To complete a project, Alice works in shifts that last \frac{3}{4} of an hour, and Bob wor… — ACT

To complete a project, Alice works in shifts that last $\frac{3}{4}$ of an hour, and Bob works in shifts that last $\frac{5}{6}$ of an hour. If they both begin their first shift at the exact same time, after how many hours will they next begin a shift at the same time?

A
$\frac{5}{8}$
B
$\frac{4}{5}$
C
$\frac{5}{4}$
D
$\frac{19}{12}$
$\frac{15}{2}$ Answer

Answer

The two workers will next begin a shift at the same time after

\frac{15}{2}

hours.

To find when both Alice and Bob will next start their shifts at the same time, we need to find the Least Common Multiple (LCM) of their shift durations,

\frac{3}{4}

and

\frac{5}{6}

hours. One way to do this is to convert the durations into minutes:

\frac{3}{4} \times 60 = 45

minutes, and

\frac{5}{6} \times 60 = 50

minutes. The prime factorizations are

45 = 3^2 \times 5

and

50 = 2 \times 5^2

. The LCM of

45

and

50

2 \times 3^2 \times 5^2 = 450

minutes. Converting this back to hours gives

\frac{450}{60} = \frac{15}{2}

hours. Alternatively, using the formula for the LCM of fractions:

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

, we get

\frac{\text{LCM}(3, 5)}{\text{GCD}(4, 6)} = \frac{15}{2}

hours.

Step-by-Step Solution

Identify the mathematical concept needed to solve the simultaneous starting times.

The next simultaneous starting time is the Least Common Multiple (LCM) of the two shift durations,

\frac{3}{4}

and

\frac{5}{6}

hours.

Because both shifts start at multiples of their respective durations, the first time they start together again must be a common multiple of both intervals.

Calculate the LCM of the fractions

\frac{3}{4}

and

\frac{5}{6}

Using the fraction LCM formula,

\text{LCM}(\frac{a}{b}, \frac{c}{d}) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)}

, we find

\text{LCM}(3, 5) = 15

and

\text{GCD}(4, 6) = 2

, yielding

\frac{15}{2}

hours.

This formula correctly scales the numerator multiples while accounting for the common division in the denominators.

Key Concept

Finding the Least Common Multiple (LCM) of fractional values or using unit conversion to find the LCM of integers.

Alternative Method

Convert the fractional hours into minutes first. Alice's shifts are

\frac{3}{4}

of an hour, which is

45

minutes. Bob's shifts are

\frac{5}{6}

of an hour, which is

50

minutes. Find the LCM of

45

and

50

by listing their multiples or using their prime factorizations:

45 = 3^2 \times 5

and

50 = 2 \times 5^2

, so their LCM is

2 \times 3^2 \times 5^2 = 450

minutes. Finally, convert

450

minutes back to hours by dividing by

60

, which simplifies to

\frac{450}{60} = \frac{15}{2}

hours.

Estimated Time:2m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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