Chapter 37: Chapter 37

Problem 1144

Previous question

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

Short Answer

Expert verified

The repeating decimal $14.23\overline{23}$ can be written as a quotient of two integers $\frac{1409}{99}$.

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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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Chapter 37: Chapter 37

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

Short Answer

Step by step solution

Unlock all solutions

Step 1: Focus on the repeating decimal part

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

Step 3: Subtract the original x from the multiplied x

Step 4: Solve for x

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

Step 6: Simplify the fraction

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