Class 9 Maths · Original EZT practice
Number Systems (Class 9): original worked practice
A decimal can keep going and still be rational. Learn to distinguish a repeating tail from a pattern that never settles, then check roots and fractions without trusting a calculator display.
8 original worked examples, 8 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 17 October 2026:
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Reduce before classifying a decimal
Choose the method: For a fraction in lowest terms, a terminating decimal needs no prime factors in its positive denominator except 2 and 5.
EZT practice question
Compare the decimal behaviour of 21/120 and 21/88.
Show the worked solution
- Reduce 21/120 to 7/40. Since 40 = 2³ × 5, this decimal terminates.
- 21/88 is already in lowest terms. Since 88 = 2³ × 11, its decimal is non-terminating recurring.
Answer: 7/40 = 0.175; 21/88 is non-terminating recurring.
Check: 0.175 × 120 = 21. The factor 11 remains in the second denominator.
Watch for this: Inspect the denominator after cancelling common factors, not before.
Now try it yourself
Does 39/150 terminate? Give its decimal.
Check your answer
Yes: 39/150 = 13/50 = 0.26.
The reduced denominator 50 = 2 × 5² has only the allowed primes.
Separate the non-repeating start
Choose the method: Shift the decimal until the two copies have identical recurring tails, then subtract.
EZT practice question
Write 0.1(36) as a reduced fraction. Parentheses mean that 36 repeats forever.
Show the worked solution
- Let x = 0.1363636… . Then 10x = 1.363636… and 1000x = 136.363636… .
- Subtract: 990x = 135. Reduce 135/990 by 45.
Answer: 3/22
Check: 3 ÷ 22 = 0.1363636… . Multiplying the answer by 990 gives 135.
Watch for this: 100x − x does not cancel the tails here: the first digit is not part of the repeating block.
Now try it yourself
Convert 0.2(54) to a reduced fraction.
Check your answer
14/55
1000x − 10x = 254.5454… − 2.5454… = 252, so x = 252/990 = 14/55.
Use squared bounds to place a root
Choose the method: For positive numbers, squaring preserves order. Use exact inequalities before decimals.
EZT practice question
Give an irrational number between 4.2 and 4.3, with a reason.
Show the worked solution
- 4.2² = 17.64 and 4.3² = 18.49.
- 18 is between these values, so 4.2 < √18 < 4.3. Since 18 is not a perfect square, √18 is irrational.
Answer: √18
Check: √18 = 3√2, and a non-zero rational multiple of an irrational number is irrational.
Watch for this: A rounded decimal alone does not prove irrationality or exact interval membership.
Now try it yourself
Give an irrational number between 5.1 and 5.2.
Check your answer
√27
5.1² = 26.01 < 27 < 27.04 = 5.2²; 27 is not a perfect square.
Do not guess the type of a sum
Choose the method: Two irrational numbers can add to either a rational or an irrational number. Look for cancellation.
EZT practice question
Let a = 4 + √7 and b = 4 − √7. Classify a and b, then their sum and product.
Show the worked solution
- Both a and b are irrational: a rational value would make √7 rational after subtracting 4.
- a + b = 8 because the root terms cancel.
- ab = 4² − (√7)² = 16 − 7 = 9.
Answer: a and b are irrational; their sum 8 and product 9 are rational.
Check: (4 + √7)(4 − √7) expands to 16 − 4√7 + 4√7 − 7.
Watch for this: Irrational + irrational is not automatically irrational.
Now try it yourself
Repeat for c = 6 + √11 and d = 6 − √11.
Check your answer
Both are irrational; c + d = 12 and cd = 25.
The opposite root terms cancel in the sum and the difference-of-squares product.
Define the infinite pattern, not just its start
Choose the method: A decimal is rational exactly when its digits eventually repeat a fixed block.
EZT practice question
A decimal is defined as 0.101001000100001…: successive 1s have one, two, three, four, and then ever more zeros between them. Explain why it is irrational.
Show the worked solution
- There are infinitely many 1s, so the decimal does not end in zeros.
- In a repeating tail containing a 1, the gaps between 1s are bounded by a fixed multiple of the repeat length.
- Here the gaps grow without bound. No fixed block can repeat eventually.
Answer: It is non-terminating and not eventually periodic, so it is irrational.
Check: The argument uses the rule for every later digit, not a finite calculator display.
Watch for this: Writing a few digits followed by dots without a rule is not an irrationality proof.
Now try it yourself
A second decimal has a 2 followed successively by two, four, six, eight, and then increasing even numbers of zeros. Is it rational?
Check your answer
No; it is irrational.
The nonzero digit occurs infinitely often with unbounded gaps, so an eventually repeating tail is impossible.
Build the root length with a right triangle
Choose the method: A right triangle with legs u and v has hypotenuse √(u² + v²). Transfer the hypotenuse length from zero.
EZT practice question
Describe a number-line construction for √13 using legs 3 and 2.
Show the worked solution
- Mark O = 0 and A = 3 on the number line. At A draw a perpendicular segment AB of length 2.
- Join O to B. Pythagoras gives OB² = 3² + 2² = 13.
- With centre O and radius OB, mark the intersection on the positive number line.
Answer: The positive mark has coordinate √13, between 3 and 4.
Check: 9 < 13 < 16, which checks the interval without claiming a ruler drawing is exact.
Watch for this: Transfer the hypotenuse, not the length-2 perpendicular.
Now try it yourself
Which root length comes from perpendicular legs 4 and 1, and between which integers is it?
Check your answer
√17, between 4 and 5.
4² + 1² = 17 and 16 < 17 < 25.
Choose a conjugate to remove a root
Choose the method: For nonzero denominators, multiply numerator and denominator by the same expression. A conjugate creates a difference of squares.
EZT practice question
Rationalise 2/(√11 − √3).
Show the worked solution
- Multiply by (√11 + √3)/(√11 + √3).
- The denominator becomes 11 − 3 = 8. Cancel the factor 2.
Answer: (√11 + √3)/4
Check: Multiplying this answer by √11 − √3 gives (11 − 3)/4 = 2.
Watch for this: Use the opposite sign in the conjugate and multiply the whole numerator.
Now try it yourself
Rationalise 3/(√14 + √5).
Check your answer
(√14 − √5)/3
The conjugate gives denominator 14 − 5 = 9; cancel 3/9.
State the base restriction before combining powers
Choose the method: Use positive real bases for the fractional-power rules here; this avoids undefined real roots and sign traps.
EZT practice question
Simplify 16^(3/4) × 16^(1/4) ÷ 16^(1/2).
Show the worked solution
- The base 16 is positive, so add and subtract the exponents.
- 3/4 + 1/4 − 1/2 = 1/2.
Answer: 4
Check: 16^(3/4) = 8, 16^(1/4) = 2 and 16^(1/2) = 4; 8 × 2 ÷ 4 = 4.
Watch for this: These statements are not permission to use every fractional-power identity for negative bases.
Now try it yourself
Simplify 81^(1/2) ÷ 81^(1/4).
Check your answer
3
The exponent is 1/2 − 1/4 = 1/4, and the positive fourth root of 81 is 3.
Try a mixed set
Choose a method or explanation before opening each answer.
1. A student sees 0.428571 on a screen. Does this display alone prove the underlying number is rational?
Show answer and reason
No.
It could be an exact terminating value or a rounded beginning of many different numbers; the definition matters.
2. Compare 0.9 recurring with 1. Is there a positive gap?
Show answer and reason
They are equal; there is no gap.
For x = 0.999… , 10x − x = 9, so 9x = 9 and x = 1.
3. Find a rational number between √6 and √7 without decimal approximations.
Show answer and reason
5/2
6 < 25/4 < 7, so √6 < 5/2 < √7.
4. Is √48 − 4√3 rational?
Show answer and reason
Yes, it is zero.
√48 = √(16 × 3) = 4√3, so the terms cancel.
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Questions about this practice page
- Is this the current Class 9 Chapter 1?
- No. This page uses the legacy Number Systems chapter title. The current linked NCERT Chapter 1 is about coordinates. Match topics to your own book.
- Does a pattern make a decimal rational?
- Only an eventually repeating fixed block guarantees rationality. A rule can generate a non-repeating decimal too.
- Are square roots always irrational?
- No. √25 = 5. A positive integer has a rational square root precisely when it is a perfect square.
- Is 22/7 exactly π?
- No. It is a rational approximation to π, which is irrational. Use the value specified in a problem when calculating.
- Are these copied textbook answers?
- No. These are original EZT questions and explanations, not a complete exercise key or an official syllabus.