Three automated drones patrol a large forest reserve and return to their base station for … — State PSC Exam

Three automated drones patrol a large forest reserve and return to their base station for a rapid data sync at regular intervals. Drone A returns every $3.\overline{3}$ hours, Drone B returns every $2.5$ hours, and Drone C returns every $2.\overline{7}$ hours. If all three drones depart from the base station simultaneously, how many hours will it take for all three to meet at the base station again for a simultaneous data sync?

$50$ Cevap
B
$\frac{5}{18}$
C
$5$
D
$\frac{25}{9}$

Cevap

It will take 50 hours for all three drones to meet simultaneously at the base station.

The time required for all drones to meet again is the LCM of their individual interval times. Converting the intervals to fractions gives 10/3, 5/2, and 25/9. The LCM of fractions is computed by dividing the LCM of the numerators (10, 5, 25) by the HCF of the denominators (3, 2, 9). The LCM of the numerators is 50 and the HCF of the denominators is 1, resulting in exactly 50 hours.

Adım Adım Çözüm

Convert the recurring and terminating decimals into fractions in their simplest form.

Drone A:

3.\overline{3} = 3 \frac{1}{3} = \frac{10}{3}

. Drone B:

2.5 = \frac{5}{2}

. Drone C:

2.\overline{7} = 2 \frac{7}{9} = \frac{25}{9}

Fractional forms are required to accurately compute the lowest common multiple of non-integer time intervals.

Identify the mathematical operation needed to find the time of the next simultaneous meeting.

We must find the Least Common Multiple (LCM) of the three fractional time intervals.

The LCM represents the smallest time value that is a perfect multiple of all individual drone intervals.

State the formula for the LCM of fractions.

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

This is the standard arithmetic rule for finding the LCM of multiple fractions.

Calculate the LCM of the numerators.

The numerators are

10, 5, 25

. Their prime factorizations are

10 = 2 \times 5

5 = 5

, and

25 = 5^2

. The

\text{LCM}(10, 5, 25) = 2 \times 5^2 = 50

The LCM must include the highest power of each prime factor present in the numerators.

Calculate the HCF of the denominators.

The denominators are

3, 2, 9

. Since they do not share any common prime factors, their

\text{HCF}(3, 2, 9) = 1

The Highest Common Factor is the largest integer that divides all the denominators evenly.

Compute the final LCM of the fractions.

\text{LCM} = \frac{50}{1} = 50

hours.

Applying the values from Steps 4 and 5 into the formula from Step 3 yields the final synchronization time.

Anahtar Kavram

LCM of Fractions and Recurring Decimals

Tahmini Süre:2m 0s

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