Give a rational number between 0.103 and 0.112111211112 . . . .

Explanation

To find a rational number between 0.103 and 0.112111211112..., we can follow these steps:

### Step 1: Understand the Numbers
We have two numbers:
- The first number is 0.103
- The second number is 0.112111211112... (which is a repeating decimal)

### Step 2: Convert the Repeating Decimal to a Fraction
The repeating decimal 0.112111211112... can be expressed as a fraction. Let's denote it as \( x \):
\[ x = 0.112111211112... \]

To convert this to a fraction, we can use the following method:
1. Multiply \( x \) by 1000 (to move the decimal point three places to the right):
\[ 1000x = 112.111211112... \]
2. Now, subtract the original \( x \) from this equation:
\[ 1000x - x = 112.111211112... - 0.112111211112... \]
\[ 999x = 112 \]
3. Solve for \( x \):
\[ x = \frac{112}{999} \]

### Step 3: Find a Rational Number Between 0.103 and 0.112111211112...
Now we need to find a rational number between 0.103 and \( \frac{112}{999} \).

1. Convert 0.103 to a fraction:
\[ 0.103 = \frac{103}{1000} \]

2. We need to find a rational number that lies between \( \frac{103}{1000} \) and \( \frac{112}{999} \).

### Step 4: Finding a Rational Number
To find a rational number between these two fractions, we can take the average:
\[ \text{Average} = \frac{\frac{103}{1000} + \frac{112}{999}}{2} \]

To add these fractions, we need a common denominator. The least common multiple of 1000 and 999 is 999000.

Convert both fractions:
- \( \frac{103}{1000} = \frac{103 \times 999}{1000 \times 999} = \frac{102097}{999000} \)
- \( \frac{112}{999} = \frac{112 \times 1000}{999 \times 1000} = \frac{112000}{999000} \)

Now add them:
\[ \frac{102097}{999000} + \frac{112000}{999000} = \frac{214097}{999000} \]

Now divide by 2:
\[ \text{Average} = \frac{214097}{999000} \div 2 = \frac{214097}{1998000} \]

### Step 5: Conclusion
Thus, a rational number between 0.103 and 0.112111211112... is:
\[ \frac{214097}{1998000} \]