Fractions and Decimals

Fractions and Decimals

Fractions and decimals are two ways to represent parts of a whole. Understanding how to work with both is essential in quantitative methods. This topic will cover the definitions, operations, and conversions between fractions and decimals.

Understanding Fractions

A fraction consists of two parts: a numerator and a denominator. The numerator is the number above the line, and the denominator is the number below the line. For example, in the fraction ¾, 3 is the numerator and 4 is the denominator. This fraction represents three parts of a whole that is divided into four equal parts.

Types of Fractions

• Proper Fractions: The numerator is less than the denominator (e.g., ⅖).
• Improper Fractions: The numerator is greater than or equal to the denominator (e.g., 5/4).
• Mixed Numbers: A whole number combined with a proper fraction (e.g., 2⅗).

Operations with Fractions

Addition of Fractions

To add fractions, they must have a common denominator. If they do not, you first find the least common denominator (LCD).

Example: Adding ⅓ and ¼

1. Find the LCD of 3 and 4, which is 12.
2. Convert each fraction:
• ⅓ = (1 × 4)/(3 × 4) = 4/12
• ¼ = (1 × 3)/(4 × 3) = 3/12
3. Add the fractions: 4/12 + 3/12 = 7/12.

Subtraction of Fractions

Subtraction follows the same rules as addition. Ensure the fractions have a common denominator.

Example: Subtracting ⅖ from ¾

1. Find the LCD of 2 and 4, which is 4.
2. Convert the fractions:
• ¾ = (3 × 1)/(4 × 1) = 3/4
• ⅖ = (2 × 2)/(5 × 2) = 4/10 (not the same denominator, so we take 4/10 to 20/50)
3. Now, convert ¾ to 20/50: 3/4 = (3 × 5)/(4 × 5) = 15/20.
4. Subtract: 15/20 - 4/10 = 15/20 - 8/20 = 7/20.

Multiplication of Fractions

To multiply fractions, multiply the numerators together and the denominators together.

Example: Multiplying ⅗ by ⅖

1. Multiply the numerators: 3 × 2 = 6.
2. Multiply the denominators: 5 × 5 = 25.
3. The result is: 6/25.

Division of Fractions

To divide fractions, multiply by the reciprocal of the second fraction.

Example: Dividing ⅗ by ⅖

1. Find the reciprocal of ⅖, which is ⅖ = 5/2.
2. Now, multiply: ⅗ × 5/2.
3. Multiply the numerators: 3 × 5 = 15.
4. Multiply the denominators: 5 × 2 = 10.
5. The result is: 15/10, which simplifies to 3/2 or 1⅗.

Understanding Decimals

A decimal is another way to represent fractions. It uses a point (.) to separate the whole number part from the fractional part. For example, 0.75 represents 75/100.

Types of Decimals

• Terminating Decimals: These decimals have a finite number of digits (e.g., 0.5).
• Recurring Decimals: These decimals have one or more digits that repeat indefinitely (e.g., 0.333...).

Operations with Decimals

Addition and Subtraction of Decimals

To add or subtract decimals, align the decimal points.

Example: Adding 0.75 and 0.25

1. Align the numbers:

  0.75
+ 0.25
------

2. Add: 0.75 + 0.25 = 1.00.

Multiplication of Decimals

To multiply decimals, multiply as if they are whole numbers. Then, count the total number of decimal places in both numbers and place the decimal point in the product accordingly.

Example: Multiplying 0.4 by 0.6

1. Multiply as whole numbers: 4 × 6 = 24.
2. Count decimal places: 0.4 has 1 decimal place and 0.6 has 1 decimal place, for a total of 2 decimal places.
3. Place the decimal point: 24 becomes 0.24.

Division of Decimals

To divide decimals, move the decimal point in the divisor to the right until it is a whole number. Move the decimal point in the dividend the same number of places.

Example: Dividing 0.75 by 0.25

1. Convert 0.25 to a whole number by moving the decimal two places to the right, which makes it 25.
2. Move 0.75 the same number of places: 75.
3. Now divide: 75 ÷ 25 = 3.

Converting Between Fractions and Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

Example: Converting ¾ to a decimal

1. Divide 3 by 4: 3 ÷ 4 = 0.75.

To convert a decimal to a fraction, write the decimal as a fraction with a denominator of 1, and then multiply the numerator and denominator by 10 for each digit after the decimal point.

Example: Converting 0.6 to a fraction

1. Write as a fraction: 0.6/1.
2. Multiply by 10: (0.6 × 10)/(1 × 10) = 6/10.
3. Simplify: 6/10 = 3/5.

Summary

• A fraction consists of a numerator and a denominator.
• Operations with fractions include addition, subtraction, multiplication, and division.
• A decimal is a way to represent fractions using a point.
• Operations with decimals include addition, subtraction, multiplication, and division.
• You can convert between fractions and decimals by dividing or multiplying.

Check your understanding

• What is the least common denominator of 6 and 8?
• Convert the fraction 5/8 to a decimal.
• What is 1/2 + 1/4?
• Multiply 0.3 by 0.4 and express the result as a decimal.