Three bells P, Q, and R are rung periodically in a school. P is rung every 20 minutes; Q is rung every 30 minutes and R is rung every 50 minutes.
If all the three bells are rung at 12:00 PM, when will the three bells ring together again the next time?
Problem Analysis: Bell Ringing Schedule
The problem asks for the next time three bells, P, Q, and R, will ring simultaneously. We are given the ringing intervals for each bell:
- Bell P: every $20$ minutes
- Bell Q: every $30$ minutes
- Bell R: every $50$ minutes
They all ring together at $12:00$ PM. To find the next time they ring together, we need to find the Least Common Multiple (LCM) of their ringing intervals.
Calculating the LCM
Find the prime factorization of each interval:
- $20 = 2 \times 2 \times 5 = 2^2 \times 5^1$
- $30 = 2 \times 3 \times 5 = 2^1 \times 3^1 \times 5^1$
- $50 = 2 \times 5 \times 5 = 2^1 \times 5^2$
The LCM is found by taking the highest power of each prime factor present in any factorization:
LCM$(20, 30, 50) = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300$ minutes.
Determining the Next Ring Time
The LCM, $300$ minutes, represents the time interval after which all bells will ring together again.
Convert this interval into hours:
$300 \text{ minutes} = \frac{300}{60} \text{ hours} = 5 \text{ hours}$
The bells rang together at $12:00$ PM. They will ring together again $5$ hours later.
Next Ring Time = $12:00 \text{ PM} + 5 \text{ hours} = 5:00 \text{ PM}$.
Conclusion
The three bells P, Q, and R will ring together again at 5:00 PM.
Important Questions from Numerical Computation
In an engineering college of 10,000 students, 1,500 like neither their core branches nor other branches. The number of students who like their core branches is 1/4th of the number of students who like other branches. The number of students who like both their core and other branches is 500.
The number of students who like their core branches is$A$ is an ($n \times n$) matrix. Consider the following two statements
Statement 1: Columns of matrix $A$ are linearly independent
Statement 2: Inverse of matrix $A$ exists
Which one of the following statements is TRUE?
- Consider the function $G(x, y, z) = 0$. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function $G(x, y, z)$ exist everywhere. Then, the value of $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is _________ (in integer)
- Levenshtein distance is used to measure the minimum edit distance between two strings by counting the minimum number of editing operations (such as, insertions, deletions, substitutions) required to transform one string to another. Consider an alternative version of the Levenshtein distance in which each insertion and each deletion has a cost of 1 and substitutions are not allowed.
Based on this alternative version, the Levenshtein distance between the strings, word and work is ______ (Answer in integer). - In a given collection of documents, let $N$ be the total number of documents and let $d$ be the number of documents in which the term $t$ occurs.
Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$?
