Unit 1: Numbers & Quantification | Class 12 Applied Maths | Boundless Maths
Unit 1 of 8 11 Marks CBSE 2026–27 Highest-Weightage Unit

Numbers & Quantification
Unit 1 — Free Study Resources

CBSE Class 12 Applied Mathematics · Free MCQs, Solved Examples & Case Studies

Unit 1 carries 11 marks in the CBSE board exam — the highest-weightage unit. Everything you need to master it: 15 interactive MCQs with instant feedback, 8 step-by-step solved examples, and 2 board-pattern case studies. Covering Modulo Arithmetic, Congruence Modulo, Boats & Streams, Pipes & Cisterns, Alligation & Mixture, Races & Games, and Numerical Inequalities. All content is aligned to the CBSE 2026–27 syllabus and board exam pattern. Once you've attempted the questions and self-assessed your answers, tap ✨ My Report to see your personalised performance breakdown.

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Unit 1 · 11 Marks

Topics & Key Formulas

Six topics examined across MCQs, short answers, and case studies. Memorise the formulas below — they appear in multiple question types every year.

1. Modulo Arithmetic

Remainder operations, clock arithmetic, cyclic power problems

\(a \equiv r \pmod{n}\) means \(n \mid (a - r)\) \(a \bmod n =\) remainder when \(a \div n\)

2. Congruence Modulo

Unit digit cycles, large power problems, clock problems

Cycle of \(7^n \bmod 10\): 7, 9, 3, 1 (period 4) Position = exponent \(\bmod\) period

3. Alligation & Mixture

Weighted averages, dilution, repeated replacement problems

Alligation ratio: \(\dfrac{d - m}{m - c}\) (dearer : cheaper) Replacement: \(n\!\left(1 - \dfrac{x}{n}\right)^k\)

4. Boats & Streams

Upstream/downstream speed, time-distance river problems

\(\text{Down} = B + S \;\; \text{Up} = B - S\) \(B = \dfrac{D+U}{2},\quad S = \dfrac{D-U}{2}\)

5. Pipes & Cisterns

Filling/emptying rates, combined work, net rate calculations

Net rate \(= \sum \text{inlets} - \sum \text{outlets}\) Time \(= \dfrac{1}{\text{Net rate}}\)

6. Numerical Inequalities

Absolute value inequalities, sign analysis, solution intervals

\(|x| < a \;\Leftrightarrow\; -a < x < a\) Flip sign when dividing by \(-ve\)
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  • Unit 1 — Numbers & Quantification
  • Unit 2 — Algebra (Matrices)
  • Unit 3 — Calculus
  • Unit 4 — Probability Distributions
  • Unit 5 — Inferential Statistics
  • Unit 6 — Index Numbers & Time Data
  • Unit 7 — Financial Mathematics
  • Unit 8 — Linear Programming
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Interactive Practice

Practice MCQs — Unit 1

Select your answer, then click Show Answer to check and reveal the full explanation. All questions are based on CBSE past papers and board exam pattern.

Q1Modular Arithmetic
What does \(5^6 \pmod{7}\) equal?
5
2
1
4
Step 1 — Find \(5^3 \bmod 7\):
\(5^3 = 125\). \(125 = 7 \times 17 + 6\), so \(5^3 \equiv -1 \pmod{7}\).

Step 2 — Square the result:
\(5^6 = (5^3)^2 \equiv (-1)^2 = 1 \pmod{7}\)

This avoids computing \(5^6 = 15625\) directly — always reduce the base first.
Q2Modular Arithmetic
On a 12-hour clock, what is \((8 \times 14)\)?
4 O'clock
8 O'clock
6 O'clock
2 O'clock
Step 1 — Multiply: \(8 \times 14 = 112\).

Step 2 — Apply mod 12 (clock has 12 positions):
\(112 = 9 \times 12 + 4\), so \(112 \equiv 4 \pmod{12}\).

Result: 4 O'clock. On a 12-hour clock, any number reduces to its remainder when divided by 12.
Q3Races & Games
In a kilometre race, A gives B a start of 50 m and C a start of 69 m. What start can B give C?
17 m
18 m
19 m
20 m
A runs 1000 m → B runs 950 m → C runs 931 m.
Ratio A : B : C = 1000 : 950 : 931

When B runs 1000 m:
C runs \(= \dfrac{931}{950} \times 1000 = 980\) m.

