Exercise 1.3 — Decimal Expansion

What Does Exercise 1.3 Cover?

Exercise 1.3 of Class 10 Mathematics — Chapter 1: Real Numbers — explores the relationship between rational numbers and their decimal expansions. The central idea is simple but powerful: the way a fraction's denominator is built from prime factors tells you exactly whether its decimal will terminate or repeat forever. This topic is directly tested in CBSE, Telangana, and Andhra Pradesh board exams, both as theory questions and as classification problems.

The Key Theorem — When Does a Decimal Terminate?

A rational number p/q (in its simplest form) has a terminating decimal if and only if the prime factorisation of the denominator q contains no prime factor other than 2 and 5. In other words, q must be of the form 2^m × 5ⁿ, where m and n are non-negative integers.

If q has any other prime factor — such as 3, 7, 11, 13, or 17 — the decimal will be non-terminating and repeating. This rule works because our decimal system is based on 10 = 2 × 5, so only denominators built from 2s and 5s can be converted exactly into a power of 10.

Question 1 — Converting Fractions to Decimals by Division

The first question asks you to actually perform division and classify each decimal. Here are the results:

• 3/8 = 0.375 — terminating (8 = 2³, only factor is 2)
• 229/400 = 0.5725 — terminating (400 = 2⁴ × 5², only 2s and 5s)
• 4⅕ = 21/5 = 4.2 — terminating (denominator is simply 5¹)
• 2/11 = 0.181818… — non-terminating, repeating (11 is a prime other than 2 or 5)
• 8/125 = 0.064 — terminating (125 = 5³, only factor is 5)

Question 2 — Classifying Without Doing Division

This is where the theorem saves you time. Instead of dividing, you simply factorise the denominator and check its prime factors. Always simplify the fraction first before checking.

• 13/3125: 3125 = 5⁵ → only 5s → terminating
• 11/12: 12 = 2² × 3 → has factor 3 → non-terminating, repeating
• 64/455: 455 = 5 × 7 × 13 → has factors 7 and 13 → non-terminating, repeating
• 15/1600: Simplify to 3/320; 320 = 2⁶ × 5 × ... wait — 320 = 2² × 5 × 17 after simplification → has 17 → non-terminating, repeating
• 29/343: 343 = 7³ → has factor 7 → non-terminating, repeating
• 23/(2³ × 5²): denominator is purely 2s and 5s → terminating
• 129/(2² × 5⁷ × 7⁵): has factor 7 → non-terminating, repeating
• 9/15: Simplify to 3/5; 5 = 5¹ → terminating
• 36/100: Simplify to 9/25; 25 = 5² → terminating
• 77/210: Simplify to 11/30; 30 = 2 × 3 × 5 → has factor 3 → non-terminating, repeating

Question 3 — Converting to Decimals Without Division

When the denominator is of the form 2^m × 5ⁿ, you can convert the fraction to a decimal by multiplying numerator and denominator by whatever factor is needed to make the denominator a power of 10. This avoids long division entirely.

• 13/25: Multiply top and bottom by 4 → 52/100 = 0.52
• 15/16: 16 = 2⁴; multiply by 5⁴ = 625 → 9375/10000 = 0.9375
• 23/(2³ × 5²): Multiply by 5 → 115/1000 = 0.115
• 7218/(3² × 5²): Simplify 7218/9 = 802; then 802/25 → multiply by 4 → 3208/100 = 32.08
• 143/110: Simplify to 13/10 = 1.3

The trick is to identify what power of 2 or 5 is missing from the denominator and multiply both numerator and denominator by exactly that amount.

Question 4 — Converting Decimals Back to p/q Form

The final question reverses the process. For terminating decimals, place the digits over the appropriate power of 10 (10, 100, 1000, etc. depending on the number of decimal places). The denominator will always be of the form 2^m × 5ⁿ. For non-terminating repeating decimals, the denominator of the simplified fraction will always contain a prime factor other than 2 or 5 — commonly factors like 3, 9, 11, or 33. For example, 0.63̄ = 63/99 = 7/11, and 43.12̄ simplifies to a fraction with denominator 3 × 11.

Common Mistakes in Exercise 1.3

• Forgetting to simplify the fraction first in Question 2 — checking the denominator before simplifying can give a wrong answer (as in 9/15, which simplifies to 3/5, a terminating decimal).
• Confusing the rule: it is the denominator of the fully simplified fraction that must be checked, not the original denominator.
• In Question 3, multiplying only the denominator by the missing factor and forgetting to multiply the numerator as well — this changes the value of the fraction.
• Assuming that a decimal with many digits must be non-terminating — 0.9375 has four decimal places but terminates perfectly.

What to Study Next

Exercise 1.3 completes the core content of Chapter 1. You are now ready to look at how irrational numbers behave — they have decimal expansions that are non-terminating and non-repeating, which is fundamentally different from the repeating decimals of rationals. Revisit the foundations in the Real Numbers Introduction, see how prime factorisation is applied in Exercise 1.2, or explore related number concepts in Class 9 Real Numbers.