Three runners are running in a circular track, and they complete one round in 20, 30 and 35 minutes respectively. When will they next meet at the starting point ?
After 7 hours
Understanding the Runners on a Circular Track Problem
This problem involves three runners moving around a circular track, each completing a round in a different amount of time. We need to find out when they will all meet again at the exact starting point from which they began their run simultaneously.
To meet again at the starting point, each runner must have completed a whole number of rounds. This means the time elapsed must be a multiple of each runner's individual round completion time. The next time they all meet at the starting point will be the smallest time that is a multiple of all three times. This smallest common multiple is known as the Least Common Multiple (LCM).
Calculating the Least Common Multiple (LCM)
The times taken by the three runners to complete one round are:
- Runner 1: 20 minutes
- Runner 2: 30 minutes
- Runner 3: 35 minutes
We need to find the LCM of 20, 30, and 35 minutes.
One common method to find the LCM is by using the prime factorization of each number.
Prime Factorization of the Times
- Prime factorization of 20: \(20 = 2 \times 10 = 2 \times 2 \times 5 = 2^2 \times 5^1\)
- Prime factorization of 30: \(30 = 2 \times 15 = 2 \times 3 \times 5 = 2^1 \times 3^1 \times 5^1\)
- Prime factorization of 35: \(35 = 5 \times 7 = 5^1 \times 7^1\)
Finding the LCM using Prime Factors
To find the LCM, we take the highest power of all the prime factors that appear in any of the numbers:
- Highest power of 2 is \(2^2\) (from 20)
- Highest power of 3 is \(3^1\) (from 30)
- Highest power of 5 is \(5^1\) (from 20, 30, and 35)
- Highest power of 7 is \(7^1\) (from 35)
LCM = \(2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7\)
LCM = \(12 \times 35\)
LCM = \(420\)
Interpreting the Result: Time to Meet
The LCM of 20, 30, and 35 minutes is 420 minutes. This means that 420 minutes is the shortest amount of time after which all three runners will have completed a whole number of rounds and will be back at the starting point simultaneously.
Let's convert this time into hours and minutes:
\(420 \text{ minutes} = \frac{420}{60} \text{ hours}\)
\(420 \text{ minutes} = 7 \text{ hours}\)
So, the three runners will next meet at the starting point after 7 hours.
Checking the Options
Let's look at the given options:
| Option | Time | Time in Minutes | Is it a multiple of 20, 30, and 35? |
|---|---|---|---|
| 1 | 3 hours 30 minutes | \((3 \times 60) + 30 = 180 + 30 = 210\) minutes | 210 is a multiple of 30 (\(210 = 7 \times 30\)) and 35 (\(210 = 6 \times 35\)), but not 20 (210 / 20 is not a whole number). |
| 2 | 4 hours 30 minutes | \((4 \times 60) + 30 = 240 + 30 = 270\) minutes | 270 is a multiple of 30 (\(270 = 9 \times 30\)), but not 20 or 35. |
| 3 | 3 hours | \(3 \times 60 = 180\) minutes | 180 is a multiple of 20 (\(180 = 9 \times 20\)) and 30 (\(180 = 6 \times 30\)), but not 35. |
| 4 | 7 hours | \(7 \times 60 = 420\) minutes | 420 is a multiple of 20 (\(420 = 21 \times 20\)), 30 (\(420 = 14 \times 30\)), and 35 (\(420 = 12 \times 35\)). This is the LCM. |
The calculation confirms that 7 hours is the correct time when all three runners will next meet at the starting point.
Revision Table: Key Concepts
| Concept | Explanation | Relevance to Problem |
|---|---|---|
| Circular Track Problem | Problems where objects move in a loop and we need to find when they meet. | The scenario involves runners on a circular track. |
| Starting Point | The common location from where all runners begin and need to return to meet. | Runners must meet at the starting point, meaning time must be a multiple of each runner's lap time. |
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more integers. | Used to find the earliest time when multiple events (completing a round) occurring at different frequencies will coincide. |
| Prime Factorization | Breaking down a number into its prime number components. | A method for finding the LCM of multiple numbers. |
Additional Information: Applications of LCM
The concept of LCM is not just useful for runner problems! It has many real-world applications:
- Scheduling: Finding when events that repeat at different intervals will next happen at the same time (e.g., bus schedules, train schedules, meeting cycles).
- Tiling Problems: Finding the smallest square that can be tiled by rectangles of specific dimensions.
- Gears and Cycles: Determining when two rotating gears or cycles with different numbers of teeth or steps will return to their original relative positions.
- Fractions: Finding the least common denominator (LCD) when adding or subtracting fractions, which is the LCM of the denominators.
In all these cases, LCM helps find the smallest synchronized point or quantity.
Similar Questions
If (x + k) is the HCF of x 2+ 5x + 6 and x 2+ 8x + 15, then what is the value of k?
Let L be the LCM and H be the HCF of two given numbers. L and H are in the ratio 3 ∶ 2. If the sum of the two numbers is 45, then what is the product of the numbers?
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HCF of two numbers is 12. Which one of the following can never be their LCM?
X, Y and Z start at some point and same time in the same direction to run around a circular stadium. X completes a round in 252 seconds, Y in 308 seconds and Z in 198 seconds. After what time will they meet again at the starting point?
What is the LCM of 1/3, 5/6, 2/9, 4/27?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
The sum of two numbers is 1215 and their HCF is 81. How many such pairs of numbers can be formed?
The LCM of two numbers in 48. Their ratio is 2:3 What is the sum of the numbers?
