Problem 51 - Essential Calculus

Question

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0.0 \overline{75} $$

Step-by-Step Solution

Answer

The geometric series sum of the repeating decimal \(0.0\overline{75}\) is represented as the ratio of two integers, \(\frac{2}{25}\).

Step 1: Assign the repeating decimal to a variable

Let \(x = 0.0\overline{75}\). When we multiply \(x\) by 100, we get \(100x = 7.5\overline{75}\). This positions the recurring number 75 in the same places.

Step 2: Form the equation by subtraction

Subtracting \(x\) from \(100x\) gives us \(99x = 7.5\overline{75} - 0.0\overline{75}\), which simplifies to \(99x = 7.5\).

Step 3: Express the sum as a ratio of integers

The value of \(x\) can be found by dividing 7.5 by 99, yielding \(x = \frac{7.5}{99}\). Note that this fraction must be further reduced to lowest terms to express the sum as a ratio of integers.

Step 4: Reduce the fraction to its simplest form

The fraction \(\frac{7.5}{99}\) can be written as \(\frac{750}{9900}\). By continuing to reduce this we get \(x = \frac{2}{25}\) as the simplest form of the fraction.

Key Concepts

Geometric Series

Understanding how a repeating decimal translates into a geometric series is a fundamental concept in mathematics. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, known as the ratio. In the context of repeating decimals, the geometric series is formed by recognizing that the repeating part can be broken down into an infinite sum of fractions, each of which is a term in the series.

For example, the repeating decimal 0.0\(\overline{75}\) can be thought of as \(0.75 + 0.0075 + 0.000075 + ...\), and so on. We can see that each term is 1/100th of the preceding term—thus, the ratio is 1/100. The first term (the one before multiplication by the ratio begins) is 0.75 in this case. Using the formula for the sum of an infinite geometric series, \(S = \frac{a}{1 - r}\) where \(S\) is the sum, \(a\) is the first term, and \(r\) is the ratio, we can solve for the repeating decimal as a neat fraction.

Ratio of Integers

When we talk about expressing a decimal as the ratio of two integers, we're essentially looking to convert it into a fraction. An integer is a number with no fractional part - it is whole, positive or negative, including zero. A ratio is a comparison between two quantities, showing the number of times one value contains or is contained within the other.

The repeating decimal problem requires us to find two integers whose ratio exactly equals the decimal. In the given exercise, we did just that when we expressed 0.0\(\overline{75}\) as \(\frac{7.5}{99}\) and then further transformed it into \(\frac{750}{9900}\), which is indeed a ratio of two integers. While this is correct, it is not in its simplest form. Simplifying the fraction is a crucial step to finding the most reduced ratio of integers that still represents the same value.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its smallest, simplest form—essentially, making the numerator and the denominator as small as possible while still retaining the same ratio. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator, then divide both by this number.

In our exercise, the fraction \(\frac{750}{9900}\) needs simplification. To do this, we note that the GCD of 750 and 9900 is 150. Dividing both the top and bottom by 150, we get \(\frac{750 \/ 150}{9900 \/ 150} = \frac{5}{66}\). However, further simplification is needed since 5 and 66 have a common divisor of 1. Thus, the fraction is already in its simplest form, and now we have a neat and tidy ratio of integers that represents the original repeating decimal.