https://file.aiquickdraw.com/mathbot/file/172599478946484rixyiz.jpg

The image asks to express the recurring decimal \( 1.52\overline{52} \) (where the digits "52" repeat) as a fraction in its simplest form.

Let's solve it step by step:

### Step 1: Define the repeating decimal
Let:
\[
x = 1.52\overline{52}
\]

### Step 2: Eliminate the decimal part
To remove the repeating decimal, multiply \(x\) by a power of 10. Since the repeating part has two digits ("52"), multiply by 100:

\[
100x = 152.52\overline{52}
\]

Now subtract the original equation from this new equation:

\[
100x - x = 152.52\overline{52} - 1.52\overline{52}
\]

\[
99x = 151
\]

### Step 3: Solve for \(x\)
Now divide both sides by 99:

\[
x = \frac{151}{99}
\]

### Step 4: Simplify the fraction
To simplify \( \frac{151}{99} \), we check if the greatest common divisor (GCD) of 151 and 99 is 1. Since 151 is a prime number and 99 does not divide it, the fraction is already in its simplest form.

Thus, the recurring decimal \( 1.52\overline{52} \) as a fraction in its simplest form is:

\[
\frac{151}{99}
\]

Do you have any questions or want more details on this process?

Here are five related questions to expand on this:
1. How do you convert a general repeating decimal into a fraction?
2. What is the significance of using powers of 10 in eliminating the decimal?
3. Can you simplify \( \frac{151}{99} \) further if it wasn’t already in its simplest form?
4. How would the process change if only one digit repeats (e.g., \( 1.\overline{5} \))?
5. Why does subtracting the two equations remove the repeating part of the decimal?

**Tip**: Always multiply by a power of 10 that matches the length of the repeating sequence to effectively eliminate the repeating decimal.

Express the recurring decimal 1.52 as a fraction in its simplest form.

Learn how to convert the recurring decimal 1.52\overline{52} to a fraction in its simplest form. This solution uses basic algebra to eliminate the decimal and provide a simplified fraction. Suitable for students in Grades 8-10.

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