Sally divided a two-digit number by another two-digit number.

Her answer was 3.18181818 ......

Find two numbers that Sally could have used.

Input note: write your answer as a fraction.

To solve this problem, we need to find two two-digit numbers where the result of dividing one by the other is \(3.18181818\ldots\), which is a repeating decimal.

### Step-by-Step Solution

1. **Understanding the Decimal:**
   \[
   3.18181818\ldots = 3 + 0.181818\ldots
   \]
   The decimal part \(0.181818\ldots\) repeats every two digits. Let's denote this repeating decimal by \(x\).

2. **Convert the Repeating Decimal to a Fraction:**
   Let \(x = 0.181818\ldots\). Then, \(100x = 18.181818\ldots\).

   Subtracting the original \(x\) from \(100x\), we get:
   \[
   100x - x = 18.181818\ldots - 0.181818\ldots
   \]
   \[
   99x = 18
   \]
   \[
   x = \frac{18}{99}
   \]

   Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 9:
   \[
   x = \frac{18 \div 9}{99 \div 9} = \frac{2}{11}
   \]

3. **Combining with the Integer Part:**
   Since the original number was \(3 + 0.181818\ldots\), we write:
   \[
   3.181818\ldots = 3 + \frac{2}{11}
   \]
   To convert this to an improper fraction:
   \[
   3 + \frac{2}{11} = \frac{3 \times 11 + 2}{11} = \frac{33 + 2}{11} = \frac{35}{11}
   \]

4. **Identify the Two-Digit Numbers:**
   We now know that the fraction \(\frac{35}{11}\) represents the division of two two-digit numbers.

   - **Numerator** (dividend): 35
   - **Denominator** (divisor): 11

5. **Verify the Solution:**
   When we divide 35 by 11:
   \[
   35 \div 11 = 3.181818\ldots
   \]
   This matches the repeating decimal Sally got.

### Answer

The two numbers Sally could have used are:

\[
\frac{35}{11}
\]

Would you like more details or have any questions?

---

Here are some related questions:

1. What is the process for converting other repeating decimals into fractions?
2. How can we determine the greatest common divisor (GCD) of two numbers?
3. Why does dividing two numbers result in a repeating decimal?
4. How can we verify if a decimal is repeating or terminating?
5. What other pairs of two-digit numbers would result in similar repeating decimals?

**Tip:** Remember that a repeating decimal represents a fraction whose denominator has a prime factor other than 2 or 5.

Learn how to find two-digit numbers that, when divided, result in a repeating decimal such as 3.18181818. Detailed step-by-step solution provided with fractions. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation