Question

Walter, Rita, and teo want to decorate a multipurpose room in a school for a dance. Walter and Ria can do the job in 4 hours while Teo can do the job in 3 hours. If all of them work together, how long will it take to decorate the room?

Answer 1

To determine how long it will take for Walter, Rita, and Teo to decorate the room together, we first calculate their individual work rates:

1. Walter and Rita together can complete the job in 4 hours. Thus, their combined work rate is:
\[
\text{Rate}_{WR} = \frac{1 \text{ job}}{4 \text{ hours}} = \frac{1}{4} \text{ jobs per hour}
\]

2. Teo can complete the job in 3 hours. Thus, his work rate is:
\[
\text{Rate}_{T} = \frac{1 \text{ job}}{3 \text{ hours}} = \frac{1}{3} \text{ jobs per hour}
\]

Next, we add their work rates to find their combined work rate when all three work together:
\[
\text{Rate}_{total} = \text{Rate}_{WR} + \text{Rate}_{T} = \frac{1}{4} + \frac{1}{3}
\]

To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.

Now we convert the rates to have a common denominator:
\[
\frac{1}{4} = \frac{3}{12}
\]
\[
\frac{1}{3} = \frac{4}{12}
\]

Now we can add the fractions:
\[
\text{Rate}_{total} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \text{ jobs per hour}
\]

Now, to find out how long it will take them to complete 1 job together, we take the reciprocal of their combined work rate:
\[
\text{Time} = \frac{1 \text{ job}}{\text{Rate}_{total}} = \frac{1}{\frac{7}{12}} = \frac{12}{7} \text{ hours}
\]

To convert this into a more understandable time:
\[
\frac{12}{7} \text{ hours} \approx 1.714 \text{ hours}
\]

This is equivalent to:
1 hour and \(\frac{12 - 7}{7} \times 60 \approx 1 \text{ hour and } 43 \text{ minutes}\).

Thus, it will take Walter, Rita, and Teo approximately 1 hour and 43 minutes to complete the decoration together.