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?

Let Abbie's painting rate be \( A \) rooms per minute. According to the problem:

- Abbie paints **twice as fast as Beth**, so Beth's rate is \( \frac{A}{2} \) rooms per minute.
- Abbie paints **three times as fast as Cathie**, so Cathie's rate is \( \frac{A}{3} \) rooms per minute.

When all three work together, their combined rate is the sum of their individual rates:

\[
A + \frac{A}{2} + \frac{A}{3}
\]

To combine these fractions, find the least common denominator:

\[
A + \frac{A}{2} + \frac{A}{3} = A \left(1 + \frac{1}{2} + \frac{1}{3}\right)
\]

The least common denominator of 2 and 3 is 6, so:

\[
1 + \frac{1}{2} + \frac{1}{3} = \frac{6}{6} + \frac{3}{6} + \frac{2}{6} = \frac{11}{6}
\]

Thus, their combined rate is:

\[
A \times \frac{11}{6} = \frac{11A}{6}
\]

Since they can paint the room together in 60 minutes, their combined rate is \( \frac{1}{60} \) rooms per minute. Therefore, we can set up the following equation:

\[
\frac{11A}{6} = \frac{1}{60}
\]

Now, solve for \( A \):

\[
A = \frac{1}{60} \times \frac{6}{11} = \frac{6}{660} = \frac{1}{110}
\]

So, Abbie’s rate is \( \frac{1}{110} \) rooms per minute. To find how long it would take her to paint the room alone, take the reciprocal of her rate:

\[
\text{Time for Abbie alone} = \frac{1}{\frac{1}{110}} = 110 \text{ minutes}
\]

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:

1. How long would it take Beth to paint the room alone?
2. How long would it take Cathie to paint the room alone?
3. 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?
4. How would the total time change if a fourth person, who paints as fast as Abbie, joined the group?
5. 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!

Solve the problem where Abbie paints twice as fast as Beth and three times as fast as Cathie. Learn how to calculate the time it would take Abbie to paint a room alone after working with others for 60 minutes. This solution explores rate problems and algebra, suitable for Grades 8-10.

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