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Difficulty: MediumHCF and LCM

A pharmaceutical laboratory uses three automated dispensers to add chemical reagents into a continuous reaction vessel. Dispenser X adds a drop every 149\frac{14}{9} seconds, Dispenser Y every 356\frac{35}{6} seconds, and Dispenser Z every 4912\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?

  1. A
    367\frac{36}{7}
  2. 4903\frac{490}{3}Answer
  3. C
    736\frac{7}{36}
  4. D
    24518\frac{245}{18}

Answer

The minimum time interval is 4903\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→49014, 35, 49 \rightarrow 490) and dividing it by the HCF of the denominators (9,6,12→39, 6, 12 \rightarrow 3), we get 4903\frac{490}{3} seconds.

Step-by-Step Solution

1
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.
2
State the formula for the LCM of fractions.
LCM of fractions = LCM of numeratorsHCF of denominators\frac{\text{LCM of numerators}}{\text{HCF of denominators}}.
This standard formula ensures the resulting value is a multiple of each original fraction.
3
Calculate the LCM of the numerators.
The numerators are 1414, 3535, and 4949. Their prime factorizations are 14=2×714 = 2 \times 7, 35=5×735 = 5 \times 7, and 49=7249 = 7^2. The LCM is 2×5×72=4902 \times 5 \times 7^2 = 490.
The LCM must include the highest power of all prime factors present in the numbers.
4
Calculate the HCF of the denominators.
The denominators are 99, 66, and 1212. Their prime factorizations are 9=329 = 3^2, 6=2×36 = 2 \times 3, and 12=22×312 = 2^2 \times 3. The highest common factor is 33.
The HCF is the largest positive integer that divides each of the numbers without leaving a remainder.
5
Apply the formula to find the final LCM.
LCM = 4903\frac{490}{3}.
Substituting the calculated numerator LCM and denominator HCF into the fraction formula.

Key Concept

Lowest Common Multiple (LCM) of fractions for periodic events
Estimated Time:1m 30s
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Topics

HCF and LCM

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