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
- An organization allows its employees to work independently on consultancy projects but charges an overhead on the consulting fee. The overhead is 20% of the consulting fee, if the fee is up to . 5,00,000. For higher fees, the overhead is . 1,00,000 plus 10% of the amount by which the fee exceeds . 5,00,000. The government charges a Goods and Services Tax of 18% on the total amount (the consulting fee plus the overhead). An employee of the organization charges this entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client cannot pay more than . 10,00,000, what is the maximum consulting fee that the employee can charge?
Three frictionless pulleys with rope attachment are in a static equilibrium as shown in the figure. The mass $m_1$ and $m_2$, in kg, respectively are
If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
A five-member truss system is shown in the figure. The maximum vertical force P in kN that can be applied so that loads on the member CD and BC do NOT exceed 50 kN and 30 kN, respectively is _____________(rounded off to 2 decimal places)
The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
