What is 1.5 repeating as a fraction

What is 1.5 repeating as a fraction

The correct Answer and Explanation is :

To convert (1.5) repeating (written as (1.555…)) into a fraction, follow these steps:

Step 1: Define the repeating decimal

Let (x = 1.555…).

Step 2: Eliminate the repeating part

To isolate the repeating part, we can multiply both sides of the equation by (10):

[
10x = 15.555…
]

Step 3: Set up the equation

Now we have two equations:

1. (x = 1.555…)
2. (10x = 15.555…)

Next, subtract the first equation from the second:

[
10x – x = 15.555… – 1.555…
]

This simplifies to:

[
9x = 14
]

Step 4: Solve for (x)

Now, divide both sides by (9):

[
x = \frac{14}{9}
]

Step 5: Conclusion

Thus, (1.5) repeating can be expressed as the fraction (\frac{14}{9}).

Explanation

Repeating Decimals and Fractions:
Repeating decimals, like (1.555…), represent an infinite series of digits that repeat. When converting them into fractions, the goal is to express the repeating decimal as a ratio of two integers.

Understanding the Process:

1. Define the Variable: Starting with the decimal helps isolate it for conversion.
2. Multiply to Shift the Decimal: By multiplying by (10), you move the decimal point and align the repeating parts, allowing for subtraction.
3. Subtraction for Simplification: This step eliminates the repeating portion, simplifying the equation to one where (x) can be isolated.
4. Final Fraction: The final step yields a clear fraction that represents the decimal precisely.

Verification:
To ensure the conversion is accurate, divide (14) by (9), which results in (1.555…), confirming that the fractional representation is correct. This method of converting decimals to fractions is applicable to any repeating decimal and provides a clear pathway to express such values mathematically.