Terminating Decimals to Fractions

Converting Terminating Decimals to Fractions

Decimals and fractions are two fundamental ways to represent parts of a whole in mathematics. Understanding how to convert terminating decimals to fractions is essential for mastering various mathematical concepts and real-life applications such as measurements, finance, and data analysis. This lesson provides a clear and organized approach to converting terminating decimals to fractions.

Understanding Terminating Decimals and Fractions

Terminating Decimals: A terminating decimal is a decimal number that has a finite number of digits after the decimal point. Examples include 0.5, 2.75, and 3.125. These decimals come to an end and do not repeat indefinitely.

Fractions: A fraction represents a part of a whole, written in the form numerator/denominator. For instance, 1/2, 3/4, and 5/8 are fractions where the numerator indicates how many parts are taken, and the denominator indicates the total number of equal parts.

Steps to Convert a Terminating Decimal to a Fraction

Converting a terminating decimal to a fraction involves a systematic approach. Follow these steps to ensure accuracy:

Identify the Decimal Place:

• Determine how many digits are to the right of the decimal point. This number will help in identifying the denominator of the fraction.

Write the Decimal as a Fraction:

• Place the decimal number over its corresponding power of ten based on the number of decimal places.
  
  
• For example, if there are two digits after the decimal, the denominator is 100.

Simplify the Fraction:

• Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

A Couple of Examples Showing how to Convert Terminating Decimals to Fractions

Example #1:

Let’s convert the terminating decimal 0.75 to a fraction.

1. Identify the Decimal Place:

• 0.75 has two digits after the decimal point.

2. Write as a Fraction:

• Since there are two digits, the denominator is 100.
• Therefore, 0.75 = 75/100.

3. Simplify the Fraction:

Find the GCD of 75 and 100, which is 25.
Divide both numerator and denominator by 25:
75 ÷ 25 = 3
100 ÷ 25 = 4
Thus, 75/100 = 3/4.

Example #2:

Let’s convert the terminating decimal 0.014 to a fraction.

1. Identify the Decimal Place:

• 0.014 has three digits after the decimal point.

2. Write as a Fraction:

• Since there are three digits, the denominator is 1000.
• Therefore, 0.014 = 14/1000.

3. Simplify the Fraction:

Find the GCD of 14 and 1000, which is 2.
Divide both numerator and denominator by 2:
14 ÷ 2 = 7
1000 ÷ 2 = 500
Thus, 14/1000 = 7/500.

Example #3:

Let’s convert the terminating decimal 2.125 to a fraction.

1. Identify the Decimal Place:

• 2.125 has three digits after the decimal point.

2. Write as a Fraction:

• Since there are three digits, the denominator is 1000.
• Therefore, 2.125 = 2125/1000.

3. Simplify the Fraction:

Find the GCD of 2125 and 1000, which is 125.
Divide both numerator and denominator by 125:
2125 ÷ 125 = 17
1000 ÷ 125 = 8
Thus, 2125/1000 = 17/8.

Another Way to Convert Terminating Decimals to Fractions

Example #2 revisited:

Convert 0.014 to a fraction.

Follow all the steps above again. First, convert 0.014 to scientific notation. 

0.014 = 14 × 10^-3

10^-3 = 1 / 10³, so 10^-3 = 1 / 10³

In general, 10^-n = 1 / 10ⁿ

14 × 10^-3 = 14 × 1 / 10³

14 × 10^-3 = (14 / 1) × (1 / 10³)

14 × 10^-3 = (14 × 1) / (1 × 10³)

14 × 10^-3 = 14 / 10³

14 × 10^-3 = 14 / 1000

14 × 10^-3 = 7 / 500