Find a rational number which when expressed as a decimal will have `1.2bar(56)` as its expansion.

Answer

To find a rational number that has the decimal expansion \(1.2\overline{56}\), we can follow these steps:

### Step 1: Understand the Decimal Expansion
The decimal \(1.2\overline{56}\) means that the digits "56" repeat indefinitely. We can express this as:
\[ 1.256565656\ldots \]

### Step 2: Separate the Whole and Decimal Parts
We can separate the whole number part from the decimal part:
\[ 1.2\overline{56} = 1 + 0.2\overline{56} \]

### Step 3: Convert the Repeating Decimal to a Fraction
Let \(x = 0.2\overline{56}\). Then we can express \(x\) as:
\[ x = 0.256565656\ldots \]

### Step 4: Isolate the Repeating Part
To isolate the repeating part, we can multiply \(x\) by \(1000\) (since "56" has two digits):
\[ 1000x = 256.565656\ldots \]
Now, subtract \(x\) from \(1000x\):
\[ 1000x - x = 256.565656\ldots - 0.256565656\ldots \]
This simplifies to:
\[ 999x = 256.3 \]

### Step 5: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{256.3}{999} \]

### Step 6: Convert \(256.3\) to a Fraction
Convert \(256.3\) into a fraction:
\[ 256.3 = \frac{2563}{10} \]
Thus:
\[ x = \frac{2563}{10 \times 999} = \frac{2563}{9990} \]

### Step 7: Combine with the Whole Number Part
Now, we can combine this with the whole number part:
\[ 1 + x = 1 + \frac{2563}{9990} = \frac{9990}{9990} + \frac{2563}{9990} = \frac{9990 + 2563}{9990} = \frac{12553}{9990} \]

### Step 8: Simplify the Fraction
Next, we can simplify \(\frac{12553}{9990}\) if possible. We find the GCD of 12553 and 9990, which is 1, so the fraction is already in its simplest form.

### Final Answer
Thus, the rational number that corresponds to the decimal \(1.2\overline{56}\) is:
\[ \frac{12553}{9990} \]