This problem involves understanding the concept of work rates and how to combine them to find the total time taken to complete a task. Aziz and Nasser are painting a wall, each with their own individual painting speed. We need to determine how long it will take them to paint the entire wall together....
Analyzing the Individual Work Rates
Aziz paints 1/6 of the wall in one hour. This means his work rate is 1/6 of the wall per hour.
Nasser paints 1/4 of the wall in one hour. His work rate is 1/4 of the wall per hour.
Combining the Work Rates
To find their combined work rate, we need to add their individual work rates:
Combined work rate = Aziz's work rate + Nasser's work rate
Combined work rate = 1/6 + 1/4
To add fractions, we need a common denominator. The least common denominator for 6 and 4 is 12:
Combined work rate = (1/6) * (2/2) + (1/4) * (3/3)
Combined work rate = 2/12 + 3/12
Combined work rate = 5/12
This means together, Aziz and Nasser can paint 5/12 of the wall in one hour.
Finding the Total Time
If they paint 5/12 of the wall in one hour, to find the total time to paint the entire wall, we need to find the reciprocal of their combined work rate:
Total time = 1 / Combined work rate
Total time = 1 / (5/12)
Total time = 12/5 hours
Converting this fraction into hours and minutes:
12/5 hours = 2.4 hours
0.4 hours is equal to 0.4 * 60 = 24 minutes.
The Answer
Therefore, Aziz and Nasser will spend a total of 2 hours and 24 minutes to paint the entire wall together.
Key Takeaways
This problem demonstrates the following concepts:
- Work rate: The amount of work completed per unit of time.
- Combined work rate: The sum of individual work rates when multiple individuals work together.
- Reciprocal relationship: The reciprocal of the combined work rate gives the total time to complete the task.
Additional Tips
Here are some additional tips for solving similar problems:
- Identify the individual work rates.
- Find a common denominator for the work rates before adding them.
- Remember that the reciprocal of the combined work rate gives the total time.
- Convert the time into hours and minutes as needed.