Question:
The value of \(\frac{\frac{1}{2} \times \frac{3}{4} \div \frac{2}{5}\text{ of } 1\frac{1}{4}}{5\frac{1}{2} \div 3\frac{2}{3}\text{ of }\frac{1}{2} – 3\frac{1}{2} + 7\frac{1}{2} \div \frac{2}{5}\text{ of } 3\frac{3}{4}}\) is:
\(\frac{\frac{1}{2} \times \frac{3}{4} \div \frac{2}{5}\text{ of } 1\frac{1}{4}}{5\frac{1}{2} \div 3\frac{2}{3}\text{ of }\frac{1}{2} – 3\frac{1}{2} + 7\frac{1}{2} \div \frac{2}{5}\text{ of } 3\frac{3}{4}}\) का मान ज्ञात कीजिए।
Options:
- \(\frac{3}{2}\)
- \(\frac{1}{6}\)
- \(\frac{1}{2}\)
- \(\frac{5}{4}\)
- \(\frac{3}{2}\)
- \(\frac{1}{6}\)
- \(\frac{1}{2}\)
- \(\frac{5}{4}\)
Correct Answer: b)
Explanation:
To evaluate the given expression, apply the BODMAS rule (Brackets, Orders/Of, Division, Multiplication, Addition, Subtraction) separately to the numerator and denominator:
Step 1: Simplify the Numerator:
\[ \text{Numerator} = \frac{1}{2} \times \frac{3}{4} \div \left(\frac{2}{5} \text{ of } 1\frac{1}{4}\right) \]
First, convert the mixed fraction into an improper fraction:
\[ 1\frac{1}{4} = \frac{5}{4} \]
Evaluate the ‘of’ operation:
\[ \frac{2}{5} \times \frac{5}{4} = \frac{2}{4} = \frac{1}{2} \]
Now perform the division:
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3}{2} \]
Then multiply:
\[ \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \]
So, \(\text{Numerator} = \frac{3}{4}\).
Step 2: Simplify the Denominator:
\[ \text{Denominator} = 5\frac{1}{2} \div \left(3\frac{2}{3} \text{ of } \frac{1}{2}\right) – 3\frac{1}{2} + 7\frac{1}{2} \div \left(\frac{2}{5} \text{ of } 3\frac{3}{4}\right) \]
Convert all mixed fractions:
\[ 5\frac{1}{2} = \frac{11}{2}, \quad 3\frac{2}{3} = \frac{11}{3}, \quad 3\frac{1}{2} = \frac{7}{2}, \quad 7\frac{1}{2} = \frac{15}{2}, \quad 3\frac{3}{4} = \frac{15}{4} \]
Evaluate ‘of’ in the first term:
\[ 3\frac{2}{3} \text{ of } \frac{1}{2} = \frac{11}{3} \times \frac{1}{2} = \frac{11}{6} \]
Divide for the first term:
\[ \frac{11}{2} \div \frac{11}{6} = \frac{11}{2} \times \frac{6}{11} = 3 \]
The second term is:
\[ -3\frac{1}{2} = -\frac{7}{2} \]
Evaluate ‘of’ in the third term:
\[ \frac{2}{5} \text{ of } \frac{15}{4} = \frac{2}{5} \times \frac{15}{4} = \frac{3}{2} \]
Divide for the third term:
\[ \frac{15}{2} \div \frac{3}{2} = \frac{15}{2} \times \frac{2}{3} = 5 \]
Combine all parts of the denominator:
\[ \text{Denominator} = 3 – \frac{7}{2} + 5 = 8 – \frac{7}{2} = \frac{16 – 7}{2} = \frac{9}{2} \]
Step 3: Compute the Final Fraction:
\[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{3}{4}}{\frac{9}{2}} = \frac{3}{4} \times \frac{2}{9} = \frac{6}{36} = \frac{1}{6} \]
Therefore, the correct answer is option b).
दिए गए व्यंजक का मान ज्ञात करने के लिए, अंश और हर पर BODMAS नियम लागू करते हैं:
चरण 1: अंश को सरल करें:
\[ \text{अंश} = \frac{1}{2} \times \frac{3}{4} \div \left(\frac{2}{5} \text{ of } 1\frac{1}{4}\right) \]
मिश्रित भिन्न को विषम भिन्न में बदलें:
\[ 1\frac{1}{4} = \frac{5}{4} \]
पहले ‘का’ (of) को हल करें:
\[ \frac{2}{5} \times \frac{5}{4} = \frac{1}{2} \]
अब भाग करें:
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3}{2} \]
अब गुणा करें:
\[ \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \]
अतः, \(\text{अंश} = \frac{3}{4}\)।
चरण 2: हर को सरल करें:
\[ \text{हर} = 5\frac{1}{2} \div \left(3\frac{2}{3} \text{ of } \frac{1}{2}\right) – 3\frac{1}{2} + 7\frac{1}{2} \div \left(\frac{2}{5} \text{ of } 3\frac{3}{4}\right) \]
सभी मिश्रित भिन्नों को बदलें:
\[ 5\frac{1}{2} = \frac{11}{2}, \quad 3\frac{2}{3} = \frac{11}{3}, \quad 3\frac{1}{2} = \frac{7}{2}, \quad 7\frac{1}{2} = \frac{15}{2}, \quad 3\frac{3}{4} = \frac{15}{4} \]
पहले पद में ‘का’ (of) का मान:
\[ \frac{11}{3} \times \frac{1}{2} = \frac{11}{6} \]
पहले पद का भाग:
\[ \frac{11}{2} \div \frac{11}{6} = \frac{11}{2} \times \frac{6}{11} = 3 \]
दूसरा पद:
\[ -\frac{7}{2} \]
तीसरे पद में ‘का’ (of) का मान:
\[ \frac{2}{5} \times \frac{15}{4} = \frac{3}{2} \]
तीसरे पद का भाग:
\[ \frac{15}{2} \div \frac{3}{2} = \frac{15}{2} \times \frac{2}{3} = 5 \]
हर के सभी पदों को जोड़ें:
\[ \text{हर} = 3 – \frac{7}{2} + 5 = 8 – \frac{7}{2} = \frac{9}{2} \]
चरण 3: अंतिम मान निकालें:
\[ \frac{\text{अंश}}{\text{हर}} = \frac{\frac{3}{4}}{\frac{9}{2}} = \frac{3}{4} \times \frac{2}{9} = \frac{1}{6} \]
अतः सही उत्तर विकल्प b) है।