Unit 1: Numbers & Quantification - Free Study Resources | Boundless Maths

📊 Unit 1: Numbers & Quantification

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Topics Covered in Unit 1

Master these 6 Important topics

1. Modulo Arithmetic

Remainder operations, clock arithmetic, divisibility rules, congruence properties

2. Congruence Modulo

Modular equations, linear congruences, solving systems of congruences

3. Alligation & Mixture

Weighted averages, mixture problems, price calculations, dilution

4. Boats & Streams

Upstream/downstream speed, relative speed, time-distance river problems

5. Pipes & Cisterns

Filling/emptying rates, combined work, time to fill/empty tanks

6. Races & Games

Head start, distance/time advantage, speed ratios in competitive scenarios

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Numbers & Quantification
11 Marks · One-Shot Video

Numerical Inequalities — Complete One-Shot

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Modulo Arithmetic Boats & Streams Alligation & Mixtures Pipes & Cisterns Races & Games
MCQs (15) Short Answer Questions (8) Case Studies (2)
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Practice MCQs with Answers

Click "Show Answer" to reveal explanations

Question 1
56 (mod 7) is
  • A5
  • B2
  • C1
  • D4
✓ Correct Answer: C (1)

Explanation: 56 ≡ (53)2 ≡ (−1)2 ≡ 1 (mod 7)

Question 2
(8 × 14) in 12 hours clock is:
  • A4 O'clock
  • B8 O'clock
  • C6 O'clock
  • D2 O'clock
✓ Correct Answer: A (4 O'clock)

Explanation: 8 × 14 = 112 (mod 12) = 4

Question 3
In a kilometre race, A, B and C are three participants. A can give B a start of 50 m and C a start of 69 m. In the same race, the start which B can allow for C is
  • A17 m
  • B18 m
  • C19 m
  • D20 m
✓ Correct Answer: D (20 m)

Explanation:
A : B : C = 1000 : 950 : 931
If B covers 1000 m, then C covers = 931/950 × 1000 = 980 m
B can allow a start of (1000 – 980)m = 20 m to C

Question 4
At a game of billiards, A can give 15 points to B in 60 and A can give 20 points to C in 60. How many points can B give to C in a game of 90?
  • A20 points
  • B10 points
  • C12 points
  • D18 points
✓ Correct Answer: B (10 points)

Explanation:
A : B = 60 : 45 = 4 : 3
A : C = 60 : 40 = 3 : 2
Therefore, B : C = 3 : 2
In a game of 90 points, B gives C = 90 × (3-2)/3 = 10 points

Question 5CBSE 2022
If 100 ≡ k (mod 7), then the least positive value of k is
  • A2
  • B3
  • C6
  • D4
✓ Correct Answer: A (2)

Explanation:
100 = 7 × 14 + 2
So, 100 ≡ 2 (mod 7), therefore k = 2

Question 6CBSE 2022
20 litres of a mixture contains milk and water in the ratio 3:1. The amount of milk, in litres, to be added to the mixture so as to have milk and water in the ratio 4:1 is
  • A7 litres
  • B4 litres
  • C5 litres
  • D6 litres
✓ Correct Answer: C (5 litres)

Explanation:
Initial: Milk = 15 L, Water = 5 L. Let x litres of milk be added.
(15 + x)/5 = 4/1 → x = 5 litres

Question 7CBSE 2022
Pipes A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the time taken to fill the tank is
  • A2 hours
  • B3⅗ hours
  • C3 hours
  • D3⅓ hours
✓ Correct Answer: D (3⅓ hours)

Explanation:
Net rate = 1/5 + 1/6 − 1/12 = 12/60 + 10/60 − 5/60 = 17/60
Time = 60/17 ≈ 3⅓ hours

Question 8CUET 2022
The number at unit place of number 17123 is
  • A1
  • B3
  • C7
  • D9
✓ Correct Answer: B (3)

Explanation:
Pattern of unit digits for powers of 7: 7, 9, 3, 1 (cycle of 4).
123 ÷ 4 = 30 remainder 3 → unit digit of 73 = 3

Question 9CUET 2022
Match list I with list II:

List I
A. 7b ≡ b (mod 9)   B. 2b ≡ b (mod 15)
C. 4b ≡ b (mod 10)   D. 8b ≡ b (mod 12)

