Problem 7 - Prealgebra

Question

Complete the following tables by converting each fraction to a decimal. Fraction \(\frac{1}{6} \quad \frac{2}{6} \quad \frac{3}{6} \quad \frac{4}{6} \quad \frac{5}{6} \quad \frac{6}{6}\) Decimal

Step-by-Step Solution

Answer

0.1667, 0.3333, 0.5, 0.6667, 0.8333, 1.0

Step 1 - Convert the First Fraction

To convert the fraction \( \frac{1}{6} \) into a decimal, divide 1 by 6. The result is 0.1666..., which is a repeating decimal, often denoted as 0.1667 for rounding to four decimal places.

Step 2 - Convert the Second Fraction

For \( \frac{2}{6} \), simplify the fraction first. This equals \( \frac{1}{3} \). Dividing 1 by 3 gives you 0.3333..., or 0.3333 in decimal form when rounded.

Step 3 - Convert the Third Fraction

The fraction \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \). Dividing 1 by 2 results in 0.5, which doesn't need further rounding as it's exact.

Step 4 - Convert the Fourth Fraction

\( \frac{4}{6} \) simplifies to \( \frac{2}{3} \). Dividing 2 by 3 results in 0.6666..., or rounded to four decimal places, it's 0.6667.

Step 5 - Convert the Fifth Fraction

Convert \( \frac{5}{6} \) by dividing 5 by 6. This gives 0.8333..., or rounded to four decimal places, it's 0.8333.

Step 6 - Convert the Sixth Fraction

The last fraction \( \frac{6}{6} \) simplifies to 1 since any number divided by itself is 1.

Key Concepts

Understanding Repeating Decimals

When you divide certain fractions, sometimes you will see a pattern in the decimals that repeats endlessly. These are called repeating decimals. For instance, when you look at the conversion of \( \frac{1}{6} \) to a decimal, it results in 0.1666..., where the digit '6' repeats forever. Similarly, the fraction \( \frac{2}{3} \) becomes 0.6666..., where the '6' repeats.
These repeating patterns can be represented with a bar over the digits that repeat, like this: 0.\overline{6} for \( \frac{1}{6} \), and 0.\overline{6} for \( \frac{2}{3} \). This helps in understanding that the sequence of numbers continues on and does not change. Here’s how the concept of repeating decimals applies to some of the fractions:

• \( \frac{1}{3} \rightarrow 0.3333... \rightarrow 0.\overline{3} \)
• \( \frac{5}{6} \rightarrow 0.8333... \rightarrow 0.\overline{3} \), rounded to 0.8333 for simplicity.

Repeated patterns are a normal aspect of certain divisions, and recognizing them can greatly help in working with such decimals in math problems.

The Importance of Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms, making them easier to work with. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
For example, the fraction \( \frac{2}{6} \) can be simplified by dividing both 2 and 6 by their GCD, which is 2, resulting in \( \frac{1}{3} \). Similarly, \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \) by dividing both by 3.
Here’s why simplification is helpful:

• Makes calculations easier and faster.
• Helps easily convert a fraction to a decimal, as seen with \( \frac{1}{2} \rightarrow 0.5 \).
• Ensures consistency in mathematical operations and comparisons.

When you simplify fractions, you often uncover simpler patterns in decimals, making mathematical insights more intuitive.

Mastering Division for Fractions and Decimals

Division is a fundamental operation used to convert fractions into decimals. The process involves dividing the numerator by the denominator.
For instance, to convert the fraction \( \frac{4}{6} \) into a decimal, first simplify it to \( \frac{2}{3} \), then perform the division: 2 divided by 3 results in 0.6666..., a repeating decimal.
Steps for division in fraction to decimal conversion:

• Simplify the fraction, if possible, to its lowest terms.
• Divide the numerator by the denominator.
• Recognize the pattern in the division and identify repeating decimals.
• Round off when necessary to suit the level of precision you need.

By understanding division, you not only convert fractions to decimals but also enhance your ability to handle more complex math problems involving rational numbers.