Question

part 3

1. please convert the following recurring decimals into fractional form.

(17.6dot{7} = 17 \frac{67}{90})

Explanation:

🆕 New Concept Discovered: Converting Recurring Decimals to Fractions
Turn repeating decimals into exact fractions using algebra.

Step 1: Separate the whole number and the decimal part

To convert \(17.6\dot{7}\) into a mixed number, we can separate the whole number from the repeating decimal part:

$$17.6\dot{7} = 17 + 0.6\dot{7}$$

The repeating decimal \(0.6\dot{7}\) means the digit \(7\) repeats forever:

$$0.6777...$$

Step 2: Convert the repeating decimal to a fraction

Let \(x\) represent the repeating decimal part:

$$x = 0.6777...$$

Multiply both sides by \(10\) to move the non-repeating part past the decimal point:

$$10x = 6.777... \quad \text{(Equation 1)}$$

Multiply both sides of the original equation by \(100\) to move the first repeating cycle past the decimal point:

$$100x = 67.777... \quad \text{(Equation 2)}$$

Subtract Equation 1 from Equation 2 to eliminate the infinite repeating decimal part:

$$100x - 10x = 67.777... - 6.777...$$

$$90x = 61$$

$$x = \frac{61}{90}$$

Step 3: Combine with the whole number

Now, combine the whole number \(17\) with the fraction we found:

$$17.6\dot{7} = 17\frac{61}{90}$$

Looking at the handwritten attempt in the image, the student wrote \(17\frac{67}{90}\). This is incorrect because they forgot to subtract the non-repeating digit (\(6\)) from the numerator. The correct numerator is \(67 - 6 = 61\).

Answer:

$$17\frac{61}{90}$$