Start B gives C: \(1000 - 980 = \mathbf{20}\) m.
Q4Races & Games
At billiards, A gives 15 points to B in 60 and 20 points to C in 60. How many points can B give C in a game of 90?
20 points
10 points
12 points
18 points
A:B = 60:45 = 4:3, A:C = 60:40 = 3:2
Common base: A:B = 12:9, A:C = 12:8 → B:C = 9:8

When B = 90: C \(= \dfrac{8}{9} \times 90 = 80\).
B gives C = \(90 - 80 = \mathbf{10}\) points.
Q5CBSE 2022Modular Arithmetic
If \(100 \equiv k \pmod{7}\), what is the least positive value of \(k\)?
2
3
6
4
\(100 = 7 \times 14 + 2\) → remainder = 2.
So \(100 \equiv 2 \pmod{7}\) and \(k = 2\).

Verify: \(100 - 2 = 98 = 7 \times 14\) ✓
Q6CBSE 2022Alligation & Mixture
20 litres of a mixture contains milk and water in the ratio 3:1. How much milk should be added so the ratio becomes 4:1?
7 litres
4 litres
5 litres
6 litres
Current: Milk = 15 L, Water = 5 L.
Let \(x\) litres milk added: \(\dfrac{15+x}{5} = 4 \Rightarrow x = 5\) litres.

Verify: 20 : 5 = 4:1 ✓
Q7CBSE 2022Pipes & Cisterns
Pipes A and B can fill a tank in 5 hours and 6 hours respectively, while pipe C can empty it in 12 hours. How long will it take to fill the tank if all three are opened together?
2 hours
3⅗ hours
3 hours
3⅓ hours
Net rate \(= \dfrac{1}{5} + \dfrac{1}{6} - \dfrac{1}{12} = \dfrac{12+10-5}{60} = \dfrac{17}{60}\) tank/hour

Time \(= \dfrac{60}{17} \approx 3\dfrac{9}{17} \approx 3\dfrac{1}{3}\) hours.
Q8CUET 2022Unit Digit Cycles
What is the digit at the unit place of \(17^{123}\)?
1
3
7
9
Unit digit of 17 = 7. Cycle of \(7^n \bmod 10\): 7, 9, 3, 1 (period 4).
\(123 \div 4 = 30\) remainder 3 → 3rd value = 3.

Unit digit of \(17^{123}\) = 3.
Q9CUET 2022Unit Digit Cycles
Match: A. \(7^b \equiv b \pmod{9}\)   B. \(2^b \equiv b \pmod{15}\)   C. \(4^b \equiv b \pmod{10}\)   D. \(8^b \equiv b \pmod{12}\)
List II: I. b=4   II. b=6   III. b=2   IV. b=5
A–IV, B–III, C–II, D–I
A–II, B–III, C–I, D–IV
A–I, B–II, C–III, D–IV
A–III, B–I, C–IV, D–II
Per CBSE official key: A–II (b=6), B–III (b=2), C–I (b=4), D–IV (b=5).

Verify C: \(4^4 = 256 \equiv 6 \pmod{10}\). Cycle of 4: 4,6,4,6... even powers → 6. ✓
This is a matching-type question — learn the pattern of each base's cycle.
Q10CUET 2022Alligation & Mixture
A mixture contains milk and water in the ratio 8:x. Adding 3 litres of water to 33 litres of the mixture makes the ratio 2:1. What is the value of \(x\)?
3
4
2
11
New total = 36 L, ratio 2:1 → Milk = 24 L (unchanged), Water = 12 L.
\(\dfrac{8}{8+x} \times 33 = 24 \Rightarrow 8+x = 11 \Rightarrow x = 3\)

Verify: Original 8:3, milk = 24 L. After +3 water: 24:12 = 2:1 ✓
Q11Inequalities
If \(|x| < 3\), what is the solution set for \(x\)?
\(3 < x < -3\)
\(-3 < x < 3\)
\(0 \le x < 3\)
\(x > 3\)
Rule: \(|x| < a\) (where \(a > 0\)) \(\Leftrightarrow -a < x < a\).

So \(|x| < 3 \Rightarrow -3 < x < 3\).

