To determine how long it will take Amaya and Mirna to empty the dishwasher together, we can use the concept of work rates. Let's work through the problem step-by-step:
1. Determine Individual Work Rates:
- Amaya takes 13 minutes to empty the dishwasher. Therefore, her work rate is:
[tex]\[
\text{Amaya's work rate} = \frac{1 \text{ dishwasher}}{13 \text{ minutes}}
\][/tex]
- Mirna takes 16 minutes to empty the dishwasher. Therefore, her work rate is:
[tex]\[
\text{Mirna's work rate} = \frac{1 \text{ dishwasher}}{16 \text{ minutes}}
\][/tex]
2. Calculate Combined Work Rate:
- When both of them work together, their combined work rate is the sum of their individual work rates. Therefore:
[tex]\[
\text{Combined work rate} = \frac{1}{13} + \frac{1}{16}
\][/tex]
3. Add the Rates:
- To add these fractions, find a common denominator:
[tex]\[
\frac{1}{13} + \frac{1}{16} = \frac{16}{208} + \frac{13}{208} = \frac{29}{208}
\][/tex]
- Therefore, the combined work rate is:
[tex]\[
\frac{29}{208} \text{ dishwashers per minute}
\][/tex]
4. Calculate the Combined Time:
- The time it takes for both to empty one dishwasher working together is the reciprocal of the combined work rate:
[tex]\[
\text{Time together} = \frac{1}{\frac{29}{208}} = \frac{208}{29}
\][/tex]
5. Simplify and Round to the Nearest Tenth:
- Simplifying the fraction:
[tex]\[
\frac{208}{29} \approx 7.17241379
\][/tex]
- Rounding to the nearest tenth:
[tex]\[
7.2
\][/tex]
Therefore, if Amaya and Mirna work together, it will take them approximately 7.2 minutes to empty the dishwasher.
