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Let's break down the problem step-by-step based on the image you uploaded:

The question asks us to find two expressions for the quantity \( 10M - M \) and set those two expressions equal to each other. We're also told that \( M = 0.4999\ldots \), and a hint is provided that one expression is simply \( 9M \).

### Step 1: Express \( M \) in terms of fractions or another simpler form.

Given:
\[
M = 0.4999\ldots
\]
This repeating decimal can be expressed as a fraction. We know:
\[
M = 0.4\overline{9} = \frac{5}{10} = \frac{1}{2}
\]
So, \( M = \frac{1}{2} \).

### Step 2: Expression for \( 10M \)

Multiplying \( M \) by 10:
\[
10M = 10 \times \frac{1}{2} = 5
\]

### Step 3: Expression for \( 10M - M \)

The expression \( 10M - M \) can now be calculated:
\[
10M - M = 5 - \frac{1}{2} = \frac{10}{2} - \frac{1}{2} = \frac{9}{2}
\]

### Step 4: Compare to the given expression \( 9M \)

We were given that \( 9M = 9 \times \frac{1}{2} = \frac{9}{2} \).

### Step 5: Solve the equation

We set the two expressions equal to each other:
\[
10M - M = 9M
\]
Both sides are \( \frac{9}{2} \), confirming the equation is valid.

### Conclusion

Thus, we discover that \( M = 0.4999\ldots \) is just another way of writing \( \frac{1}{2} \), demonstrating something interesting about repeating decimals.

---

Would you like further details or explanations on this? Here are five related questions to expand on this concept:

1. How do you convert any repeating decimal into a fraction?
2. Why does \( 0.999\ldots = 1 \), and how is it similar to the situation in this problem?
3. Can you generalize the process of solving \( 10M - M \) for any repeating decimal?
4. What is the significance of \( 9M \) in this type of problem involving repeating decimals?
5. How do repeating decimals like \( 0.4999\ldots \) relate to limits in calculus?

**Tip:** When converting repeating decimals to fractions, set the decimal equal to a variable and multiply both sides by powers of 10 to eliminate the repeating part.

Suppose M = 0.4999... Then what does 10M equal? Find two expressions for the quantity 10M - M and set those two expressions equal to each other.

Learn how to find two expressions for 10M - M when M equals 0.4999... and solve the equation to reveal the relationship between repeating decimals and fractions. Suitable for Grades 8-10, this problem illustrates how to convert repeating decimals to fractions and solve simple algebraic equations.

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