Geometric meaning: All real numbers within distance 3 from zero.
Q12Inequalities
If \(x\) is real and \(\dfrac{x+2}{-7} > 0\), what is the solution set for \(x\)?
\(x \in (-\infty, -5)\)
\(x \in (5, \infty)\)
\(x \in (-5, \infty)\)
\(x \in (-\infty, -2)\)
Denominator = \(-7\) (always negative). For fraction to be positive, numerator must also be negative:
\(x + 2 < 0 \Rightarrow x < -2\)

Solution: \(x \in (-\infty,\,-2)\).
Q13CBSE 2022Inequalities
What is the solution of \(\dfrac{3x-1}{2x+3} \geq 0\)?
\((-\infty, -\tfrac{3}{2}) \cup [\tfrac{1}{3}, \infty)\)
\((-\infty, -\tfrac{3}{2})\)
\((-\tfrac{3}{2}, \tfrac{1}{3})\)
No solution
Critical points: \(x = \tfrac{1}{3}\) (numerator = 0), \(x = -\tfrac{3}{2}\) (denominator = 0, excluded).

Sign analysis:
\(x < -\tfrac{3}{2}\): (−)/(−) = (+) ✓
\(-\tfrac{3}{2} < x < \tfrac{1}{3}\): (−)/(+) = (−) ✗
\(x \geq \tfrac{1}{3}\): (+)/(+) = (+) ✓ (include \(x=\tfrac{1}{3}\))

Solution: \((-\infty,\,-\tfrac{3}{2}) \cup [\tfrac{1}{3},\,\infty)\)
Q14CBSE 2022Inequalities
What is the solution set of \(|x - 5| < 3\)?
\((-\infty, 2)\)
\((2, 8)\)
\((8, \infty)\)
\((-\infty, 8)\)
\(|x - a| < r \Leftrightarrow a - r < x < a + r\)

With \(a=5, r=3\): \(5-3 < x < 5+3 \Rightarrow 2 < x < 8\)

Solution: \((2, 8)\)
Q15CUET 2022Inequalities
For the system \(3x + 2y \geq 24\) and \(x + y \leq 10\), in what range does the solution for \(x\) lie?
\((2, \infty)\)
\((2, 8)\)
\([4, \infty)\)
\((-\infty, 8)\)
From (ii): \(y \leq 10-x\). Substitute into (i):
\(3x + 2(10-x) \geq 24 \Rightarrow x + 20 \geq 24 \Rightarrow x \geq 4\)

Solution: \([4, \infty)\)
Check: \(x=4, y=6\) → \(12+12=24 \geq 24\) ✓ and \(4+6=10 \leq 10\) ✓
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Step-by-Step Solutions

Short Answer Solved Examples

2-mark and 3-mark questions with complete working. Click Show Solution to reveal each answer.

