What is 0.46 repeating as a fraction?

To convert the repeating decimal 0.46 (where the 6 repeats indefinitely) into a fraction, follow these steps:

Step 1: Set Up the Equation

Let x equal the repeating decimal:

x = 0.466666...

Step 2: Eliminate the Repeating Part

To isolate the repeating part, multiply both sides of the equation by 10:

10x = 4.66666...

Step 3: Subtract the Original Equation

Now, subtract the original equation from this new equation:

10x - x = 4.66666... - 0.46666...

This simplifies to:

9x = 4.2

Step 4: Solve for x

Now, divide both sides by 9:

x = 4.2 / 9

Step 5: Convert to Fraction

To express 4.2 as a fraction, note that 4.2 is the same as 42/10. Therefore:

x = (42/10) / 9 = 42 / 90

Step 6: Simplify the Fraction

Now, simplify 42/90 by finding the greatest common divisor (GCD), which is 6:

• 42 ÷ 6 = 7
• 90 ÷ 6 = 15

This gives us:

x = 7/15

Final Result

Thus, the repeating decimal 0.46 repeating can be expressed as the fraction 7/15.