Three cyclists start riding laps around a closed circular track at the same time from the … — ACT

Three cyclists start riding laps around a closed circular track at the same time from the same starting line. Cyclist A completes a lap in $\frac{2}{3}$ of a minute, Cyclist B completes a lap in $\frac{3}{4}$ of a minute, and Cyclist C completes a lap in $\frac{5}{6}$ of a minute. If they maintain these constant rates, after how many minutes will all three cyclists cross the starting line together again?

A
$12$
$30$ Answer
C
$\frac{5}{2}$
D
$\frac{1}{12}$
E
$\frac{10}{13}$

Answer

30 minutes

To find when the three cyclists meet at the starting line again, we calculate the least common multiple (LCM) of their lap times:

\frac{2}{3}

\frac{3}{4}

, and

\frac{5}{6}

minutes. Using the formula

\text{LCM}(\text{fractions}) = \frac{\text{LCM of numerators}}{\text{GCD of denominators}}

, the LCM of

2

3

, and

5

30

, and the GCD of

3

4

, and

6

1

. The LCM is

\frac{30}{1} = 30

minutes. At this time, each cyclist will have completed an integer number of laps (

45

40

, and

36

laps, respectively).

Step-by-Step Solution

Identify the required mathematical concept.

We need to find the least common multiple (LCM) of the three lap times:

\frac{2}{3}

\frac{3}{4}

, and

\frac{5}{6}

minutes.

The next time all three cyclists cross the starting line together is the smallest positive time that is an integer multiple of each individual lap time.

Apply the formula for the LCM of a set of fractions.

\text{LCM}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{LCM}(a, c, e)}{\text{GCD}(b, d, f)}

The LCM of fractions is determined by dividing the LCM of the numerators by the greatest common divisor (GCD) of the denominators.

Calculate the LCM of the numerators and the GCD of the denominators.

The LCM of

2

3

, and

5

30

. The GCD of

3

4

, and

6

1

For the numerators, since

2

3

, and

5

are prime numbers, their LCM is

2 \times 3 \times 5 = 30

. For the denominators, the only positive integer that divides

3

4

, and

6

1

Compute the final LCM value.

\frac{30}{1} = 30\text{ minutes}

Dividing the numerator LCM by the denominator GCD gives the least common multiple of the fractions.

Key Concept

Calculating the least common multiple (LCM) of a set of fractions to solve a periodic scheduling word problem.

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