Converting Between Fractions and Decimals

Introduction

Key Concepts

1. Understanding Fractions and Decimals

A fraction represents a part of a whole and is expressed as one integer over another, such as $\frac{3}{4}$. The numerator indicates how many parts we have, while the denominator shows the total number of equal parts the whole is divided into.

A decimal is another way to represent fractions, especially those with denominators that are powers of ten. Decimals are based on the base-10 system and are written with a decimal point separating the whole number from the fractional part, such as 0.75.

2. Converting Fractions to Decimals

Converting a fraction to a decimal involves division. The numerator is divided by the denominator to obtain the decimal equivalent.

For example, to convert $\frac{3}{4}$ to a decimal:

Another example: $\frac{5}{8}$

3. Converting Decimals to Fractions

To convert a decimal to a fraction, identify the place value of the last digit. For instance, in 0.75, the last digit is in the hundredths place.

Steps to convert:

1. Write the decimal as a fraction with the denominator as the place value. For 0.75, it becomes $\frac{75}{100}$.
2. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD). Here, GCD of 75 and 100 is 25.
3. $$ \frac{75 \div 25}{100 \div 25} = \frac{3}{4} $$

Therefore, 0.75 is equivalent to $\frac{3}{4}$.

4. Equivalent Forms

Fractions and decimals can represent the same value in different forms. Recognizing equivalent forms is essential for comparing and performing operations with them.

Examples:

• $\frac{1}{2} = 0.5$
• $\frac{2}{5} = 0.4$
• $\frac{7}{10} = 0.7$

5. Repeating Decimals

Some fractions result in repeating decimals, where a digit or a group of digits repeats indefinitely.

For example, $\frac{1}{3}$ converts to $0.\overline{3}$, where the bar indicates that 3 repeats forever.

To express repeating decimals as fractions, algebraic methods can be used:

Let $x = 0.\overline{3}$

6. Terminating Decimals

Fractions that have denominators with prime factors of only 2 or 5 result in terminating decimals.

For example, $\frac{1}{4} = 0.25$ and $\frac{3}{5} = 0.6$.

This is because 4 and 5 are factors of 10, the base of the decimal system.

7. Practical Applications

Converting between fractions and decimals is essential in various real-life contexts, such as in finance for calculating interest rates, in science for measurements, and in everyday scenarios like cooking and shopping.

For instance, understanding discounts in shopping often requires converting percentages (which are related to decimals) to fractions to calculate the final price.

8. Tools and Strategies for Conversion

Several strategies can aid in the conversion process:

• Long Division: Traditional method for converting fractions to decimals.
• Prime Factorization: Helps in simplifying fractions and determining if a decimal is terminating or repeating.
• Use of Calculators: For quick and accurate conversions, especially with complex fractions.

9. Common Mistakes to Avoid

Students often make errors such as:

• Incorrectly simplifying fractions.
• Misplacing the decimal point during conversion.
• Confusing repeating decimals with terminating ones.

Ensuring a clear understanding of the underlying principles helps mitigate these mistakes.

10. Practice Problems

To reinforce the concepts, here are some practice problems:

1. Convert $\frac{7}{8}$ to a decimal.
2. Convert 0.125 to a fraction.
3. Express $\frac{5}{6}$ as a decimal.
4. Convert 0.\overline{6} to a fraction.

Answers:

• $\frac{7}{8} = 0.875$
• 0.125 = $\frac{1}{8}$
• $\frac{5}{6} = 0.\overline{8333}$
• 0.\overline{6} = $\frac{2}{3}$

Comparison Table

Summary and Key Takeaways