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Q7:
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?
Solution :
Given:
Time taken by Sonia to complete one round = $18$ minutes.
Time taken by Ravi to complete one round = $12$ minutes.
Both start from the same point, at the same time, and move in the same direction.
To Find:
The time after which they will meet again at the starting point.
Step 1: Understanding the Mathematical Requirement
Since Sonia and Ravi are moving in a circular path and we need to find the earliest time they meet at the starting point, we are looking for the smallest time interval that is a multiple of both $18$ and $12$. This is defined as the Least Common Multiple (LCM) of the two time durations.
Step 2: Prime Factorization of the Given Numbers
To find the LCM, we first express each number as a product of its prime factors.
For $18$:
$18 = 2 \times 9 = 2 \times 3 \times 3 = 2^1 \times 3^2$
For $12$:
$12 = 2 \times 6 = 2 \times 2 \times 3 = 2^2 \times 3^1$
Step 3: Calculating the Least Common Multiple (LCM)
The LCM of two or more numbers is the product of the highest power of each prime factor involved in the numbers.
Prime factors involved: $2$ and $3$.
Highest power of $2$: $2^2$ (from $12$).
Highest power of $3$: $3^2$ (from $18$).
Therefore, $\text{LCM}(18, 12) = 2^2 \times 3^2$.
$\text{LCM}(18, 12) = 4 \times 9 = 36$.
Step 4: Verification
Sonia completes rounds at: $18, 36, 54, \dots$ minutes.
Ravi completes rounds at: $12, 24, 36, 48, \dots$ minutes.
The first common multiple is $36$.
Final Answer: They will meet again at the starting point after 36 minutes.
More Questions from Class 10 Mathematics Real numbers EXERCISE 1.1
- Q1(i): Express each number as a product of its prime factors: (i) 140
- Q1(ii): Express each number as a product of its prime factors: (ii) 156
- Q1(iii): Express each number as a product of its prime factors: (iii) 3825
- Q1(iv): Express each number as a product of its prime factors: (iv) 5005
- Q1(v): Express each number as a product of its prime factors: (v) 7429
- Q2(i): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (i) 26 and 91
- Q2(ii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (ii) 510 and 92
- Q2(iii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (iii) 336 and 54
- Q3(i): Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21
- Q3(ii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29
- Q3(iii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (iii) 8, 9 and 25
- Q4: Given that HCF (306, 657) = 9, find LCM (306, 657).
- Q5: Check whether $6^n$ can end with the digit 0 for any natural number $n$.
- Q6: Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
CBSE Solutions for Class 10 Mathematics Real numbers
Chapters in CBSE - Class 10 Mathematics
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