📝Self-assess your answer after revealing each solution below — your responses build your free Unit 1 AI Performance Report. Tap ✨ My Report (bottom-right) once you're done.
Q1CBSE 2024 CompttModular Arithmetic
Evaluate \((137 + 995) \pmod{12}\).
Reduce 137 mod 12: \(137 = 11 \times 12 + 5\) → \(137 \equiv 5 \pmod{12}\)
Reduce 995 mod 12: \(995 = 82 \times 12 + 11\) → \(995 \equiv 11 \pmod{12}\)
Add: \(5 + 11 = 16 = 1 \times 12 + 4\) → \(16 \equiv 4 \pmod{12}\)
\((137 + 995) \bmod 12 = \mathbf{4}\)
Self-assess: ✓ Saved
Q2CBSE 2024 CompttUnit Digit Cycles
Find the unit's digit of \(12^{12}\).
Unit digit of 12 = 2. Find unit digit of \(2^{12}\).
Cycle of \(2^n\): 2, 4, 8, 6 (period 4)
\(12 \div 4 = 3\) remainder 0 → use 4th value = 6
Verify: \(2^{12} = 4096\) → unit digit 6 ✓
Unit digit of \(12^{12}\) = 6
Self-assess: ✓ Saved
Q3CBSE 2023Repeated Replacement
A bottle is full of dettol. One-third is taken out and replaced with water, repeated three times. Find the final ratio of dettol to water.
Formula: Remaining = \(n\!\left(1-\dfrac{x}{n}\right)^k\). Here \(n=1\), \(x=\tfrac{1}{3}\), \(k=3\).
Dettol remaining: \(\left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}\)
Water: \(1 - \dfrac{8}{27} = \dfrac{19}{27}\)
Dettol : Water = 8 : 19
Self-assess: ✓ Saved
Q4CBSE 2023, 2024Repeated Replacement
A container has 50 L of juice. 5 L is taken out and replaced by water, repeated 5 times. How much juice remains?
\(n=50\), \(x=5\), \(k=5\). Retention fraction = \(1 - \tfrac{5}{50} = 0.9\)
Juice remaining = \(50 \times (0.9)^5 = 50 \times 0.59049 = 29.5245\) L
Juice remaining ≈ 29.52 litres
Self-assess: ✓ Saved
Q5CBSE 2023 CompttBoats & Streams
A person rows 5 km/h in still water. It takes him thrice as long to row upstream as downstream. Find the speed of the stream.
Let stream speed = \(x\). Down = \(5+x\), Up = \(5-x\).
Time equation: \(\dfrac{d}{5-x} = 3 \times \dfrac{d}{5+x}\)
\(5+x = 3(5-x) \Rightarrow 4x = 10 \Rightarrow x = 2.5\) km/h
Verify: Down = 7.5, Up = 2.5. Time ratio = 7.5/2.5 = 3 ✓
Speed of stream = 2.5 km/h
Self-assess: ✓ Saved
Q6Unit Digit Cycles
Find the remainder when \(17^{23}\) is divided by 5.
\(17 \equiv 2 \pmod{5}\) → find \(2^{23} \bmod 5\)
Cycle: \(2^1\!\to\!2,\, 2^2\!\to\!4,\, 2^3\!\to\!3,\, 2^4\!\to\!1\) (period 4)
\(23 \div 4 = 5\) remainder 3 → 3rd value = 3
Remainder = 3
Self-assess: ✓ Saved
Q7Repeated Replacement
A container has 60 L of milk. 6 L is removed and replaced with water three times. How much milk remains?
\(n=60\), \(x=6\), \(k=3\). Retention = \(1-\tfrac{6}{60} = 0.9\)
Milk remaining = \(60 \times (0.9)^3 = 60 \times 0.729 = 43.74\) L
Milk remaining = 43.74 litres
Self-assess: ✓ Saved
Q8Boats & Streams
A man rows 18 km/h in still water. It takes him twice as long to row upstream as downstream. Find the speed of the stream.
Let stream = \(s\). Down = \(18+s\), Up = \(18-s\).
\(\dfrac{d}{18-s} = 2 \cdot \dfrac{d}{18+s} \Rightarrow 18+s = 2(18-s) \Rightarrow 3s = 18\)
\(s = 6\) km/h. Verify: 24/12 = 2 ✓
Speed of stream = 6 km/h
Self-assess: ✓ Saved

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4-Mark Questions

Case Study Questions

Board-pattern 4-mark case studies. Read the context carefully, then click Show Solution under each part.

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Case Study 1: E-Commerce Delivery System

Type A vehicle: 40 km/h · 8 deliveries/trip · ₹500/trip
Type B vehicle: 50 km/h · 6 deliveries/trip · ₹600/trip
The company needs to complete 200 deliveries. Average route = 30 km.
(i) How many trips if only Type A vehicles are used? What is the total cost?
Trips = 200 ÷ 8 = 25 trips
Cost = 25 × ₹500 = ₹12,500
Trips = 25  ·  Total cost = ₹12,500
Self-assess: ✓ Saved
(ii) How many trips are needed and what is the total cost if only Type B vehicles are used?
Trips = ⌈200 ÷ 6⌉ = 34 trips (partial trip still required)
Cost = 34 × ₹600 = ₹20,400
Trips = 34  ·  Total cost = ₹20,400
Self-assess: ✓ Saved
(iii) If time matters most, which vehicle type should be prioritized?
Type A total time: 25 × (30 ÷ 40) = 18.75 hours
Type B total time: 34 × (30 ÷ 50) = 20.40 hours
Type B is faster per trip (0.6 h vs 0.75 h) but needs more trips — making total time longer.
Type A should be prioritized — 18.75 h total vs 20.40 h for Type B
Self-assess: ✓ Saved
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Case Study 2: Water Tank Management System

