Three electronic metronomes are set to tick at regular intervals of \frac{2}{3} of a secon… — State PSC Exam

Three electronic metronomes are set to tick at regular intervals of $\frac{2}{3}$ of a second, $\frac{3}{4}$ of a second, and $\frac{4}{5}$ of a second, respectively. If they all tick simultaneously at a given moment, after how many seconds will they next tick together simultaneously?

12Answer
B
$\frac{1}{60}$
C
$\frac{1}{5}$
D
1

Answer

The correct answer accurately uses the formula for the LCM of fractions, which is the LCM of the numerators (

2, 3, 4

) divided by the HCF of the denominators (

3, 4, 5

). This results in

12 \div 1 = 12

seconds.

Step-by-Step Solution

Identify the mathematical operation required.

To find when the metronomes tick together next, we must find the Least Common Multiple (LCM) of their interval times:

\frac{2}{3}

\frac{3}{4}

, and

\frac{4}{5}

The LCM of multiple time intervals gives the smallest total time at which all periodic events align.

Apply the formula for the LCM of fractions.

The formula is:

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

This standard formula ensures the resulting value is a multiple of all the given fractional intervals.

Calculate the LCM of the numerators.

The numerators are

2, 3,

and

4

. Their LCM is

12

12

is the smallest number perfectly divisible by

2

3

, and

4

Calculate the HCF of the denominators.

The denominators are

3, 4,

and

5

. Their Highest Common Factor (HCF) is

1

3, 4,

and

5

are co-prime integers with no common divisor other than

1

Determine the final LCM.

LCM = \frac{12}{1} = 12

seconds.

Substituting the calculated values into the formula yields the final answer.

Key Concept

LCM of Fractions

Estimated Time:45s

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Answer

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Key Concept

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