Question

1. please convert the recurring decimal to a fraction in its simplest form.

denominator = 9s number of repeating digit, 0s number of non-repeating digit

numerator = whole decimal part minus the non-repeating part

(0.11dot{6} = \frac{\text{ }}{\text{ }})

Explanation:

🆕 New Concept Discovered: Converting Repeating Decimals to Fractions
How to turn repeating decimals into fractions using place value rules.

Step 1: Identify the parts of the decimal

The given decimal is \(0.11\dot{6}\), which means the digit \(6\) repeats forever:

$$0.11666...$$

• Repeating part: \(6\) (1 digit)
• Non-repeating decimal part: \(11\) (2 digits)
• Whole decimal part: \(116\)

Step 2: Find the denominator

Following the rule in the image:

• The denominator consists of 9s followed by 0s.
• Number of 9s = number of repeating digits = \(1\) (since only \(6\) repeats).
• Number of 0s = number of non-repeating decimal digits = \(2\) (since \(11\) does not repeat).

$$\text{Denominator} = 900$$

Step 3: Find the numerator

Following the rule in the image:

• Subtract the non-repeating part from the whole decimal part.

$$\text{Numerator} = 116 - 11 = 105$$

Step 4: Write and simplify the fraction

Combine the numerator and denominator:

$$\frac{105}{900}$$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(15\):

$$\frac{105 \div 15}{900 \div 15} = \frac{7}{60}$$

Answer:

$$\frac{7}{60}$$