Every afternoon Amaya and Mirna have to empty the dishwasher. When Amaya
does it, she takes 13 minutes, but Mirna takes 16 minutes doing it by herself. How
long will it take them to empty the dishwasher if they work together? (Round to the
nearest tenth.)



Answer :

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.