The problem asks for the greatest possible speed, let's call it s (in km/h), such that the time taken to travel two different distances, 13.3 km and 20.9 km, results in whole numbers.
Finding the Speed Condition
We know the relationship between speed, distance, and time: Time = Distance / Speed.
- For the first distance: Time_1 = 13.3 km / s km/h = \(n_1\) hours, where \(n_1\) is a whole number.
- For the second distance: Time_2 = 20.9 km / s km/h = \(n_2\) hours, where \(n_2\) is a whole number.
This implies that the speed s must be a value such that when it divides both 13.3 and 20.9, the results (\(n_1\) and \(n_2\)) are integers.
Using Decimal Representation
Let's rewrite the distances as fractions to work with whole numbers:
- \(13.3 = \frac{133}{10}\)
- \(20.9 = \frac{209}{10}\)
The conditions become:
- \(\frac{133/10}{s} = n_1 \implies s = \frac{133}{10 \times n_1}\)
- \(\frac{209/10}{s} = n_2 \implies s = \frac{209}{10 \times n_2}\)
For s to be the greatest possible speed, \(n_1\) and \(n_2\) must be the smallest possible positive integers that satisfy the relationship derived from equating the expressions for s:
\(\frac{133}{10 \times n_1} = \frac{209}{10 \times n_2}\)
\(\frac{133}{n_1} = \frac{209}{n_2}\)
Rearranging gives:
\(133 \times n_2 = 209 \times n_1 \implies \frac{n_2}{n_1} = \frac{209}{133}\)
Calculating the Greatest Common Divisor (GCD)
To find the simplest ratio \(\frac{n_2}{n_1}\), we need to find the greatest common divisor (GCD) of 209 and 133.
- Prime factorization of 133: \(133 = 7 \times 19\)
- Prime factorization of 209: \(209 = 11 \times 19\)
- The GCD(133, 209) is 19.
Simplify the ratio:
\(\frac{n_2}{n_1} = \frac{209 \div 19}{133 \div 19} = \frac{11}{7}\)
The smallest whole numbers for \(n_1\) and \(n_2\) are \(n_1=7\) and \(n_2=11\).
Determining the Maximum Speed
Now, substitute these smallest integer times back into the speed equations:
- Using \(n_1=7\): \(s = \frac{13.3}{n_1} = \frac{13.3}{7} = 1.9\) km/h.
- Using \(n_2=11\): \(s = \frac{20.9}{n_2} = \frac{20.9}{11} = 1.9\) km/h.
Both calculations yield the same speed, 1.9 km/h. Since we used the smallest possible whole number times derived from the GCD, this speed is the greatest possible speed.
Final Answer Verification
Let's check if this speed yields whole number times:
- Time for 13.3 km: \(t_1 = \frac{13.3 \text{ km}}{1.9 \text{ km/h}} = 7\) hours (a whole number).
- Time for 20.9 km: \(t_2 = \frac{20.9 \text{ km}}{1.9 \text{ km/h}} = 11\) hours (a whole number).
The speed 1.9 km/h satisfies the conditions.
Similar Questions
- Two trains A and B leave Delhi for Hyderabad at 7:00 a.m. and 7:50 a.m. on the same day and travel at 80 kmph and 100 kmph respectively. After how many kilometers from Delhi will the two trains be together?
- The speeds of four cars are \(2u, 3u, 4u\) and \(xu\) and the time taken by them to cover the same distance is \(xt, 4t, 3t\) and \(2t\) respectively, where \(x, u, t\) are real numbers. What is the value of \(x\)?
Two towers \(A\) and \(B\) of height 23 m and 11 m respectively, stand 9 m apart. A straight rod is joined to the two tops of the towers. A monkey sitting on the top of \(A\), climbs the rod to reach the top of \(B\). If the monkey takes 5 minutes to reach the other end, what is the average speed of the monkey?
What is the remainder when 2P - 1 is divided by p, where p > 5 is a prime number?
If sin a + cos a = √2, where 0 < a < π/2, then what is sin³ a cos³ a equal to?
- A train P starts from station X for station Y and another train Q starts from station Y for station X at the same time. After passing each other, P takes 4 hours to reach Y and Q takes 1 hour to reach X. If the average speed of P is 60 km/h, then what is the average speed of Q ?
- Without stoppages, a train travels at an average speed of 60 km/h and with stoppages it covers the same distance at an average speed of 48 km/h. How many minutes per hour does the train stop ?
- Two trains X and Y are travelling in the same direction at 100 km/hr and 60 km/hr respectively. Train X crosses a man in train Y in 9 seconds. What is the length of train X?
- Two persons X and Y leave place P for place Q at 7:00 a.m. and 7:10 a.m. respectively along the same path. X walks at a speed of 4.8 km/hr and Y walks at a speed of 6 km/hr. How many kilometres from place P will X meet Y?
Important Questions from Speed Time and Distance
A train travelling at a speed of 72 km/hr crosses a post in 20 seconds. If it crosses another train travelling at a speed of 54 km/hr in the same direction in 1 minute 45 seconds, then the difference in length between the two trains is
Rajiv's boat can travel along the current at the 8 km/hour and against the current at the rate 6 km/hour. Find the time taken by the boat to sail 28 km in still water.
Rohit and Dinesh are 64 km apart. Rohit can walk at a speed of 15 km/hr and Dinesh at the speed of 17 km/hr. In how many hours will they meet if they are travelling towards each other?
Two trains running in opposite directions cross a man standing on the platform in 25 seconds and 32 seconds respectively and they cross each other in 30 seconds. The ratio of their speed is:
A worker covers a distance of 81 km in 11 hours. He travels partly on foot at 4.5 km/h and partly on bicycle at 15 km/h. What is the distance covered on the cycle?
