To find how long it will take for Hannah and Destiny to paint the room together, we calculate their combined work rate and use it to determine the total time required.

Step 1: Determine Individual Work Rates

• Hannah's rate is $\frac{1}{16}$ of the room per hour.
• Destiny's rate is $\frac{1}{12}$ of the room per hour.

Step 2: Combine Their Work Rates

The combined rate of both working together is the sum of their individual rates: $\text{Combined rate} = \frac{1}{16} + \frac{1}{12}$

To add these fractions, find a common denominator (LCM of 16 and 12 is 48): $\frac{1}{16} = \frac{3}{48}, \quad \frac{1}{12} = \frac{4}{48}$ $\text{Combined rate} = \frac{3}{48} + \frac{4}{48} = \frac{7}{48}$

Step 3: Calculate Total Time

The time required to paint the room together is the reciprocal of their combined rate: $\text{Time} = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{7}{48}} = \frac{48}{7} \text{ hours}$

Step 4: Simplify

$\frac{48}{7} \approx 6.86 \text{ hours} \text{ or } 6 \text{ hours and 51 minutes (approx.)}.$

Final Answer:

It will take approximately 6 hours and 51 minutes for Hannah and Destiny to paint the room together.

Would you like a detailed explanation of any step or clarification?

Related Questions:

1. How do you calculate combined work rates when there are more than two people?
2. What is the formula for calculating the total time when multiple people are working together?
3. How would the time change if one person works twice as fast as the other?
4. Can this method be applied to other collaborative tasks, such as typing or assembly?
5. What is the advantage of using LCM in adding rates?

Tip:

Always double-check the units of time and rates to ensure consistency in work problems!