CAT Terminating and Non- terminating decimal numbers - Practice Questions & MCQ

Introduction

Numbers can be classified into various categories based on their properties.
In the number system, decimals are classified as terminating or non-terminating.
Terminating decimals are the ones that end after a certain number of decimal places.
Non-terminating decimals, on the other hand, go on forever without repeating or ending.

Terminating Decimal Numbers

A terminating decimal number is a decimal number that ends at a particular point.
When we write a terminating decimal in decimal form, it does not go beyond a certain number of decimal places.

For example:

• 1/2 = 0.5 (terminating)
• 3/4 = 0.75 (terminating)
• 2/5 = 0.4 (terminating)
• 7/8 = 0.875 (terminating)

Non-terminating Decimal Numbers

A non-terminating decimal number is a decimal number that continues indefinitely without repeating.
Non-terminating decimals don't have a final digit, and they go on infinitely.

For example:

• 1/3 = 0.3333... (non-terminating)
• 1/7 = 0.142857142857... (non-terminating)
• 5/9 = 0.5555... (non-terminating)

Solved Example:

Question 1: Convert 0.16 into a fraction.

Solution:

0.16 can be expressed as 16/100, which can be further simplified by dividing the numerator and the denominator by their greatest common divisor (4), leading to 4/25.

 

Question 2: Find the value of (10.4444...)². 

Solution:

10.4444... can be expressed as a rational number. Let's write it as:

10.4444... = 10 + 0.4444...

Let's denote x = 0.4444...

Therefore, 10x = 4.4444..., which implies 10x - x = 4.

So, 9x = 4, which gives us x = 4/9.

Therefore, 10.4444... = 10 + 4/9 = 90/9 + 4/9 = 94/9.

Now, (10.4444...)² = (94/9)² = 8836/81.

Question 3: Simplify the expression 0.333... + 0.1666...

Solution:

Both 0.333... and 0.1666... are repeating decimals that can be converted to fractions.

Let's denote x = 0.333..., so 3x = 0.999..., which implies 3x = 1 (since 0.999... equals 1). So, x = 1/3.

Now, let's denote y = 0.1666..., so 6y = 0.999..., which implies 6y = 1. So, y = 1/6.

So, 0.333... + 0.1666... = 1/3 + 1/6 = 2/6 + 1/6 = 1/2.

Tip and Tricks

Tip 1: To determine if a decimal is terminating or non-terminating, find its fraction form. If the denominator only has prime factors of 2 and/or 5, the decimal is terminated.

Tip 2: Non-terminating decimals can sometimes be converted into fractions through algebraic manipulation. For example, if x = 0.3333..., then multiplying both sides by 10 gives 10x = 3.3333... Subtracting x from both sides gives 9x = 3, and dividing both sides by 9 gives x = 1/3.

Tip 3: When dealing with non-terminating decimals, try to spot any patterns and use those patterns to simplify or calculate the decimal value.