Problem 98 - Elementary Algebra

Question

(a) How can we tell that \(\frac{5}{16}\) will produce a terminating decimal? (b) How can we tell that \(\frac{7}{15}\) will not produce a terminating decimal? (c) Determine which of the following will produce a terminating decimal: \(\frac{7}{8}, \frac{11}{16}, \frac{5}{12}, \frac{7}{24}, \frac{11}{75}, \frac{13}{32}, \frac{17}{40}\), \(\frac{11}{30}, \frac{9}{20}, \frac{3}{64} .\)

Step-by-Step Solution

Answer

Fractions \( \frac{5}{16}, \frac{7}{8}, \frac{11}{16}, \frac{13}{32}, \frac{17}{40}, \frac{9}{20}, \frac{3}{64} \) produce terminating decimals.

Step 1: Understanding Terminating Decimals

A fraction in its simplest form will have a terminating decimal if its denominator (after simplifying) is composed only of the prime factors 2 and/or 5. This means the denominator can be expressed as a power of 2, a power of 5, or a product of powers of 2 and 5.

Step 2: Evaluate \( \frac{5}{16} \)

The fraction \( \frac{5}{16} \) is already in its simplest form. The denominator 16 can be expressed as \( 2^4 \), which involves only the prime factor 2. Therefore, \( \frac{5}{16} \) will produce a terminating decimal.

Step 3: Evaluate \( \frac{7}{15} \)

The fraction \( \frac{7}{15} \) is in its simplest form. The denominator 15 can be expressed as \( 3 imes 5 \), which includes the prime factor 3. Since the denominator involves a prime factor other than 2 or 5, \( \frac{7}{15} \) will produce a non-terminating decimal.

Step 4: Test Denominators for Termination

Check the denominator of each fraction given:- \( \frac{7}{8} \): Denominator 8 is \( 2^3 \). Terminating.- \( \frac{11}{16} \): Denominator 16 is \( 2^4 \). Terminating.- \( \frac{5}{12} \): Denominator 12 is \( 2^2 \times 3 \). Non-terminating.- \( \frac{7}{24} \): Denominator 24 is \( 2^3 \times 3 \). Non-terminating.- \( \frac{11}{75} \): Denominator 75 is \( 3 imes 5^2 \). Non-terminating.- \( \frac{13}{32} \): Denominator 32 is \( 2^5 \). Terminating.- \( \frac{17}{40} \): Denominator 40 is \( 2^3 \times 5 \). Terminating.- \( \frac{11}{30} \): Denominator 30 is \( 2 imes 3 imes 5 \). Non-terminating.- \( \frac{9}{20} \): Denominator 20 is \( 2^2 \times 5 \). Terminating.- \( \frac{3}{64} \): Denominator 64 is \( 2^6 \). Terminating.

Key Concepts

Prime Factors

Prime factors are the building blocks of numbers. They are the prime numbers that multiply together to give the original number. Finding prime factors is crucial in understanding whether a fraction's decimal representation will be terminating or not.
The prime numbers are 2, 3, 5, 7, and so on.
For the purpose of identifying terminating decimals, you focus on the factors 2 and 5.

• Numbers like 8 or 16 are composed only of the prime factor 2 as in, 8 = 2³ and 16 = 2⁴.
• Other numbers, such as 15, which equals 3 x 5, include a factor of 3. This means they aren't solely made up of the factors 2 or 5.

When dealing with fractions, looking at the denominator's prime factors helps determine if its decimal representation is terminating.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form. This means using the greatest common divisor of the numerator and the denominator to divide both, making the fraction as simple as possible. A simplified fraction can make it easier to determine the decimal nature.
Let's take the fraction \(\frac{8}{12}\). To simplify, calculate the GCD of 8 and 12, which is 4. You divide both the numerator and the denominator by 4 to get \(\frac{2}{3}\).
Checking the simplest form of the denominator gives clues about the factors involved. Whether or not it's simplified correctly is crucial because it affects the analysis of its decimal representation.

• For example, \(\frac{10}{100}\) simplifies to \(\frac{1}{10}\) whose factor is just 5, indicating a terminating decimal.

Decimal Representation

Decimal representation tells you how a fraction is expressed in decimal form. This could be terminating or non-terminating depending on the prime factors of the denominator after simplification.
For fractions whose simplified denominator consists only of 2 and/or 5 as prime factors, the decimal representation will be terminating. This occurs because these factors can be seamlessly divided into parts of a whole without any remainder.

• For instance, \(\frac{1}{8}\) results in a decimal of 0.125, clearly terminating after three decimal places.

When the denominator involves prime factors beyond 2 and 5, like 3 or 7, decimals become repeating or non-terminating. Translating a fraction into its decimal form helps in visualizing these numerical behaviors extensively.

Non-terminating Decimal

Non-terminating decimals never come to an end. They either continue indefinitely or repeat at intervals. A common feature is their connection with prime factors beyond 2 and 5, as seen in denominators.
If the simplified denominator of a fraction is completely reducible to 2s and 5s, you will get a terminating decimal. But if there's a 3, 7, or another prime factor involved, expect a non-terminating decimal.

• Fractions like \(\frac{2}{3}\) or \(\frac{1}{7}\) translate to decimals like 0.666... or 0.142857..., clearly demonstrating non-termination or repetition.

Understanding these properties helps clarify why certain fractions result in endless decimals and how prime factors drive these outcomes.