Examkin

Question

Difficulty: EasyHCF and LCM

Two automated watering sprinklers in a greenhouse operate on continuous cycles. Sprinkler X activates every 125\frac{12}{5} minutes, and Sprinkler Y activates every 1615\frac{16}{15} minutes. If both sprinklers activate simultaneously at a given moment, what is the minimum time interval, in minutes, before they activate together again?

  1. 485\frac{48}{5}Answer
  2. B
    415\frac{4}{15}
  3. C
    165\frac{16}{5}
  4. D
    45\frac{4}{5}

Answer

The minimum time interval is 485\frac{48}{5} minutes.
To find when two periodic events will happen together again, we must calculate the Least Common Multiple (LCM) of their periods. For fractions, the LCM is found by dividing the LCM of the numerators (12 and 16, which is 48) by the HCF of the denominators (5 and 15, which is 5). This results in 485\frac{48}{5}.

Step-by-Step Solution

1
Identify that the problem requires finding the Least Common Multiple (LCM) of the two time intervals to determine when the periodic events will align again.
Calculate the LCM of 125\frac{12}{5} and 1615\frac{16}{15}.
Simultaneous events occurring at regular intervals coincide at the lowest common multiple of their respective periods.
2
Apply the formula for the LCM of fractions.
Formula: LCM of numeratorsHCF of denominators\frac{\text{LCM of numerators}}{\text{HCF of denominators}}.
This is the standard mathematical rule for finding the LCM of fractional values.
3
Find the LCM of the numerators (12 and 16) and the HCF of the denominators (5 and 15).
LCM(12, 16) = 48. HCF(5, 15) = 5.
The smallest positive number divisible by both 12 and 16 is 48. The largest positive number that divides both 5 and 15 is 5.
4
Substitute the calculated values back into the fraction formula.
485\frac{48}{5}
Combining the calculated numerator and denominator yields the final time interval.

Key Concept

Calculating the Least Common Multiple (LCM) of fractions to solve simultaneous event problems.
Rate this question

Topics

HCF and LCM

More questions from this topic