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Chapter 7: Problem 22

Express $$ 5.1372647264 \ldots $$ as a fraction; here the digits 7264 repeat forever.

Short Answer

Expert verified
The repeating decimal number \(5.1372647264 \dots\) can be expressed as the fraction \(\dfrac{50859}{9900}\).

Step by step solution

01

Assign a variable to the repeating decimal number

Let the variable x represent the repeating decimal number: \(x = 5.1372647264 \dots \) The repeating part is 7264.
02

Multiply the variable by an appropriate power of 10 to isolate the repeating part

We need to isolate the repeating part of the decimal. To do this, we will multiply the equation by a power of 10. Since the repeating part starts after two decimal places, we will multiply both sides of the equation by \(10^2 = 100\): \(100x = 513.72647264 \dots \)
03

Multiply the variable by another appropriate power of 10 to align the repeating part

Now, we need to align the repeating part of the decimal so that by subtracting, we can eliminate the repeating part. In this case, since the repeating part has four digits (7264), we will multiply both sides of the equation by \(10^4 = 10000\): \(10000x = 51372.647264 \dots \)
04

Subtract the two equations to eliminate the repeating decimals

Now, subtract the equation from Step 2 from the equation in Step 3 to eliminate the repeating decimals: \(10000x - 100x = 51372.647264 \dots - 513.72647264 \dots \) \(9900x = 50859\)
05

Solve for the variable

Divide both sides of the equation by 9900 to solve for x: \(x = \dfrac{50859}{9900}\) Thus, the repeating decimal number \(5.1372647264 \dots\) can be expressed as the fraction \(\dfrac{50859}{9900}\).

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