NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 7
NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 7
Question:
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Given:
- Sonia takes 18 minutes to complete one round.
- Ravi takes 12 minutes to complete one round.
To Find:
The time after which both Sonia and Ravi will meet again at the starting point.
Formula:
The required time is the Least Common Multiple (LCM) of the times taken by Sonia and Ravi to complete one round.
If Sonia's time is \( a \) and Ravi's time is \( b \), then
$$ \text{Time to meet again} = \operatorname{LCM}(a, b) $$
Definitions:
- Least Common Multiple (LCM): The smallest positive integer that is exactly divisible by each of the given numbers.
- Prime Factorization: Writing a number as a product of its prime numbers.
Solution:
Step 1: Write the times as numbers to find their LCM.
$$ a = 18,\quad b = 12 $$Step 2: Find the prime factorization of each number.
$$ 18 = 2 \times 3 \times 3 = 2 \times 3^2 $$ $$ 12 = 2 \times 2 \times 3 = 2^2 \times 3 $$Step 3: Take the highest power of each prime factor to find the LCM.
Highest power of 2: \( 2^2 \)
Highest power of 3: \( 3^2 \)
Step 4: Calculate the product.
$$ 2^2 = 4 $$ $$ 3^2 = 9 $$ $$ 4 \times 9 = 36 $$Step 5: Interpret the result.
The LCM is 36. So, Sonia and Ravi will meet again at the starting point after 36 minutes.
Result:
Final Answer: Sonia and Ravi will meet again at the starting point after 36 minutes.
$$ \boxed{36\ \text{minutes}} $$
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