A city's public transport network features three distinct tram lines that operate on conti… — State PSC Exam

A city's public transport network features three distinct tram lines that operate on continuous circular routes departing from a central station. Tram Line 1 completes its route every $\frac{45}{4}$ minutes. Tram Line 2 completes its route every $\frac{25}{2}$ minutes, and Tram Line 3 takes $\frac{75}{8}$ minutes per loop. If all three trams depart from the central station simultaneously, how many minutes will it take for them to depart together again for the first time?

Answer: 112.5 minutes

Answer

It will take 112.5 minutes for all three trams to depart together again.

The correct answer is found by taking the Least Common Multiple of the fractional times. By finding the LCM of the numerators (225) and dividing it by the Highest Common Factor of the denominators (2), we get 225/2, which evaluates to exactly 112.5 minutes.

Step-by-Step Solution

Determine the mathematical operation required to find when the events will synchronize.

Identify the need to calculate the Least Common Multiple (LCM) of the fractions

\frac{45}{4}

\frac{25}{2}

, and

\frac{75}{8}

The trams will meet again at a time that is a common multiple of their individual loop durations. The 'first time' indicates the least common multiple is needed.

Apply the rule for calculating the LCM of fractional values.

Use the formula:

\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

To synchronize fractional frequencies, the numerators must reach a common multiple while strictly maintaining the largest common baseline unit defined by the denominators.

Calculate the LCM of the numerators: 45, 25, and 75.

The LCM of 45, 25, and 75 is 225.

Prime factorization:

45 = 3^2 \times 5

;

25 = 5^2

;

75 = 3 \times 5^2

. Taking the highest powers gives

3^2 \times 5^2 = 9 \times 25 = 225

Calculate the HCF of the denominators: 4, 2, and 8.

The HCF of 4, 2, and 8 is 2.

2 is the largest integer that can divide 4, 2, and 8 without leaving a remainder.

Compute the final synchronized time.

Divide the LCM of numerators by the HCF of denominators:

\frac{225}{2} = 112.5

Applying the values to the fraction LCM formula yields the exact time in minutes.

Key Concept

Calculating the Least Common Multiple (LCM) for fractions to solve simultaneous event problems.

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