A pharmaceutical laboratory uses three automated dispensers to add chemical reagents into … — State PSC Exam

A pharmaceutical laboratory uses three automated dispensers to add chemical reagents into a continuous reaction vessel. Dispenser X adds a drop every $\frac{14}{9}$ seconds, Dispenser Y every $\frac{35}{6}$ seconds, and Dispenser Z every $\frac{49}{12}$ seconds. If all three dispensers release a drop simultaneously, what is the minimum time interval (in seconds) until they all release a drop together again?

A
$\frac{36}{7}$
$\frac{490}{3}$ Cevap
C
$\frac{7}{36}$
D
$\frac{245}{18}$

Cevap

The minimum time interval is

\frac{490}{3}

seconds.

The correct answer is found by applying the LCM formula for fractions. Since the events repeat over time, their next simultaneous occurrence happens at a time that is a common multiple of all three periods. By calculating the LCM of the numerators (

14, 35, 49 \rightarrow 490

) and dividing it by the HCF of the denominators (

9, 6, 12 \rightarrow 3

), we get

\frac{490}{3}

seconds.

Adım Adım Çözüm

Identify the mathematical operation required.

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

Simultaneous periodic events repeat exactly at the LCM of their individual periods.

State the formula for the LCM of fractions.

LCM of fractions =

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

This standard formula ensures the resulting value is a multiple of each original fraction.

Calculate the LCM of the numerators.

The numerators are

14

35

, and

49

. Their prime factorizations are

14 = 2 \times 7

35 = 5 \times 7

, and

49 = 7^2

. The LCM is

2 \times 5 \times 7^2 = 490

The LCM must include the highest power of all prime factors present in the numbers.

Calculate the HCF of the denominators.

The denominators are

9

6

, and

12

. Their prime factorizations are

9 = 3^2

6 = 2 \times 3

, and

12 = 2^2 \times 3

. The highest common factor is

3

The HCF is the largest positive integer that divides each of the numbers without leaving a remainder.

Apply the formula to find the final LCM.

LCM =

\frac{490}{3}

Substituting the calculated numerator LCM and denominator HCF into the fraction formula.

Anahtar Kavram

Lowest Common Multiple (LCM) of fractions for periodic events

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru