A robotics engineering team is programming three distinct automated paint dispensers on an… — State PSC Exam

A robotics engineering team is programming three distinct automated paint dispensers on an assembly line. They are calibrated to release a burst of paint every $\frac{4}{5}$ , $\frac{8}{15}$ , and $\frac{12}{25}$ of a second, respectively. Let $L$ be the least common multiple (LCM) of these three time intervals, representing the exact time when all three dispensers fire simultaneously. Let $H$ be the highest common factor (HCF) of the intervals, representing the largest fundamental time unit that can perfectly measure all three intervals. What is the value of the ratio $\frac{L}{H}$ ?

$90$ Answer
B
$\frac{1}{90}$
C
$\frac{2}{5}$
D
$\frac{32}{125}$

Answer

The value of the ratio

L/H

is 90.

To solve the problem, we first find the LCM and HCF of the fractions

\frac{4}{5}

\frac{8}{15}

, and

\frac{12}{25}

. The formula for the LCM of fractions is

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

. The numerators are 4, 8, and 12, which have an LCM of 24. The denominators are 5, 15, and 25, which have an HCF of 5. Thus,

L = \frac{24}{5}

. The formula for the HCF of fractions is

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

. The HCF of 4, 8, and 12 is 4. The LCM of 5, 15, and 25 is 75. Thus,

H = \frac{4}{75}

. Finally, dividing

L

H

gives

\frac{24}{5} \div \frac{4}{75}

, which equals

\frac{24}{5} \times \frac{75}{4} = 6 \times 15 = 90

Step-by-Step Solution

Calculate

L

, the LCM of the three fractions.

L = \frac{24}{5}

The LCM of fractions is the LCM of their numerators divided by the HCF of their denominators. LCM(4, 8, 12) = 24 and HCF(5, 15, 25) = 5.

Calculate

H

, the HCF of the three fractions.

H = \frac{4}{75}

The HCF of fractions is the HCF of their numerators divided by the LCM of their denominators. HCF(4, 8, 12) = 4 and LCM(5, 15, 25) = 75.

Calculate the ratio

\frac{L}{H}

\frac{24/5}{4/75} = \frac{24}{5} \times \frac{75}{4} = 6 \times 15 = 90

To divide by a fraction, multiply by its reciprocal. Simplifying the resulting expression yields 90.

Key Concept

Calculating the LCM and HCF of fractional values

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