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Problem 1144

# Write the repeating decimal $$14.23$$ as a quotient of two integers, $\mathrm{p} / \mathrm{q}$

Expert verified
The repeating decimal $$14.23\overline{23}$$ can be written as a quotient of two integers $$\frac{1409}{99}$$.
See the step by step solution

## Step 1: Focus on the repeating decimal part

Let's call the repeating decimal part x and set it to 0.23: $x=0.23\overline{23}$

## Step 2: Multiply x by 100 to shift the decimal point

To get rid of the decimal point, we will multiply x by 100: $100x=23\overline{23}$

## Step 3: Subtract the original x from the multiplied x

$100x - x = 99x$ $23\overline{23} - 0.23\overline{23} = 23$

## Step 4: Solve for x

Now, we can solve for x: $99x = 23$ $x = \frac{23}{99}$

## Step 5: Add the whole part to the repeating decimal part

Now that we have found the repeating decimal part as a fraction, we can add the whole part (14) to it: $\frac{23}{99} + 14 = \frac{23 + 14 \times 99}{99}$

## Step 6: Simplify the fraction

Simplify the fraction to find the final answer: $\frac{23 + 1386}{99} = \frac{1409}{99}$ So, the repeating decimal $$14.23\overline{23}$$ can be written as a quotient of two integers $$\frac{1409}{99}$$.

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