Math Problem Statement
Abbie paints twice as fast as Beth and three times as fast as Cathie. If it takes them 60 minutes to paint a living room with all three working together, how long would it take Abbie if she works alone?
Solution
Let Abbie's painting rate be rooms per minute. According to the problem:
- Abbie paints twice as fast as Beth, so Beth's rate is rooms per minute.
- Abbie paints three times as fast as Cathie, so Cathie's rate is rooms per minute.
When all three work together, their combined rate is the sum of their individual rates:
To combine these fractions, find the least common denominator:
The least common denominator of 2 and 3 is 6, so:
Thus, their combined rate is:
Since they can paint the room together in 60 minutes, their combined rate is rooms per minute. Therefore, we can set up the following equation:
Now, solve for :
So, Abbie’s rate is rooms per minute. To find how long it would take her to paint the room alone, take the reciprocal of her rate:
Therefore, it would take Abbie 110 minutes to paint the room if she works alone.
Would you like more details or have any questions about this? Here are five related questions:
- How long would it take Beth to paint the room alone?
- How long would it take Cathie to paint the room alone?
- If Cathie improved her speed and painted at the same rate as Beth, how long would it take the three to paint the room together?
- How would the total time change if a fourth person, who paints as fast as Abbie, joined the group?
- What if Abbie painted four times faster than Cathie instead of three times faster—how would that affect the outcome?
Tip: When dealing with combined rates, always remember to add rates directly, not times, to find how fast people work together!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rate problems
Fractions
Algebra
Formulas
Combined work rate formula: 1/T = 1/t1 + 1/t2 + 1/t3
Abbie's rate formula: A = 1/time taken
Rate addition: A + A/2 + A/3 = Combined rate
Theorems
Rate addition for combined work problems
Suitable Grade Level
Grades 8-10