Question

Two buses start from the same spot for the same circular route. One is a BRTS bus returning in 35 minutes. The other is a regular express bus taking 42 minutes to return. After how many minutes will they meet again at the same initial spot?

Answer 1

Hint: Here we take the time when the buses meet as a variable. Using the information given in the question we form two equations which state that the time when the buses meet is divisible by the time taken by the BRTS bus to complete one round and time taken by Regular express bus to complete one round. Using the concept of LCM we find the value of the variable.
* LCM is the least common multiple of two or more numbers. We write each number in the form of its prime factors and write the LCM of the numbers as multiplication of prime factors with their highest powers.

Complete step-by-step answer:
We are given a BRTS bus that takes 35 minutes to complete one round.
Also, a regular express bus takes 42 minutes to complete one round.
Let us assume the T as the time when both the buses meet.
Since, the routes for the buses are circular, we can write
\[n \times \]time taken by bus to cover one round \[ = \]total time taken for n round.
Let the time taken for BRTS bus to complete n rounds is \[{T_1}\]
\[ \Rightarrow n \times 35 = {T_1}\]
This states that \[{T_1}\]is divisible by 35.
Similarly, let the time taken for regular express bus to complete n rounds is \[{T_2}\]
\[ \Rightarrow n \times 43 = {T_2}\]
This states that \[{T_2}\]is divisible by 42.
We have to find after how many minutes the two buses meet, so we take the time \[{T_1} = {T_2} = T\]
So, T is divisible by both 35 and 42.
Now we find the LCM of the numbers 35 and 42 which will give us the value of time after which the buses meet.
We can write
\[
  35 = 5 \times 7 \\
  42 = 2 \times 3 \times 7 \\
 \]
From the prime factorization we see that the highest power of prime number 2 is 1, 3 is 1, 5 is 1 and 7 ia 1.
\[ \Rightarrow \]LCM \[ = 2 \times 3 \times 5 \times 7\]
\[ \Rightarrow \]LCM \[ = 210\]

So, the time after which the two buses meet is 210 minutes.

Note: Students might get confused of why we take LCM of two given time values and not HCF, keep in mind whenever we have to find the common value where the buses meet or any other object intersects, we always take LCM because that gives us the least or the minimum value of time when the two objects meet.