.
Finally, evaluate the entire fraction:
$\frac{104/5}{104/15} = \frac{104}{5} \times \frac{15}{104} = \frac{15}{5} = 3$.
Hence, option (b) is the correct answer.
पहले सभी मिश्रित भिन्नों को अनुचित भिन्नों (improper fractions) में बदलें:
– $3\frac{2}{3} = \frac{11}{3}$
– $1\frac{3}{4} = \frac{7}{4}$
– $5\frac{1}{4} = \frac{21}{4}$
– $3\frac{3}{5} = \frac{18}{5}$
– $4\frac{1}{2} = \frac{9}{2}$
– $1\frac{1}{3} = \frac{4}{3}$
– $5\frac{1}{5} = \frac{26}{5}$
अब, अंदर से शुरू करते हुए चरण-दर-चरण सरल करें:
1. सबसे आंतरिक कोष्ठक: $\frac{18}{5} \div \frac{9}{2} = \frac{18}{5} \times \frac{2}{9} = \frac{4}{5}$।
2. मँझला कोष्ठक: $\frac{21}{4} \times \frac{4}{5} = \frac{21}{5}$।
3. बड़ा कोष्ठक: $\frac{7}{4} \div \frac{21}{5} = \frac{7}{4} \times \frac{5}{21} = \frac{5}{12}$।
4. 11/3 से भाग दें: $\frac{11}{3} \div \frac{5}{12} = \frac{11}{3} \times \frac{12}{5} = \frac{44}{5}$।
5. अंश (Numerator) हो जाता है: $12 + \frac{44}{5} = \frac{60 + 44}{5} = \frac{104}{5}$।
6. हर (Denominator) है: $\frac{4}{3} \times \frac{26}{5} = \frac{104}{15}$।
अंत में, संपूर्ण भिन्न का मान ज्ञात करें:
$\frac{104/5}{104/15} = \frac{104}{5} \times \frac{15}{104} = \frac{15}{5} = 3$।
अतः, विकल्प (b) सही उत्तर है।