In a highly automated manufacturing plant, three different robotic assembly lines are sync… — State PSC Exam

In a highly automated manufacturing plant, three different robotic assembly lines are synchronized by a central control system. Line A completes a specific micro-calibration cycle every $\frac{4}{15}$ of a minute. Line B completes its calibration cycle every $\frac{6}{25}$ of a minute, and Line C completes its cycle every $\frac{8}{35}$ of a minute. If all three lines complete a calibration cycle simultaneously at a given moment, what is the shortest time interval that must elapse before all three lines complete their calibration cycles together again?

A
$\frac{2}{525}$ minutes
B
$\frac{8}{175}$ minutes
$\frac{24}{5}$ minutesCevap
D
$\frac{2}{5}$ minutes

Cevap

The shortest time interval is

\frac{24}{5}

minutes.

To find when independent periodic events next occur simultaneously, we must calculate their Lowest Common Multiple (LCM). For fractions, the LCM is found by dividing the LCM of the numerators by the HCF of the denominators. The LCM of the numerators (

4, 6, 8

) is

24

, and the HCF of the denominators (

15, 25, 35

) is

5

. Therefore, the overall LCM is

\frac{24}{5}

Adım Adım Çözüm

Identify the mathematical operation required.

We need to find the Lowest Common Multiple (LCM) of the three given time intervals (

\frac{4}{15}

\frac{6}{25}

\frac{8}{35}

) to determine when the periodic events will next coincide.

The LCM represents the smallest common interval of time into which each individual cycle can fit perfectly.

Recall the formula for finding the LCM of fractions.

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

This standard formula is required when working with periodic fractional amounts.

Find the LCM of the numerators.

The numerators are

4

6

, and

8

. The lowest common multiple of

4, 6, \text{ and } 8

24

Multiples of

8

are

8, 16, 24

24

is the smallest multiple evenly divisible by both

4

and

6

Find the Highest Common Factor (HCF) of the denominators.

The denominators are

15

25

, and

35

. The highest common factor of

15, 25, \text{ and } 35

5

The common prime factor shared by

3 \times 5

5 \times 5

, and

7 \times 5

5

Apply the values to the fraction LCM formula.

\frac{\text{LCM}(4, 6, 8)}{\text{HCF}(15, 25, 35)} = \frac{24}{5}

Combining the calculated numerator and denominator provides the final fractional interval.

Anahtar Kavram

Lowest Common Multiple (LCM) of Fractions

Soru

Cevap

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