To solve the problem, we need to determine how long it will take Vince and Janno to paint the house together.

Step 1: Work rates of Vince and Janno

• Vince can paint the house in 5 days, so his work rate is $\frac{1}{5}$ of the house per day.
• Janno can paint the house in 4 days, so his work rate is $\frac{1}{4}$ of the house per day.

Step 2: Combined work rate

When they work together, their combined work rate is the sum of their individual rates: $\text{Combined work rate} = \frac{1}{5} + \frac{1}{4}$ To add these, we find a common denominator: $\frac{1}{5} + \frac{1}{4} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}$ So, together they can complete $\frac{9}{20}$ of the house per day.

Step 3: Time to complete the job together

To find how long it will take them to complete the entire job, we take the reciprocal of the combined rate: $\text{Time to finish} = \frac{1}{\frac{9}{20}} = \frac{20}{9} \text{ days}$ This simplifies to approximately 2.22 days.

Conclusion:

Vince and Janno, working together, can finish the job in $\frac{20}{9}$ days, or approximately 2.22 days.

Would you like a deeper explanation of any step? Here are some related questions you might want to explore:

1. What is the work rate formula for similar problems?
2. How do you find a common denominator when adding fractions?
3. How does the reciprocal of a rate give time?
4. Can this method be applied to more than two workers?
5. What if one of them takes breaks, how would that affect the calculation?

Tip: When solving work rate problems, adding individual rates is a quick way to determine the total rate when people work together.