A residential society has three tanks: Tank A (5,000 L), Tank B (8,000 L), Tank C (6,000 L).
Pipe X (inlet): 250 L/h · ₹20/h   Pipe Y (inlet): 400 L/h · ₹35/h   Pipe Z (outlet): 150 L/h
(i) If both Pipe X and Pipe Y are used to fill Tank B together, how long will it take and what is the total cost?
Combined fill rate = 250 + 400 = 650 L/h
Time = 8000 ÷ 650 ≈ 12.31 hours (12 hrs 18 min)
Cost = 12.31 × (₹20 + ₹35) ≈ ₹677
Time ≈ 12 hours 18 minutes  ·  Cost ≈ ₹677
Self-assess: ✓ Saved
(ii) If Pipe Y is filling Tank A while Pipe Z is accidentally left open, how long will it take to fill, and how much extra time is wasted?
Net fill rate = 400 − 150 = 250 L/h
Time = 5000 ÷ 250 = 20 hours
Without Pipe Z: 5000 ÷ 400 = 12.5 h → extra time = 7.5 hours
Time to fill Tank A = 20 hours  ·  Extra time wasted = 7.5 hours
Self-assess: ✓ Saved
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Exam Tips — Unit 1

Mistakes students make in the board exam — and how to avoid them.

✓ Tip 1 — Memorise Unit-Digit Cycles

The unit-digit cycles for powers of 2, 3, 7 and 8 all have period 4: (2→2,4,8,6), (3→3,9,7,1), (7→7,9,3,1), (8→8,4,2,6). For digits 0, 1, 5, and 6, the unit digit never changes. Knowing these by heart lets you answer any unit-digit MCQ in under 15 seconds. This topic appears as a guaranteed 1-mark MCQ every year — don't drop these easy marks.

🔒  4 more exam tips for Unit 1 — sign analysis for inequalities, how to present pipes & cisterns for full method marks, the fastest alligation shortcut, and the repeated replacement formula trick — are included in the AI Question Bank.

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❌ Common Mistakes to Avoid in Unit 1

  • Using remainder 0 as the 0th cycle position — it should be the 4th (last) position
  • Forgetting to flip the inequality sign when dividing by a negative number
  • Including the denominator's zero in the solution set of rational inequalities
  • Not writing the repeated-replacement formula before substituting (loses method marks)
  • Confusing upstream and downstream speeds in boats & streams problems
  • Adding outlet pipe rates instead of subtracting them in pipes & cisterns
Common Questions

Frequently Asked Questions

Questions students frequently ask about Unit 1 — topics, formulas, and exam strategy.

Unit 1 – Numbers & Quantification carries 11 marks — the highest of all 8 units. Questions appear as 1-mark MCQs, 2–3 mark short answers, and 4-mark case studies. Focus on this unit heavily — scoring well here makes a significant difference to your overall marks.
Six topics: (1) Modulo Arithmetic, (2) Congruence Modulo and unit-digit cycles, (3) Alligation & Mixture and Repeated Replacement, (4) Boats & Streams, (5) Pipes & Cisterns, (6) Races & Games. Numerical Inequalities is also tested as MCQs and 3-mark problems.
Modulo arithmetic deals with the remainder when one integer is divided by another. Written as \(a \equiv r \pmod{n}\), it means \(a\) and \(r\) leave the same remainder when divided by \(n\). In Class 12, it is used to find unit digits of large powers (\(17^{123}\)), solve clock problems, and verify congruence equations.
Let \(B\) = boat speed in still water, \(S\) = stream speed. Downstream = B+S, Upstream = B−S. If upstream time is \(n\) times downstream time: set up \(\dfrac{d}{B-S} = n \cdot \dfrac{d}{B+S}\) and solve for \(S\). Always write the formula before substituting — it earns method marks.
When \(x\) litres are repeatedly removed from \(n\) litres and replaced with water, after \(k\) operations: Remaining original liquid \(= n \times \left(1 - \dfrac{x}{n}\right)^k\). Example: 60 L, 6 removed 3 times → \(60 \times (0.9)^3 = 43.74\) L.
Step 1: Use only the unit digit of the base — for \(17^{123}\), unit digit = 7. Step 2: Find the cycle: 7→9→3→1 (period 4). Step 3: \(123 \div 4\) = remainder 3 → 3rd value = 3. When remainder is 0, use the last (4th) value.
Based on past papers (2022–2026): Modulo Arithmetic/Unit Digit appears as MCQ every year. Alligation & Mixture/Repeated Replacement appears as a 3-mark short answer. Pipes & Cisterns as MCQ or short answer. Numerical Inequalities as MCQ + 3-mark problem. Boats & Streams in alternate years.
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