Decimal Fraction

Decimal Fractions: Detailed Explanation

Decimal fractions are a key concept in arithmetic and are essential for various mathematical and financial calculations. Here’s a comprehensive guide to understanding decimal fractions, including their conversion, types, operations, and practical applications.

1. Understanding Decimal Fractions

Definition: Decimal fractions are fractions where the denominator is a power of 10, which can be expressed as decimals. They provide a way to represent numbers that are not whole numbers.

Example:

• The fraction $\frac{3}{10}$ can be written as 0.3 in decimal form.
• The fraction $\frac{27}{100}$ can be written as 0.27 in decimal form.

Converting Fractions to Decimals: To convert a fraction to a decimal, divide the numerator by the denominator.

• Example: Convert $\frac{7}{4}$ to a decimal:
  • Divide 7 by 4: $7 \div 4 = 1.75$.
  • So, $\frac{7}{4} = 1.75$ in decimal form.

Converting Decimals to Fractions: To convert a decimal to a fraction, express it as a fraction with a denominator of 10, 100, or 1000, depending on the number of decimal places. Then simplify if necessary.

• Example: Convert 0.75 to a fraction:
  • Write 0.75 as $\frac{75}{100}$.
  • Simplify $\frac{75}{100}$ to $\frac{3}{4}$.

2. Types of Decimals

1. Terminating Decimals:

   • Definition: These decimals have a finite number of digits after the decimal point.
   • Example: 0.5 (which is $\frac{5}{10}$), 1.25 (which is $\frac{125}{100}$), 3.75 (which is $\frac{375}{100}$).

2. Repeating Decimals:

   • Definition: These decimals have a repeating pattern of digits after the decimal point.
   • Example: 0.333... (where 3 repeats), 1.666... (where 6 repeats).

Converting Repeating Decimals to Fractions: Use algebraic methods to convert repeating decimals to fractions.

• Example: Convert 0.666... to a fraction:
  • Let $x = 0.666...$
    
  • Then $10x = 6.666...$
    
  • Subtract the original equation from this new equation: $10x - x = 6.666... - 0.666...$
    
  • This simplifies to $9x = 6$, so $x = \frac{6}{9}$, which simplifies to $\frac{2}{3}$.

3. Operations with Decimals

1. Addition and Subtraction:

   • Procedure: Align the decimal points and perform the operation as with whole numbers. Fill in any missing decimal places with zeros.
   • Example:
     • Addition: Calculate 2.75 + 1.4
       • Align decimals: 2.75

+1.40

• Add: 2.75 + 1.40 = 4.15

• Subtraction: Calculate 5.5 - 2.3
  • Align decimals: 5.50

-2.30

• Subtract: 5.50 - 2.30 = 3.20

1. Multiplication:

   • Procedure: Multiply the numbers as if they were whole numbers. Count the total number of decimal places in both factors and place the decimal point accordingly in the product.
   • Example: Multiply 0.6 by 0.4
     • Multiply as whole numbers: 6 × 4 = 24.
     • Count decimal places in the factors (1 + 1 = 2).
     • Place the decimal point: 0.24.

2. Division:

   • Procedure: Divide as with whole numbers. Move the decimal point in the divisor and dividend to make the divisor a whole number, then place the decimal point in the quotient accordingly.
   • Example: Divide 4.5 by 0.3
     • Convert to whole numbers by multiplying both by 10: $\frac{45}{3} = 15$.
     • So, 4.5 ÷ 0.3 = 15.

4. Applications in Business

1. Financial Calculations:

   • Example: If a product costs $19.99 and is sold with a 15% profit margin, calculate the selling price:
     • Profit = 19.99 × 0.15 = 2.9985.
     • Selling Price = 19.99 + 2.9985 = 22.9885 ≈ $22.99.

2. Budgeting:

   • Example: If monthly expenses are $1234.56, budgeting for a 10% reduction involves:
     • Reduction = 1234.56 × 0.10 = 123.456.
     • New Budget = 1234.56 - 123.456 = 1111.104 ≈ $1111.10.

3. Data Analysis:

   • Example: Calculating the average score from several decimal scores. If the scores are 89.5, 92.3, and 87.8, then:
     • Average = $\frac{89.5 + 92.3 + 87.8}{3} = \frac{269.6}{3} = 89.87$.

Example Problems

1. Convert to Decimal:

   • Convert $\frac{5}{8}$ to a decimal.
   • Solution: Divide 5 by 8: $5 ÷ 8 = 0.625$.

2. Convert to Fraction:

   • Convert 0.4 to a fraction.
   • Solution: Write 0.4 as $\frac{4}{10}$, which simplifies to $\frac{2}{5}$.

3. Add Decimals:

   • Calculate 3.25 + 4.75.
   • Solution: 3.25 + 4.75 = 8.00.

4. Multiply Decimals:

   • Multiply 0.3 by 0.6.
   • Solution: 0.3 × 0.6 = 0.18.

5. Divide Decimals:

   • Divide 7.5 by 0.5.
   • Solution: 7.5 ÷ 0.5 = 15.

By mastering decimal fractions, BBA students will be well-equipped for precise mathematical calculations in various business contexts, including financial analysis, budgeting, and data interpretation.