Write 2.5 repeating as a mixed number in simplest form

Write 2.5 repeating as a mixed number in simplest form.

The Correct Answer and Explanation is :

To convert the decimal (2.5) (which is (2.5) repeating) into a mixed number, we first represent it mathematically. The decimal (2.5) repeating can be expressed as (2.\overline{5}), where the digit (5) repeats indefinitely.

Step 1: Setting Up the Equation

Let (x = 2.\overline{5}). This means that:

[
x = 2.55555\ldots
]

Step 2: Eliminating the Repeating Decimal

To eliminate the repeating part, we can multiply (x) by (10) (since the repeating digit is in the tenths place):

[
10x = 25.55555\ldots
]

Now, we have two equations:

(x = 2.55555\ldots)
(10x = 25.55555\ldots)

Step 3: Subtracting the Two Equations

Now we subtract the first equation from the second:

[
10x – x = 25.55555\ldots – 2.55555\ldots
]

This simplifies to:

[
9x = 23
]

Step 4: Solving for (x)

Next, divide both sides by (9):

[
x = \frac{23}{9}
]

Step 5: Converting to Mixed Number

To convert the improper fraction (\frac{23}{9}) into a mixed number, we perform the division:

Divide (23) by (9). The quotient is (2) (since (9 \times 2 = 18)).
The remainder is (23 – 18 = 5).

Thus, we can express (\frac{23}{9}) as:

[
2 \frac{5}{9}
]

Conclusion

Therefore, the mixed number form of (2.5) repeating (or (2.\overline{5})) is: