What is Terminating Decimals as Rational Numbers?

S3-SA4-0135

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Terminating decimals are decimals that have a finite number of digits after the decimal point, meaning they end. When we express these decimals as rational numbers, we show them in the form p/q, where p and q are integers and q is not zero.

Simple Example

Quick Example

Imagine you bought a packet of biscuits for Rs. 25.50. This is a terminating decimal because it ends after two digits (50). To write Rs. 25.50 as a rational number, we can say it's 2550/100, which simplifies to 51/2.

Worked Example

Step-by-Step

Let's convert the terminating decimal 0.75 into a rational number (p/q form).

Step 1: Write the decimal as a fraction by placing the decimal number over 1.
0.75 / 1

---Step 2: Count the number of digits after the decimal point. Here, there are two digits (7 and 5).

---Step 3: Multiply both the numerator and the denominator by 10 raised to the power of the number of decimal places. Since there are two decimal places, we multiply by 10^2 = 100.
(0.75 * 100) / (1 * 100) = 75 / 100

---Step 4: Simplify the fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 75 and 100 is 25.
75 ÷ 25 = 3
100 ÷ 25 = 4

---Answer: So, 0.75 as a rational number is 3/4.

Why It Matters

Understanding terminating decimals as rational numbers is crucial for accuracy in many fields. Data scientists use it to ensure precise calculations, engineers rely on it for exact measurements, and economists use it for financial modeling. This skill helps you build a strong foundation for future careers in AI, finance, and technology.

Common Mistakes

MISTAKE: Forgetting to simplify the fraction to its lowest terms. For example, writing 50/100 instead of 1/2 for 0.5. | CORRECTION: Always simplify the resulting fraction by dividing both numerator and denominator by their greatest common divisor until no common factors (other than 1) remain.

MISTAKE: Incorrectly counting decimal places or multiplying by the wrong power of 10. For example, multiplying 0.125 by 100 instead of 1000. | CORRECTION: Count the digits after the decimal point carefully. If there are 'n' digits, multiply by 10^n (1 followed by 'n' zeros).

MISTAKE: Confusing terminating decimals with non-terminating decimals (like 1/3 = 0.333...). | CORRECTION: Remember that terminating decimals have a definite end, while non-terminating decimals continue infinitely without repeating or repeating a pattern.

Practice Questions

Try It Yourself

QUESTION: Convert 0.4 into a rational number in its simplest form. | ANSWER: 2/5

QUESTION: Express 1.25 as a rational number (p/q). | ANSWER: 5/4

QUESTION: A tailor uses 3.75 meters of fabric for a shirt. Write this length as a rational number. | ANSWER: 15/4

MCQ

Quick Quiz

Which of the following is the correct rational number representation for 0.6?

6/10

3/5

60/100

All of the above

The Correct Answer Is:

While 6/10 and 60/100 are correct, 3/5 is the simplest form. Since all options correctly represent 0.6 (even if not in simplest form), 'All of the above' is the best answer.

Real World Connection

In the Real World

When you use UPI to pay Rs. 150.75 for your chai, that amount is a terminating decimal. Banks and payment apps convert these decimal amounts into fractions internally for precise calculations, ensuring your money transactions are always accurate. Similarly, in cricket, batting averages like 35.5 are terminating decimals which can be expressed as rational numbers for statistical analysis.

Key Vocabulary

Key Terms

TERMINATING DECIMAL: A decimal that ends after a finite number of digits. | RATIONAL NUMBER: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. | NUMERATOR: The top number in a fraction. | DENOMINATOR: The bottom number in a fraction. | SIMPLIFY: To reduce a fraction to its lowest terms.

What's Next

What to Learn Next

Great job understanding terminating decimals! Next, you should explore 'Non-Terminating Repeating Decimals as Rational Numbers'. This will show you how to convert decimals that go on forever but repeat a pattern, expanding your understanding of rational numbers even further.