Question:
A car travels for 45 minutes at a speed of 80 km/h, the next 40 minutes at a speed of 48 km/h and the next 35 minutes at a speed of \(x\text{ km/h}\). If his average speed for the whole journey is \(72\frac{1}{4}\text{ km/h}\), then what is the value of \(x\)?
एक कार 45 मिनट तक 80 km/h की चाल से, अगले 40 मिनट तक 48 km/h की चाल से और अगले 35 मिनट तक \(x\text{ km/h}\) की चाल से यात्रा करती है। यदि पूरी यात्रा के लिए उसकी औसत चाल \(72\frac{1}{4}\text{ km/h}\) है, तो \(x\) का मान क्या है?
Options:
- 96
- 90
- 80
- 84
- 96
- 90
- 80
- 84
Correct Answer: b)
Explanation:
Total time of journey = \(45 + 40 + 35 = 120\text{ minutes} = 2\text{ hours}\).
Average speed = \(72\frac{1}{4}\text{ km/h} = \frac{289}{4}\text{ km/h} = 72.25\text{ km/h}\).
Total distance = \(\text{Average speed} \times \text{Total time} = \frac{289}{4} \times 2 = \frac{289}{2} = 144.5\text{ km}\).
Distances covered:
1. First part: \(80 \times \frac{45}{60} = 80 \times \frac{3}{4} = 60\text{ km}\)
2. Second part: \(48 \times \frac{40}{60} = 48 \times \frac{2}{3} = 32\text{ km}\)
3. Third part: \(x \times \frac{35}{60} = \frac{7}{12}x\text{ km}\)
Equation for total distance:
\[60 + 32 + \frac{7}{12}x = 144.5\]
\[92 + \frac{7}{12}x = 144.5\]
\[\frac{7}{12}x = 52.5\]
\[x = 52.5 \times \frac{12}{7} = 7.5 \times 12 = 90\text{ km/h}\]
Hence, the correct option is (b).
यात्रा का कुल समय = \(45 + 40 + 35 = 120\text{ मिनट} = 2\text{ घंटे}\)।
औसत चाल = \(72\frac{1}{4}\text{ km/h} = 72.25\text{ km/h}\)।
कुल दूरी = \(72.25 \times 2 = 144.5\text{ km}\)।
अलग-अलग भागों में तय दूरी:
1. पहला भाग = \(80 \times \frac{45}{60} = 60\text{ km}\)
2. दूसरा भाग = \(48 \times \frac{40}{60} = 32\text{ km}\)
3. तीसरा भाग = \(x \times \frac{35}{60} = \frac{7}{12}x\text{ km}\)
समीकरण:
\[60 + 32 + \frac{7}{12}x = 144.5\]
\[\frac{7}{12}x = 52.5\]
\[x = \frac{52.5 \times 12}{7} = 90\text{ km/h}\]
अतः सही विकल्प (b) है।