Converting Decimal to a Fraction

Converting decimals to fractions is a valuable skill in mathematics that helps in understanding numbers and their relationships. Here, I will guide you step-by-step through the process of converting decimal numbers into fraction form.

Step 1: Understanding Decimal Places

Decimals are based on powers of ten, which means each digit's place tells you how much of a part of ten it represents. For example, in the decimal number 0.75:

• The '7' is in the tenths place (7/10)
• The '5' is in the hundredths place (5/100)

Step 2: Write the Decimal as a Fraction

To convert a decimal into a fraction, start by writing it as a fraction with 1 in the denominator:

• For example, if your decimal is 0.75, you can write it as:

  [ 0.75 = \frac{75}{100} ]

Step 3: Simplify the Fraction

Now that you have the fraction, the next step is to simplify it by finding the greatest common divisor (GCD) of the numerator and denominator. In our example:

• The GCD of 75 and 100 is 25.

Now divide both the numerator and the denominator by their GCD:

• [ \frac{75 \div 25}{100 \div 25} = \frac{3}{4} ]

Step 4: Converting Different Types of Decimals

1. Terminating Decimals: These are decimals that end after a certain number of digits (e.g., 0.5 or 0.75). You can follow the steps above.
2. Repeating Decimals: These are decimals that have a repeating pattern (e.g., 0.333... or 0.666...). For these, it is best to set up an equation:
   • For example, let’s take 0.666...
   • Let x = 0.666...
   • Multiply both sides by 10 to shift the decimal: 10x = 6.666...
   • Now subtract the original equation: [ 10x - x = 6.666... - 0.666... ]
     [ 9x = 6 ]
     • Divide by 9: [ x = \frac{6}{9} ]
     • Simplify: [ x = \frac{2}{3} ]

Step 5: Practice with Examples

• Convert 0.25 to a fraction: [ 0.25 = \frac{25}{100} = \frac{1}{4} ]
• Convert 0.333... to a fraction: [ 0.333... = \frac{1}{3} ]

Practice Makes Perfect

Try converting the following decimals into fractions:

• 0.5
• 0.875
• 0.666...

Helpful Tips

• Always start by writing the decimal over 1. This sets the foundation for conversion.
• Remember to include the place value of the decimal when determining your denominator. If there are two numbers after the decimal, use 100.
• For repeating decimals, set them as variables to help isolate them and make the equation easier.
• It may be helpful to memorize the fractions of common decimals to speed up your work in the future!

With these steps, you should now be able to confidently convert decimals into fractions! Keep practicing, and you'll find it gets easier over time.