The bells initially rang together at 9:00 a.m.. Adding 3 hours to 9:00 a.m. gives 12:00 p.m., which is noon.

three bells ring at intervals of 12 minutes and 18 minutes and 30 minutes if they ring together at 9:00 a.m. when will the ring together again

Answer: $12:00\text{ p.m. (noon)}$

Step-by-step solution

Step 1: Understand the Problem and Identify the Required Concept

The three bells ring after intervals of $12$ , $18$ , and $30$ minutes respectively. To find the minimum time when all three bells will ring together again, we need to calculate the lowest common multiple, $\text{LCM}(12, 18, 30)$ , using the prime factorisation method.

Step 2: Find Prime Factorisations

Express each given interval as a product of powers of prime numbers according to the Fundamental Theorem of Arithmetic. We have $12 = 2^2 \times 3$ , $18 = 2 \times 3^2$ , and $30 = 2 \times 3 \times 5$ .

Step 3: Compute the LCM

The LCM is the product of the greatest power of each prime factor involved in the numbers. Taking the highest powers of the primes $2$ , $3$ , and $5$ , we get $2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$ minutes.

Step 4: Convert Minutes into Hours

Since $1$ hour equals $60$ minutes, we divide $180$ by $60$ to convert the time interval into hours. $180 \div 60 = 3$ hours.

Step 5: Determine the Next Ringing Time

The bells initially rang together at $9:00\text{ a.m.}$ . Adding $3$ hours to $9:00\text{ a.m.}$ gives $12:00\text{ p.m.}$ , which is noon.

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