Sure, let's tackle the given math problem step-by-step.
Rachel can paint a room in 6 hours, and Matt can paint the same room in 4 hours. We need to find out how long it will take for them to paint the room if they work together.
### Step-by-Step Solution:
1. Calculate Individual Painting Rates
- Rachel's painting rate:
Rachel can paint 1 room in 6 hours. Therefore, in one hour, Rachel can paint [tex]\( \frac{1}{6} \)[/tex] of a room.
[tex]\[
\text{Rachel's rate} = \frac{1}{6} \text{ rooms per hour}
\][/tex]
- Matt's painting rate:
Matt can paint 1 room in 4 hours. Therefore, in one hour, Matt can paint [tex]\( \frac{1}{4} \)[/tex] of a room.
[tex]\[
\text{Matt's rate} = \frac{1}{4} \text{ rooms per hour}
\][/tex]
2. Add the Individual Rates to Get the Combined Rate
When Rachel and Matt work together, their rates add up.
[tex]\[
\text{Combined rate} = \text{Rachel's rate} + \text{Matt's rate}
\][/tex]
Substitute the values:
[tex]\[
\text{Combined rate} = \frac{1}{6} + \frac{1}{4}
\][/tex]
3. Find a Common Denominator and Add the Fractions
The common denominator for 6 and 4 is 12. Convert the fractions to have this common denominator:
[tex]\[
\frac{1}{6} = \frac{2}{12} \quad \text{and} \quad \frac{1}{4} = \frac{3}{12}
\][/tex]
Add these fractions:
[tex]\[
\frac{2}{12} + \frac{3}{12} = \frac{5}{12}
\][/tex]
Therefore, the combined rate is:
[tex]\[
\text{Combined rate} = \frac{5}{12} \text{ rooms per hour}
\][/tex]
4. Calculate the Time Required to Paint One Room Together
To find the time [tex]\(t\)[/tex] required to paint one room together, take the reciprocal of the combined rate:
[tex]\[
t = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \text{ hours}
\][/tex]
5. Simplify the Result
[tex]\[
\frac{12}{5} \text{ hours} = 2.4 \text{ hours}
\][/tex]
So, if Rachel and Matt work together, it will take them 2.4 hours to paint the room.
