https://file.aiquickdraw.com/mathbot/file/1727208507981jhgt5c9s.jpg

Let's break down the two problems shown in the image:

### Problem 9:
**Miro can paint a plank alone in 6 hours, and Miranda can paint the same plank alone in 3 hours. How long will it take if they work together?**

We need to calculate their combined rate of work. The formula for the time when two people work together is:

\[
\text{Combined time} = \frac{1}{\frac{1}{T_1} + \frac{1}{T_2}}
\]

Where:
- \( T_1 = 6 \) (Miro's time)
- \( T_2 = 3 \) (Miranda's time)

So, the combined rate is:

\[
\frac{1}{T_1} + \frac{1}{T_2} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}
\]

Thus, the time it will take them together is:

\[
\text{Time} = \frac{1}{\frac{1}{2}} = 2 \text{ hours}
\]

### Problem 10:
**What is the value of the following expression:**

\[
(1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \dots (1 + \frac{1}{99})
\]

This product is equivalent to:

\[
(1 + \frac{1}{2})(1 + \frac{1}{3})(1 + \frac{1}{4}) \dots (1 + \frac{1}{99}) = \frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \dots \frac{100}{99}
\]

Notice that most terms cancel out in the sequence, and the final product simplifies to:

\[
\frac{100}{2} = 50
\]

So, the value of the expression is **50**.

---

Would you like further details or have any other questions?

Here are 5 related questions for further practice:

1. What is the formula for combined work rates for more than two workers?
2. How would the total time change if Miro and Miranda had a third person helping them?
3. Can you explain why the product in problem 10 simplifies in such a way?
4. How would the answer change if the final fraction was \( \frac{1}{100} \) instead of \( \frac{1}{99} \)?
5. Can you find the sum of the reciprocals from \( \frac{1}{2} \) to \( \frac{1}{99} \)?

**Tip**: When solving work-rate problems, always try to express each person's contribution as a fraction of the job completed in one hour to simplify the problem.

9. Miro can paint a plank alone in 6 hours. Miranda can paint the same plank alone in 3 hours. How long will it take if they work together?
10. What is the value of the product (1 + 1/2)(1 + 1/3)(1 + 1/4) ... (1 + 1/99)?

Learn how to solve a work-rate problem where Miro and Miranda paint a plank together, and find the value of a sequential product of fractions. This lesson covers the combined time formula and product simplifications, ideal for Grades 8-10 students.

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