Jasmine\:can\:clean\:the\:house\:in\:4\:hours\:working\:alone.\:Jack\:can\:clean\:the\:house\:in\:6\:boirs:working\:alone.\:If\:they\:work\:together,\:how\:long\:would\:it\:take\:to\:clean\:the\:house? use fractional expression

To solve this problem, we can use the concept of work rates. Let's break it down:

1. **Jasmine's work rate**: Jasmine can clean the house in 4 hours, so her work rate is \(\frac{1}{4}\) of the house per hour.
  
2. **Jack's work rate**: Jack can clean the house in 6 hours, so his work rate is \(\frac{1}{6}\) of the house per hour.

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

\[
\text{Combined work rate} = \frac{1}{4} + \frac{1}{6}
\]

To add these fractions, we need a common denominator. The least common denominator of 4 and 6 is 12:

\[
\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}
\]

So, their combined work rate is:

\[
\frac{3}{12} + \frac{2}{12} = \frac{5}{12}
\]

This means that together, Jasmine and Jack can clean \(\frac{5}{12}\) of the house in one hour. To find out how long it would take them to clean the entire house, we take the reciprocal of their combined work rate:

\[
\text{Time to clean the house together} = \frac{1}{\frac{5}{12}} = \frac{12}{5} \text{ hours}
\]

So, it would take them \(\frac{12}{5}\) hours, which is equivalent to 2.4 hours, to clean the house together.

Would you like more details on this solution, or do you have any other questions?

Here are 5 related questions to explore further:

1. How long would it take Jasmine and Jack to clean the house if Jack could only work for 3 hours?
2. What if another person, Alex, joins them and can clean the house alone in 8 hours? How long would it take all three to clean the house together?
3. If Jasmine and Jack clean a different house that is twice as large, how long would it take them together?
4. How does the combined work rate change if Jasmine's efficiency improves, and she can now clean the house in 3 hours?
5. If Jasmine and Jack both work at their rates but take a 15-minute break halfway through, how does this affect the total cleaning time?

**Tip:** When solving work problems, remember that rates can be added together, and the total time is the reciprocal of the combined rate.

Jasmine can clean the house in 4 hours working alone. Jack can clean the house in 6 hours working alone. If they work together, how long would it take to clean the house?

Learn how to solve a work rate problem involving Jasmine and Jack cleaning a house together. By combining their individual work rates, you can calculate how long it takes for them to finish the task. This problem uses fractional expressions and is suitable for students in Grades 7-8.

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