Relationship Between Fractions and Decimals

The process of converting a fraction to its equivalent decimal form is called fraction-to-decimal conversion. Understanding this process helps in solving problems more efficiently.
 

We can use two methods to convert a fraction into its decimal form. They are called long division and multiples of 10 methods.

Long Division Method: In this method, we divide the numerator by the denominator.

Step 1: The numerator becomes the dividend, and the divisor is the denominator. 

        
Step 2: Compare the values of the numerator and the denominator. If the numerator is less than the denominator, then add a decimal point to the quotient and place a zero next to the dividend. 

 

Step 3: Follow the process involved in the long division method. Bring down zeros if necessary and continue until the remainder becomes zero. We can also stop the division if we notice a repeating pattern.
 

Example: Convert \(\frac{5}{6} \) to a decimal.
 

We use long division and get,
 

\(5 ÷ 6 = 0.833…\) (repeating), which can be rounded to 0.833 or 0.83 depending on context.

Multiples of 10 Method: Adjust the denominator to a power of 10 (such as 10, 100, or 1000) and then rewrite the fraction as a decimal. 

Step 1: Choose a number to multiply both the numerator and denominator so that the denominator becomes a power of 10 (10, 100, 1000, etc.). 
 

Step 2: Apply the same multiplier to the numerator and denominator.
 

Step 3: Once the denominator is a power of 10, write the fraction in decimal form. 

Example: Convert \(\frac{2}{5} \) to a decimal.
 

Multiply numerator and denominator by 2: \(2 \times 2 / 5 \times 2 = \frac{4}{10} \)
 

Decimal form: 0.4
 

So, the answer is 0.4

Every decimal number can be written as a fraction by removing the decimal point, dividing by a power of 10, and simplifying.

Step 1: Write the number without the decimal point. 
 

Step 2: Place the number over 10, 100, or 1000, depending on the number of decimal places.
 

Step 3: Reduce the fraction to its simplest form. 

 

Example: Convert 7.2 to a fraction.
 

Writing the number without the decimal point, we get 72 
 

Divide by 10: \(\frac{72}{10} \)
 

To simplify, \(\frac{72}{10}\), we should find the GCF of 72 and 10. 
 

Let us list the factors of 72 and 10.
 

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
 

Factors of 10: 1, 2, 5, 10
 

Among the factors of 72 and 10, the greatest common factor (GCF) is 2.

Now, we should divide the numerator and denominator by the GCF
 

\(72 \div 2 \div 10 \div 2 = \frac{36}{5} \)

So, \(\frac{72}{10} \) it can be simplified to \(\frac{36}{5} \)
 

Therefore, the fraction form of 7.2 is \(\frac{36}{5} \). 
 

A fraction-to-decimal chart makes conversion easy by showing common fractions with their decimal equivalents. Always remember that proper fractions have a numerator smaller than the denominator and improper fractions have a numerator equal to or greater than the denominator, so their decimal values are 1 or more.

Let’s list some common fractions and their decimal forms in a chart. 

Decimal fractions are divided into two main types: terminating and non-terminating. The second type can be divided further into non-terminating, repeating, and non-terminating, non-repeating decimals. 

Terminating Decimals: As the name suggests, the numbers after the decimal point do not go on forever, as they come to an end.
    
 Example: 0.75, 2.5, 4.125

Non-terminating Decimals: Here, the numbers after the decimal point do not end, as they continue indefinitely. Non-terminating decimals are further classified into two types:

Repeating (recurring) decimals: These are decimals in which one or more digits repeat in a regular pattern after the decimal point.


Example: 0.333… = 0.3, 0.727272… = 0.72 (approx.) or 0.727272… (repeating).

Non-repeating decimals: The decimal digits continue without a repeating pattern. These are called irrational numbers.

Example: \(𝜋 = 3.141592653…\)(π is an irrational number, non-repeating and non-terminating), 
 

\(\sqrt{2} = 1.414213\ldots \)

We can perform different mathematical operations on decimal fractions, including addition, subtraction, multiplication, and division. The following table explains these operations in detail.
 

Understanding the relationship between fractions and decimals is essential for solving math problems quickly and accurately. These tips and tricks help you convert, compare, and work with fractions and decimals with ease.
 

• Know the basic conversions: Memorize common fractions and their decimal equivalents, like \(\frac{1}{2} = 0.5 \).
   
• Use long division for conversion: Divide the numerator by the denominator to convert any fraction into a decimal.
   
• Identify repeating decimals: Recognize patterns in recurring decimals, for instance \(\frac{1}{3} = 0.333\ldots \), to make calculations easier.
   
• Round when needed: Round decimals to the required number of decimal places for estimation or practical use.
   
• Compare with place value: Use decimal equivalents to quickly compare fractions and determine which is larger or smaller.

Fractions and decimals are used in many everyday activities, from handling money to measuring ingredients and distances. Here are three common real-life applications where understanding their relationship is important.

• Money and Finance: Prices, discounts, and taxes often involve decimals and fractions. For example, if an item costs $4.50, it can be written as \(\frac{9}{2} \) in fraction form. 
   
• Cooking and Baking: Recipes use fractions (e.g., \(\frac{1}{2} \) cup of sugars) and sometimes require conversions to decimals for accurate measurements.
   
• Measurement and Construction: Lengths and distances are often given in fractions (e.g., \(\frac{3}{4} \) inch) but may need to be converted into decimals for precision tools.
   
• Time Management: Time is often divided into fractions and decimals. For example, 15 minutes is \(\frac{1}{4} \)of an hour, and 30 minutes is 0.5 hours. Understanding these helps in scheduling, billing, and tracking work hours accurately.
   
• Sports and Fitness: Fractions and decimals are used to record scores, calculate averages, or measure distances. For instance, a runner completing \(\frac{3}{4} \)of a lap or a batting average of 0.325 both rely on these concepts for precise performance tracking.