Three environmental monitoring sensors transmit data packets at regular intervals of \frac… — State PSC Exam

Three environmental monitoring sensors transmit data packets at regular intervals of $\frac{8}{3}$ seconds, $\frac{12}{5}$ seconds, and $\frac{24}{7}$ seconds, respectively. If all three sensors transmit a packet simultaneously at exactly 10:00:00 AM, how many seconds will pass before they next transmit a packet together at the exact same instant?

$24$ Answer
B
$\frac{105}{4}$
C
$\frac{4}{105}$
D
$\frac{8}{35}$

Answer

24 seconds

The sensors will sync up again at a time that is a multiple of all three individual intervals. To find this, we calculate the LCM of the fractions

\frac{8}{3}

\frac{12}{5}

, and

\frac{24}{7}

. The mathematical rule for the LCM of fractions is the LCM of the numerators divided by the HCF of the denominators. We find

\text{LCM}(8, 12, 24) = 24

and

\text{HCF}(3, 5, 7) = 1

. Thus, the total LCM is

\frac{24}{1} = 24

seconds.

Step-by-Step Solution

Identify the mathematical requirement

The time until the next simultaneous transmission is the Least Common Multiple (LCM) of the three interval times.

Simultaneous events that repeat at regular intervals align perfectly at the LCM of those individual intervals.

Apply the mathematical formula for the LCM of fractions

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

Calculating the LCM for fractions requires evaluating the numerators and denominators cross-layer using this specific formula.

Calculate the LCM of the numerators

\text{LCM}(8, 12, 24) = 24

The smallest integer that is perfectly divisible by 8, 12, and 24 is 24.

Calculate the HCF of the denominators

\text{HCF}(3, 5, 7) = 1

The numbers 3, 5, and 7 are all prime numbers and share no common divisor other than 1.

Compute the final result

\frac{24}{1} = 24

seconds

Dividing the LCM of the numerators by the HCF of the denominators gives the synchronized time interval.

Key Concept

Calculating the Lowest Common Multiple (LCM) of fractional values for synchronization problems.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic