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Class 9 Maths Ganita Manjari The World of Numbers Question Bank with Answers | MCQs, Exemplar, Olympiad & HOTS

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Class 9 Mathematics • Ganita Manjari • Chapter 3

The World of Numbers — Advanced Question Bank

A relatively difficult source-aligned question bank covering natural numbers, Śhūnya and integers, rational numbers, density, irrational numbers, proof by contradiction, construction of √n, real numbers, terminating/repeating decimals and cyclic numbers.

30 MCQs15 One-Mark15 Two-Mark15 Three-Mark10 Four-Mark5 Case Studies12 Special Exemplar12 Special Olympiad/HOTS

High-Value Revision

Number Sets

N ⊂ Z ⊂ Q ⊂ R. Irrational numbers are real but not rational.

Rational Numbers

p/q with integers p,q and q≠0. They are dense on the number line.

Decimal Test

After reduction, a rational decimal terminates iff the denominator has only factors 2 and/or 5.

Irrationals

Non-terminating, non-repeating decimals such as √2 and π.

Distance on number line: |a−b|
A rational between a and b: (a+b)/2
If q=2ᵃ5ᵇ in lowest terms, decimal length ≤ max(a,b)
Difficulty policy: Main questions are deliberately above routine recall. The special Exemplar and Olympiad/HOTS sections are newly written at those levels, not claimed as verbatim external questions.

Visual Learning — HTML Canvas

Only the four diagrams use JavaScript; answer accordions use native HTML.

Real number line with rational and irrational points.
Density through repeated averaging.
Constructing √2 from a unit right triangle.
Hierarchy of number sets.

Section A — 30 MCQs

Conceptual, numerical and reasoning-based.

1Core

The set N = {1,2,3,…} represents:

  1. A. Integers
  2. B. Natural numbers
  3. C. Rational numbers
  4. D. Real numbers
View answer / solution

Answer: B
The chapter uses N for the counting numbers {1,2,3,…}.

2Core

Natural numbers are closed under:

  1. A. subtraction
  2. B. addition
  3. C. division
  4. D. taking negatives
View answer / solution

Answer: B
The sum of two natural numbers is always a natural number.

3Source-based

Which agrees with Brahmagupta’s rules for zero?

  1. A. a+0=0
  2. B. a−0=0
  3. C. a×0=0
  4. D. a/0=a
View answer / solution

Answer: C
Multiplication by zero gives zero.

4Application

A trader has debt ₹850, profit ₹1200, then loss ₹450. Final standing:

  1. A. −₹100
  2. B. +₹100
  3. C. +₹800
  4. D. −₹500
View answer / solution

Answer: A
−850+1200−450=−100.

5Core

Which is rational?

  1. A. √2
  2. B. π
  3. C. 0.272727…
  4. D. √10
View answer / solution

Answer: C
Every repeating decimal is rational.

6Core

A rational number p/q requires:

  1. A. q=0
  2. B. q≠0
  3. C. p>q
  4. D. p,q positive
View answer / solution

Answer: B
Division by zero is undefined.

7Core

Which pair is equal?

  1. A. 2/3,3/4
  2. B. −3/5,−6/10
  3. C. 5/8,10/15
  4. D. 7/9,14/27
View answer / solution

Answer: B
Both simplify to −3/5.

8Exemplar-level

Distance between −7/4 and 5/4 is:

  1. A. 1
  2. B. 2
  3. C. 3
  4. D. 12
View answer / solution

Answer: C
|−7/4−5/4|=3.

9Core

A rational strictly between 3/5 and 4/5 is:

  1. A. 1/2
  2. B. 7/10
  3. C. 9/10
  4. D. 1
View answer / solution

Answer: B
0.7 lies between 0.6 and 0.8.

10Concept

For distinct rationals a<b, (a+b)/2 is:

  1. A. outside the interval
  2. B. irrational
  3. C. strictly between them
  4. D. equal to a
View answer / solution

Answer: C
The average lies between the endpoints.

11Core

Which is irrational?

  1. A. √81
  2. B. √12
  3. C. 0.333…
  4. D. 0.125
View answer / solution

Answer: B
√12=2√3 is irrational.

12Source-based

The chapter’s proof that √2 is irrational uses:

  1. A. induction
  2. B. contradiction
  3. C. coordinate geometry
  4. D. approximation
View answer / solution

Answer: B
Assuming √2=p/q in lowest terms forces both p and q even.

13Source-based

In that proof, after p=2k, one obtains:

  1. A. q²=4k²
  2. B. q²=2k²
  3. C. 2q=k
  4. D. q=2k²
View answer / solution

Answer: B
From 2q²=4k², divide by 2.

14Concept

Which decimal is irrational?

  1. A. 0.125
  2. B. 0.454545…
  3. C. 1.01001000100001…
  4. D. 3.141414…
View answer / solution

Answer: C
It never terminates and has no fixed repeating block.

15Exemplar-level

If a reduced denominator is 2³×5², the decimal terminates in at most:

  1. A. 2 places
  2. B. 3 places
  3. C. 5 places
  4. D. 6 places
View answer / solution

Answer: B
The larger exponent is 3.

16Core

18/125 is:

  1. A. terminating
  2. B. non-terminating repeating
  3. C. irrational
  4. D. non-repeating
View answer / solution

Answer: A
125=5³.

17Core

Which reduced denominator gives a repeating decimal?

  1. A. 40
  2. B. 125
  3. C. 72
  4. D. 250
View answer / solution

Answer: C
72 contains prime factor 3.

18Source-based

0.999… equals:

  1. A. 0.9
  2. B. a number below 1
  3. C. 1
  4. D. irrational
View answer / solution

Answer: C
Algebraically, if x=0.999…, then 9x=9.

19Source-based

The repeating block of 1/7 is:

  1. A. 123456
  2. B. 142857
  3. C. 285714
  4. D. 076923
View answer / solution

Answer: B
1/7=0.142857142857….

20Source-based

142857×5 equals:

  1. A. 714285
  2. B. 857142
  3. C. 571428
  4. D. 428571
View answer / solution

Answer: A
The digits rotate cyclically.

21Core

Which inclusion is correct?

  1. A. Q⊂Z
  2. B. I⊂Q
  3. C. N⊂Z⊂Q⊂R
  4. D. R⊂Q
View answer / solution

Answer: C
Natural ⊂ integers ⊂ rationals ⊂ reals.

22Core

Q together with irrational numbers forms:

  1. A. Z
  2. B. N
  3. C. R
  4. D. imaginary numbers
View answer / solution

Answer: C
Together they form the real numbers.

23Core

Which is not real?

  1. A. −5
  2. B. √3
  3. C. π
  4. D. √(−1)
View answer / solution

Answer: D
No real number has square −1.

24Olympiad

If a,b are non-zero rationals and a+1/b=0, then ab=

  1. A. 1
  2. B. −1
  3. C. 0
  4. D. cannot tell
View answer / solution

Answer: B
a=−1/b, so ab=−1.

25Olympiad

If x+y+z=0 and xy+yz+zx=0, then x²+y²+z²=

  1. A. 0
  2. B. 1
  3. C. xyz
  4. D. unknown
View answer / solution

Answer: A
Square x+y+z and use the second condition.

26Core

A guaranteed rational between a<b is:

  1. A. ab
  2. B. a+b
  3. C. (a+b)/2
  4. D. a−b
View answer / solution

Answer: C
The average is rational and lies between.

27Core

0.35 in lowest fractional form is:

  1. A. 35/10
  2. B. 7/20
  3. C. 3/5
  4. D. 35/99
View answer / solution

Answer: B
35/100 reduces to 7/20.

28Core

0.666… equals:

  1. A. 1/6
  2. B. 2/3
  3. C. 6/10
  4. D. 3/5
View answer / solution

Answer: B
10x−x=6 gives x=2/3.

29Core

A unit square has diagonal:

  1. A. 1
  2. B. 2
  3. C. √2
  4. D. π
View answer / solution

Answer: C
d²=1²+1².

30Exemplar-level

If a decimal ends with a non-zero fourth decimal digit, writing it as p/10⁴ gives p:

  1. A. divisible by 10
  2. B. not divisible by 10
  3. C. prime
  4. D. even
View answer / solution

Answer: B
Otherwise the last digit would not really be at the fourth place.

Section B — 15 One-Mark Questions

1Core

Write the integers from −3 to 3.

View answer / solution

{−3,−2,−1,0,1,2,3}.

2Core

Evaluate 0−(−14).

View answer / solution

14.

3Core

Evaluate (−20)÷4.

View answer / solution

−5.

4Core

Define a rational number.

View answer / solution

A number p/q with integers p,q and q≠0.

5Core

Give one rational between 1/2 and 2/3.

View answer / solution

7/12, for example.

6Core

Find |−11/6|.

View answer / solution

11/6.

7Exemplar-level

Find the distance between −2/3 and 5/6.

View answer / solution

3/2.

8Core

Classify √49.

View answer / solution

Rational; it equals 7.

9Core

Classify √7.

View answer / solution

Irrational.

10Core

State the decimal signature of an irrational number.

View answer / solution

Non-terminating and non-repeating.

11Core

Classify 13/250 without division.

View answer / solution

Terminating; 250=2×5³.

12Core

Convert 0.125 to lowest terms.

View answer / solution

1/8.

13Source-based

Write the repeating block of 1/7.

View answer / solution

142857.

14Core

Name Q together with all irrational numbers.

View answer / solution

Real numbers R.

15Concept

Does √(−1) have a real value?

View answer / solution

No.

Section C — 15 Two-Mark Questions

1Core

Show that −3/5 and −6/10 are equal.

View answer / solution

Cross-products are both 30, or −6/10 simplifies to −3/5.

2Core

Find 7/12+5/8.

View answer / solution

29/24.

3Core

Find −7/9−(−2/3).

View answer / solution

−1/9.

4Core

Find (−4/7)÷(5/14).

View answer / solution

−8/5.

5Exemplar-level

Give three rationals between −1/2 and 1/4.

View answer / solution

For example −1/4, 0, 1/8.

6Source-based

A tailor has 15 3/4 m silk; each kurta uses 2 1/4 m. How many?

View answer / solution

(63/4)÷(9/4)=7.

7Core

Classify 7/20 and state decimal places.

View answer / solution

Terminating in 2 places, because 20=2²×5.

8Core

Classify 4/15 without division.

View answer / solution

Non-terminating repeating; denominator contains 3.

9Source-based

Convert 0.454545… to p/q.

View answer / solution

Let x=0.4545…; 100x−x=45, so x=5/11.

10Source-based

Convert 0.1666… to p/q.

View answer / solution

10x=1.666…,100x=16.666…; subtract:90x=15, x=1/6.

11Source-inspired

Give two rationals between 3.1415 and 3.1416.

View answer / solution

For example 3.14151 and 3.14155.

12Concept

Why do 1/2,2/4,50/100 mark one point?

View answer / solution

They are equivalent fractions, all equal to 1/2.

13Exemplar-level

If a<b, prove (a+b)/2>a.

View answer / solution

Difference=(b−a)/2>0.

14Exemplar-level

If a<b, prove (a+b)/2<b.

View answer / solution

Difference b−(a+b)/2=(b−a)/2>0.

15Exemplar-level

A reduced denominator is 2³×5. How many decimal places at most?

View answer / solution

3 places, after supplying two 5s to make 10³.

Section D — 15 Three-Mark Questions

1Core

Verify distributivity for 2/3, 3/4 and −1/2.

View answer / solution

LHS=(2/3)(1/4)=1/6; RHS=1/2−1/3=1/6.

2Source-extension

Solve (5/6)(x+3/5)=(5/6)x+1/2.

View answer / solution

Both sides simplify identically, so every rational x satisfies it.

3Core

Locate 11/4 conceptually on the number line.

View answer / solution

11/4=2 3/4: divide [2,3] into four equal parts and take the third point after 2.

4Exemplar-level

Prove a rational exists between rationals a<b.

View answer / solution

m=(a+b)/2 is rational; m−a=(b−a)/2>0 and b−m=(b−a)/2>0.

5Exemplar-level

Prove √3 irrational by contradiction.

View answer / solution

Assume √3=p/q in lowest terms. p²=3q² implies 3|p; put p=3k. Then q²=3k², so 3|q, contradiction.

6Concept

Why is √12 irrational?

View answer / solution

√12=2√3; non-zero rational 2 times irrational √3 is irrational.

7Source-extension

Determine exact decimal type of 18/125.

View answer / solution

18/125=144/1000=0.144, terminating in 3 places.

8Source-based

A reduced denominator is 2³×5. Explain decimal length.

View answer / solution

Multiply by 5² to make 10³; it terminates within 3 places.

9Source-based

Convert 2.357777… to p/q.

View answer / solution

100x=235.777…,1000x=2357.777…; 900x=2122, x=1061/450.

10Source-based

Convert 2.45373737… to p/q.

View answer / solution

100x=245.3737…,10000x=24537.3737…; 9900x=24292, x=6073/2475.

11Source-based

Prove 0.999…=1 algebraically.

View answer / solution

x=0.999…;10x=9.999…; subtract to get 9x=9, so x=1.

12Exemplar-level

Explain a four-place terminating decimal as p/10⁴.

View answer / solution

Shift four places: x=p/10000. If fourth digit is non-zero, p is not divisible by 10; reduction may cancel 2s or 5s.

13Source-extension

If a+1/b=0, find ab.

View answer / solution

a=−1/b, so ab=−1.

14Core

Find six rationals between 3 and 4 using one denominator.

View answer / solution

3=21/7,4=28/7; use 22/7,23/7,24/7,25/7,26/7,27/7.

15Core

Find five rationals between 2/5 and 3/5.

View answer / solution

12/30 and 18/30 bound 13/30,14/30,15/30,16/30,17/30.

Section E — 10 Four-Mark Questions

1Source-based Exemplar

Prove √5 irrational by contradiction.

View answer / solution

Assume √5=p/q lowest terms. p²=5q²⇒5|p; p=5k. Then q²=5k²⇒5|q, contradicting coprimality.

2Source-Olympiad

For a=7/12,b=5/6, write exactly five rationals between them with one denominator.

View answer / solution

Use denominator 72: 42/72<43/72<44/72<45/72<46/72<47/72<60/72.

3Source-Olympiad

If x+y+z=0 and xy+yz+zx=0, show x=y=z=0.

View answer / solution

Square the first equation: x²+y²+z²+2(xy+yz+zx)=0, so x²+y²+z²=0. Each square is non-negative, hence each is zero.

4Source-Olympiad

Show (a+b)/2 lies between a<b and connect this to density.

View answer / solution

Both gaps equal positive multiples of b−a. Repeating averaging in smaller subintervals generates endlessly many rational points.

5Exemplar-level

Convert 0.1234512345… to a fraction.

View answer / solution

Let x repeat block 12345. 100000x−x=12345, so x=12345/99999=4115/33333.

6Exemplar-level

Classify 1.01001000100001… and justify.

View answer / solution

Irrational: non-terminating and gaps of zeros keep changing, so no fixed block repeats.

7Exemplar-level

A reduced denominator is 2⁴×5³. Determine maximum decimal places.

View answer / solution

Multiply by 5 to obtain 10⁴, so at most 4 places.

8Source-based

Explain construction of √5 using the square-root spiral.

View answer / solution

Build unit right triangles successively: √2, then √3, then √4, then √5, each new triangle having one leg 1 and the previous hypotenuse as the other leg.

9Source-extension

Explain why 2.47=2.46999….

View answer / solution

Because 0.999…=1, 0.00999…=0.01; hence 2.46+0.00999…=2.47.

10Source-based

List the cyclic products of 142857 by 2 to 6.

View answer / solution

285714, 428571, 571428, 714285, 857142. Each is a cyclic rotation of the same digits.

Section F — 5 Case Studies

Case 1Source-inspired Application

Ladakh Temperature and Signed Numbers

At noon temperature is 4°C; by midnight it drops 15°C; next morning it rises 7°C and then drops 3°C.

  1. Midnight?
  2. After rise?
  3. Final?
  4. Essential number set?
View case-study solutions

Midnight? −11°C.

After rise? −4°C.

Final? −7°C.

Essential number set? Integers.

Case 2Exemplar Number Line

Rational Points on a Number Line

A=−5/4 and B=11/4.

  1. Where is A?
  2. Where is B?
  3. Distance AB?
  4. Midpoint?
View case-study solutions

Where is A? Between −2 and −1.

Where is B? Between 2 and 3.

Distance AB? 4 units.

Midpoint? 3/4.

Case 3Competency / Exemplar

Decimal Detective

A=7/20, B=4/15, C=13/250, D=18/125, all reduced.

  1. Which terminate?
  2. Which repeats?
  3. Decimal places for D?
  4. Denominator test?
View case-study solutions

Which terminate? A,C,D.

Which repeats? B.

Decimal places for D? 3.

Denominator test? Only factors 2 and/or 5.

Case 4Proof-based Exemplar

The Irrational Diagonal

A square has side 1 and diagonal d.

  1. Find d.
  2. Rational?
  3. Contradiction in p/q proof?
  4. Why contradiction?
View case-study solutions

Find d. √2.

Rational? No.

Contradiction in p/q proof? Both p and q become even.

Why contradiction? p/q was assumed lowest terms.

Case 5Special Olympiad

Cyclic Number 142857

The decimal expansion of 1/7 repeats 142857.

  1. ×2?
  2. ×4?
  3. Visible property?
  4. Value of 0.142857…?
View case-study solutions

×2? 285714.

×4? 571428.

Visible property? Cyclic rotation.

Value of 0.142857…? 1/7.

Special Exemplar Challenge — 12 Questions

Non-routine exact reasoning and representation.

1Special Exemplar

Without division, classify 77/600.

View worked solution

Repeating: 600=2³×3×5² and the factor 3 remains after reduction.

2Special Exemplar

A reduced denominator is 2⁵×5². Maximum decimal places?

View worked solution

5.

3Special Exemplar

Convert 0.272727… to p/q.

View worked solution

3/11.

4Special Exemplar

Convert 1.23333… to p/q.

View worked solution

37/30.

5Special Exemplar

Find four rationals between 5/7 and 6/7.

View worked solution

26/35,27/35,28/35,29/35.

6Special Exemplar

If a<b, show (2a+b)/3 lies between.

View worked solution

Subtract each endpoint: (b−a)/3 and 2(b−a)/3 are positive.

7Special Exemplar

Classify 0.101001000100001….

View worked solution

Irrational: no fixed repeating block.

8Special Exemplar

Prove √10 irrational.

View worked solution

Assume p/q lowest terms. p²=10q² forces both 2 and 5 to divide p, then also q, contradiction.

9Special Exemplar

Find exact fraction for 2.46999….

View worked solution

2.47=247/100.

10Special Exemplar

A rational decimal ends at the 6th place. What about its reduced denominator?

View worked solution

Only factors 2 and 5, with needed exponent no larger than 6; it need not equal 10⁶.

11Special Exemplar

If x=0.123123…, find 7x.

View worked solution

x=41/333, so 7x=287/333.

12Special Exemplar

Show √18 irrational.

View worked solution

√18=3√2 and non-zero rational times irrational is irrational.

Special Olympiad / HOTS Challenge — 12 Questions

Advanced reasoning connected to the chapter.

1Special Olympiad / HOTS

Classify x=0.12345678910111213… formed by concatenating integers.

View worked solution

Irrational: it does not terminate and its digit structure cannot eventually repeat a fixed finite block.

2Special Olympiad / HOTS

Find a rational between √2 and 3/2 without decimals.

View worked solution

17/12 works because (17/12)²=289/144>2, while 17/12<18/12=3/2.

3Special Olympiad / HOTS

If x+y+z=0 and x²+y²+z²=0, prove all zero.

View worked solution

Squares are non-negative; their sum is zero only when each is zero.

4Special Olympiad / HOTS

If r is rational, prove √2+r is irrational.

View worked solution

Otherwise subtract rational r from a rational sum to make √2 rational, contradiction.

5Special Olympiad / HOTS

Prove non-zero rational × irrational is irrational.

View worked solution

If rα were rational with r≠0 rational, then α=(rα)/r would be rational.

6Special Olympiad / HOTS

Smallest n such that 7/2ⁿ has exactly 6 decimal places?

View worked solution

n=6, since 7/2ⁿ=7·5ⁿ/10ⁿ and numerator is not divisible by 10.

7Special Olympiad / HOTS

For q=2ᵃ5ᵇ, prove termination within max(a,b) places.

View worked solution

Let m=max(a,b); multiply to turn denominator into 2^m5^m=10^m.

8Special Olympiad / HOTS

Construct n rationals between a<b with one formula.

View worked solution

r_k=((n+1−k)a+kb)/(n+1), k=1,…,n.

9Special Olympiad / HOTS

What must the second decimal representation of a terminating rational look like?

View worked solution

It eventually consists of repeating 9s.

10Special Olympiad / HOTS

Prove √2+√3 irrational.

View worked solution

If rational r, squaring gives √6=(r²−5)/2 rational, contradiction.

11Special Olympiad / HOTS

Prove distance between rational a,b is rational.

View worked solution

|a−b| is rational because Q is closed under subtraction and sign change.

12Special Olympiad / HOTS

Why do products 142857×1,…,×6 keep the same digit cycle?

View worked solution

They correspond to k/7; long division cycles through the same nonzero remainders modulo 7 in shifted order.

Exam Strategy & Common Traps

TopicBest first stepCommon trap
Decimal classificationReduce the fraction first, then factor the denominator.Applying the 2-and-5 rule before reducing.
DensityUse averages or enlarge the common denominator.Believing a rational has a “next” rational.
Irrationality proofAssume p/q in lowest terms and derive a common factor.Not stating p,q are coprime.
Repeating decimalShift by powers of 10 until repeating tails align.Subtracting misaligned tails.
Number lineLocate the interval first.Ignoring signs and improper fractions.
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