List II  I. b = 4   II. b = 6   III. b = 2   IV. b = 5
  • AA–IV, B–III, C–II, D–I
  • BA–II, B–III, C–I, D–IV
  • CA–I, B–II, C–III, D–IV
  • DA–III, B–I, C–IV, D–II
✓ Correct Answer: B (A–II, B–III, C–I, D–IV)

Explanation:
A. 76 ≡ 6 (mod 9) → b=6 (II)   B. 22=4 → b=2 (III)
C. 44≡4 (mod 10) → b=4 (I)   D. 85≡5 (mod 12) → b=5 (IV)

Question 10CUET 2022
A mixture contains milk and water in the ratio 8:x. If 3 litres of water is added in 33 litres of mixture, the ratio of milk and water becomes 2:1, then the value of x is
  • A3 L
  • B4 L
  • C2 L
  • D11 L
✓ Correct Answer: A (3 L)

Explanation:
Milk = 8/(8+x) × 33. After adding 3L water, Milk : (Water+3) = 2:1. Solving gives x = 3.

Question 11
If |x| < 3, then
  • A3 < x < −3
  • B−3 < x < 3
  • C0 ≤ x < 3
  • Dx > 3
✓ Correct Answer: B (−3 < x < 3)

Explanation: |x| < 3 means the distance of x from 0 is less than 3, so −3 < x < 3.

Question 12
If x is a real number such that (x + 2)/(−7) > 0, then
  • Ax ∈ (−∞, −5)
  • Bx ∈ (5, ∞)
  • Cx ∈ (−5, ∞)
  • Dx ∈ (−∞, −2)
✓ Correct Answer: D (x ∈ (−∞, −2))

Explanation:
Denominator = −7 (negative). For fraction to be positive, numerator must also be negative: x + 2 < 0, so x < −2 → x ∈ (−∞, −2)

Question 13CBSE 2022
The solution of (3x − 1)/(2x + 3) ≥ 0 is
  • A(−∞, −3/2) ∪ [1/3, ∞)
  • B(−∞, −3/2)
  • C(−3/2, 1/3)
  • DNo solution
✓ Correct Answer: A ((−∞, −3/2) ∪ [1/3, ∞))

Explanation:
Critical points: x = 1/3 and x = −3/2.
Sign analysis: x < −3/2 → (+) ✔ | −3/2 < x < 1/3 → (−) ✘ | x > 1/3 → (+) ✔
Solution: (−∞, −3/2) ∪ [1/3, ∞)

Question 14CBSE 2022
The solution set of the inequality |x − 5| < 3 is
  • A(−∞, 2)
  • B(2, 8)
  • C(8, ∞)
  • D(−∞, 8)
✓ Correct Answer: B ((2, 8))

Explanation:
|x − 5| < 3 → −3 < x − 5 < 3 → 2 < x < 8 → Solution: (2, 8)

Question 15CUET 2022
The system of linear inequalities 3x + 2y ≥ 24 and x + y ≤ 10 has solution
  • A(2, ∞)
  • B(2, 8)
  • C(8, ∞)
  • D(−∞, 8)
✓ Correct Answer: C ((8, ∞))

Explanation:
Solving the system of inequalities by graphical/algebraic method, the feasible region gives x ∈ (8, ∞).

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Short Answer Questions with Step-by-Step Solutions

Practice 2-mark and 3-mark questions with detailed solutions

Question 1 CBSE 2024 Comptt
Evaluate (137 + 995) (mod 12).
Solution:
137 ÷ 12 = 11 remainder 5, so 137 ≡ 5 (mod 12)
995 ÷ 12 = 82 remainder 11, so 995 ≡ 11 (mod 12)
= 5 + 11 = 16 (mod 12) = 4
Final Answer: 4
Question 2 CBSE 2024 Comptt
Find the unit's digit of 1212.
Solution:
12 ≡ 2 (mod 10)
26 = 64 ≡ 4 (mod 10), so 126 ≡ 4 (mod 10)
1212 = (126)2 ≡ 42 = 16 ≡ 6 (mod 10)
Final Answer: Unit's digit is 6
Question 3 CBSE 2023
A bottle is full of dettol. One-third of its dettol is taken away and an equal amount of water is poured into the bottle. This operation is repeated three times. Find the final ratio of dettol to water in the bottle.
Solution:
After each operation, dettol remaining = previous × (2/3)
After 3 operations: dettol = 1 × (2/3)3 = 8/27
Water = 1 − 8/27 = 19/27
Final Answer: Dettol : Water = 8 : 19
Question 4 CBSE 2023, 2024
A container has 50L of juice. 5L of juice is taken out and replaced by 5L of water. This process is repeated a total of 5 times. What is the amount of juice remaining?
Solution:
Formula: Final quantity = Initial × (1 − removed/total)n
= 50 × (45/50)5 = 50 × (0.9)5
= 50 × 0.59049 = 29.5245 L
Final Answer: ≈ 29.5 litres of juice remains
Question 5 CBSE 2023 Comptt
A person can row a boat 5 km an hour in still water. It takes him thrice as long to row upstream as to row downstream. Find the rate at which the stream is flowing.
Solution:
Let speed of stream = x km/h. Upstream = (5−x), Downstream = (5+x)
Given: d/(5−x) = 3 × d/(5+x)
5 + x = 3(5 − x) → 5 + x = 15 − 3x → 4x = 10 → x = 2.5
Final Answer: Stream is flowing at 2.5 km/h
Question 6
Find the remainder when 1723 is divided by 5.
Solution:
17 ≡ 2 (mod 5), so 1723 ≡ 223 (mod 5)
Cycle of 2 mod 5: 2,4,3,1 (period 4). 23 = 4×5 + 3
223 ≡ 23 = 8 ≡ 3 (mod 5)
Final Answer: The remainder is 3
Question 7
A container contains 60 litres of milk. From this container, 6 litres of milk is taken out and replaced with water. This process is repeated two more times. How much milk remains?
Solution:
Final quantity = 60 × (54/60)3 = 60 × (0.9)3
= 60 × 0.729 = 43.74 litres
Final Answer: 43.74 litres of milk remains
Question 8
A man can row 18 km/h in still water. It takes him twice as long to row upstream as to row downstream. Find the rate of the stream.
Solution:
Let speed of stream = s km/h. d/(18−s) = 2 × d/(18+s)
18 + s = 2(18 − s) → 18 + s = 36 − 2s → 3s = 18
Final Answer: Speed of stream = 6 km/h

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Case Studies

Real-world application based questions

Case Study 1
E-Commerce Delivery System
An e-commerce company uses a fleet of delivery vehicles:

Type A: Travel at 40 km/h, complete 8 deliveries per trip, cost ₹500/trip
Type B: Travel at 50 km/h, complete 6 deliveries per trip, cost ₹600/trip

The company needs to complete 200 deliveries. The average distance per route is 30 km.
Question 1:
How many trips would be needed if only Type A vehicles are used?
✓ Answer

Solution: Type A = 8 deliveries/trip → 200 ÷ 8 = 25 trips

Question 2:
What would be the total cost if only Type B vehicles are used?
✓ Answer

Solution: Type B = 6 deliveries/trip → 200 ÷ 6 ≈ 34 trips (rounded up)
Total cost = 34 × ₹600 = ₹20,400

Question 3:
If time is more important than cost, which vehicle type should be prioritized?
✓ Answer

Solution:
Type A: 25 trips × 0.75 h = 18.75 h  |  Type B: 34 trips × 0.6 h = 20.4 h
Type A is faster overall (18.75 h vs 20.4 h).

Case Study 2
Water Tank Management System
A residential society has three water tanks:
Tank A: 5000 L   Tank B: 8000 L   Tank C: 6000 L

Pipe X: Fills at 250 L/h (cost ₹20/h)   Pipe Y: Fills at 400 L/h (cost ₹35/h)
Pipe Z (outlet): Empties at 150 L/h
Question 1:
If both inlet pipes work together to fill Tank B, how long will it take?
✓ Answer

Solution: Combined rate = 250 + 400 = 650 L/h
Time = 8000 ÷ 650 ≈ 12 hours 18 minutes

Question 2:
If Tank A is being filled by Pipe Y while outlet Pipe Z is accidentally left open, how long will it take?
✓ Answer

Solution: Net filling rate = 400 − 150 = 250 L/h
Time = 5000 ÷ 250 = 20 hours

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