\(1.5\overline { 5} = \frac { 14} { 9} \)

UpStudy Free Solution:

To convert the repeating decimal \(1.5\overline { 5} \) to a fraction, we can use the following method:

Let's denote the repeating decimal as \(x\):

\(x = 1.55555\ldots \)

Since the decimal repeats every digit, we can multiply \(x\) by 10 to shift the decimal point one place to the right:

\(10x = 15.55555\ldots \)

Now, we subtract the original \(x\) from this new equation:

\(10x - x = 15.55555\ldots - 1.55555\ldots \)

This subtraction eliminates the repeating part:

\(9x = 14\)

Now, we solve for \(x\) by dividing both sides by 9:

\(x = \frac { 14} { 9} \)

Thus, \(1.5\overline{5}\) as a fraction is:

\(\frac { 14} { 9} \)

Supplemental Knowledge

Converting repeating decimals to fractions is a fundamental technique in number theory. A repeating decimal is a decimal number that has digits that repeat forever. These can be converted into fractions using algebraic methods.

Steps to Convert a Repeating Decimal to a Fraction:

1. Define the repeating decimal as a variable \(x\).

2. Multiply the variable by a power of 10 to shift the repeating part. This step ensures that the decimal part aligns properly for subtraction.

3. Subtract the original variable from this new expression. This eliminates the repeating part, leaving a simple algebraic equation.

4. Solve for the variable. This will give the decimal in fraction form.

5. Simplify the fraction if necessary.

For instance, consider the repeating decimal \(0.\overline { 3} \):

- Let \(x = 0.\overline { 3} \).

- Multiply by 10: \(10x = 3.\overline { 3} \).

- Subtract the original equation: \(10x - x = 3.\overline { 3} - 0.\overline { 3} \) results in \(9x = 3\).

- Solve for \(x\): \(x = \frac { 3} { 9} = \frac { 1} { 3} \).

This method can be extended to more complex repeating decimals, including those with non-repeating parts before the repeating sequence.

Understanding concepts like converting repeating decimals to fractions can be a game-changer for students tackling number theory problems. If you’re finding these steps intriguing and want to dive deeper into the world of mathematics, UpStudy is here to help!