Problem 26 - Thomas Calculus

Question

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 3 . \overline{142857}=3.142857142857 \ldots $$

Step-by-Step Solution

Answer

The number is expressed as \(\frac{22}{7}\).

Step 1: Understand the Problem

We are given a repeating decimal, \(3.\overline{142857}\), and need to express it as the ratio of two integers. The repeating part is \(142857\).

Step 2: Define Variables for the Decimal

Let \(x = 3.142857142857\ldots\). This can be rewritten as \(x = 3 + 0.142857142857\ldots\). We will focus on converting \(0.\overline{142857}\) into a fraction first.

Step 3: Set Up Equation for the Decimal Expansion

Let \(y = 0.142857142857\ldots\). Notice how after one cycle of repeating, we have \(10^6 \times y = 142857.142857\ldots\).

Step 4: Subtract to Eliminate Repeating Part

Subtract \(y = 0.142857\ldots\) from \(10^6y = 142857.142857\ldots\): \[ 10^6y - y = 142857.142857 - 0.142857 \].This simplifies to \(999999y = 142857\).

Step 5: Solve for y

Divide both sides of the equation by \(999999\) to solve for \(y\):\[ y = \frac{142857}{999999} \].

Step 6: Simplify the Fraction \(\frac{142857}{999999}\)

Observe that the greatest common divisor (GCD) of 142857 and 999999 is 142857. Dividing both the numerator and the denominator by 142857,\[ y = \frac{142857 \div 142857}{999999 \div 142857} = \frac{1}{7} \].

Step 7: Express Original Number as a Fraction

We originally expressed \(x = 3 + y\) and found \(y = \frac{1}{7}\). Therefore, \[ x = 3 + \frac{1}{7} \].Express this as a single fraction: \[ x = \frac{21}{7} + \frac{1}{7} = \frac{22}{7} \].

Step 8: Conclusion

The repeating decimal \(3.\overline{142857}\) is equivalent to the fraction \(\frac{22}{7}\).

Key Concepts

Ratio of Integers

The concept of expressing numbers as the ratio of integers means representing a number as a fraction. This involves writing a number in the form \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b eq 0 \). Every rational number, which includes both repeating and terminating decimals, can be expressed in this form. When you're working with repeating decimals, like in our example of \( 3.\overline{142857} \), the goal is to convert the repeating decimal into a fraction. This process involves setting the decimal equal to a variable, using algebraic techniques to isolate the repeating part, and finally finding simpler integer values to turn the decimal into a fraction. Using ratios of integers helps us understand numbers beyond decimals, showing their exact fractional value, which can sometimes be lost in the infinite continuation of a decimal number.

Fraction Conversion

Converting a repeating decimal to a fraction involves several key steps. First, identify the repeating part of the decimal. For \( 3.\overline{142857} \), \( 142857 \) is the repeating sequence. The next step is to set this repeating decimal equal to a variable, say \( y \), and use multiplication to shift the decimal point so you can subtract and eliminate the repeating part. Consider \( y = 0.\overline{142857} \). By multiplying both sides of the equation by \( 10^6 \) (since \( 142857 \) has six digits), you align the repeating part: \[ 10^6 y = 142857.142857\ldots \]Subtract \( y \) from \( 10^6 y \), allowing the repeating parts to cancel out:\[ 999999y = 142857 \]Now, \( y \) can be solved as \( \frac{142857}{999999} \).This systematic approach enables you to convert the unwieldy repeat into tidy integers ready to be simplified into a fraction.

Simplifying Fractions

Once you have a fraction derived from a repeating decimal, simplifying it involves reducing the fraction to its simplest form. This can be done by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number. For the example given, \( y = \frac{142857}{999999} \), the GCD of 142857 and 999999 is 142857. Dividing both numbers by their GCD gives:\[ \frac{142857 \div 142857}{999999 \div 142857} = \frac{1}{7} \]This simplification shows that the repeating sequence \( 0.\overline{142857} \) is equivalent to \( \frac{1}{7} \).By understanding this process, you can handle all repeating decimals with confidence, ensuring that you reach the most reduced and elegant form of the fraction every time. Simplifying fractions not only makes them easier to work with but also reveals the inherent simplicity in seemingly complex numbers.