20 Mathematical Aptitude
This unit covers nine recurring areas of arithmetic and elementary mathematics that NTA Paper-I tests: percentages, ratio and proportion, profit and loss, simple and compound interest, average, time–speed–distance, time and work, permutation–combination, and probability.
20.1 Number System
| Type | Description | Examples |
|---|---|---|
| Natural numbers (ℕ) | Counting numbers from 1 | 1, 2, 3, … |
| Whole numbers | Natural numbers + 0 | 0, 1, 2, 3, … |
| Integers (ℤ) | Whole numbers + negatives | …, −3, −2, −1, 0, 1, 2, … |
| Rational numbers (ℚ) | Can be written as p/q (q ≠ 0) | 1/2, −3/4, 5, 0.75 |
| Irrational numbers | Cannot be written as p/q | √2, π, e |
| Real numbers (ℝ) | Rational + irrational | All numbers on a number line |
| Prime numbers | Exactly two divisors (1 and itself) | 2, 3, 5, 7, 11, 13, 17, 19, 23 |
| Composite numbers | More than two divisors | 4, 6, 8, 9, 10, 12, 14, 15, 16 |
Note: 1 is neither prime nor composite. 2 is the only even prime.
20.2 Percentages
A percentage expresses a number as a fraction of 100. x % = x/100.
| Need | Formula |
|---|---|
| Convert to fraction | x % = x/100 |
| Increase by x % | New value = Old × (1 + x/100) |
| Decrease by x % | New value = Old × (1 − x/100) |
| Successive change | New = Old × (1 + a/100) × (1 + b/100) |
| Original value | Original = New / (1 + x/100) |
| Net effect of +a % then −b % | a − b − ab/100 |
A salary is increased by 10 % and then decreased by 10 %. What is the net change?
Net = +10 − 10 − (10 × 10)/100 = 0 − 1 = −1 % (a 1 % decrease).
20.3 Ratio and Proportion
A ratio compares two quantities. a : b means a/b.
A proportion states that two ratios are equal: a : b :: c : d means a/b = c/d, i.e., ad = bc (the cross-multiplication rule).
If 8 workers can build a wall in 12 days, how many days will 6 workers take?
Workers and days are inversely proportional: 8 × 12 = 6 × x → x = 16 days.
20.4 Profit and Loss
| Concept | Formula |
|---|---|
| Profit | Selling Price (SP) − Cost Price (CP) |
| Loss | CP − SP |
| Profit % | (Profit / CP) × 100 |
| Loss % | (Loss / CP) × 100 |
| SP from CP and Profit % | SP = CP × (1 + Profit %/100) |
| CP from SP and Profit % | CP = SP / (1 + Profit %/100) |
| Discount % | (Discount / Marked Price) × 100 |
| SP after discount | SP = MP × (1 − Discount %/100) |
A book is bought for ₹120 and sold for ₹150. What is the profit %?
Profit = 150 − 120 = ₹30. Profit % = (30/120) × 100 = 25 %.
20.5 Simple and Compound Interest
| Type | Formula |
|---|---|
| Simple Interest (SI) | SI = (P × R × T) / 100 |
| Amount (SI) | A = P + SI = P × (1 + RT/100) |
| Compound Interest (CI) — annual | A = P × (1 + R/100)ᵀ |
| CI compounded n times per year | A = P × (1 + R/(100n))ⁿᵀ |
| CI | CI = A − P |
Where P = Principal, R = Rate per cent per annum, T = Time in years, A = Amount.
₹5,000 is invested at 8 % per annum for 3 years. SI?
SI = (5000 × 8 × 3) / 100 = ₹1,200.
₹10,000 at 10 % per annum compounded annually for 2 years.
A = 10000 × (1 + 10/100)² = 10000 × 1.21 = ₹12,100. CI = 12,100 − 10,000 = ₹2,100.
20.6 Average
Average = (Sum of all values) / (Number of values).
- Average of first n natural numbers = (n + 1) / 2.
- Average of first n even numbers = n + 1.
- Average of first n odd numbers = n.
- Average of consecutive numbers from a to b = (a + b) / 2.
The average age of 10 students is 14 years. If a teacher (age 40) is included, what is the new average?
Sum of ages = 14 × 10 = 140. New sum = 140 + 40 = 180. New average = 180 / 11 ≈ 16.36 years.
20.7 Time, Speed and Distance
| Need | Formula |
|---|---|
| Distance | D = Speed × Time |
| Speed | S = Distance / Time |
| Time | T = Distance / Speed |
| Convert km/h to m/s | × 5/18 |
| Convert m/s to km/h | × 18/5 |
| Average speed (equal distances at S₁ and S₂) | 2S₁S₂ / (S₁ + S₂) — harmonic mean |
| Relative speed (same direction) | S₁ − S₂ |
| Relative speed (opposite direction) | S₁ + S₂ |
A train 200 m long passes a stationary man in 20 seconds. Speed?
Speed = 200 m / 20 s = 10 m/s = 10 × 18/5 = 36 km/h.
20.8 Time and Work
| Need | Formula |
|---|---|
| If A does a work in a days, A’s 1-day work | 1/a |
| Combined 1-day work of A and B | 1/a + 1/b |
| Time for A + B together | ab / (a + b) |
| If A is k times as efficient as B and A takes a days | B takes ka days |
A can do a job in 12 days, B in 8 days. Together?
1-day work = 1/12 + 1/8 = (2 + 3)/24 = 5/24. Time = 24/5 = 4.8 days.
20.9 Permutation and Combination
- Permutation — arrangement; order matters.
- Combination — selection; order does not matter.
| Need | Formula |
|---|---|
| Permutation of n distinct objects taken r at a time | nPr = n! / (n − r)! |
| Combination of n distinct objects taken r at a time | nCr = n! / (r! × (n − r)!) |
| Permutation of n objects (all together) | n! |
| Permutation of n objects with repetition (k₁ of one type, k₂ of another, …) | n! / (k₁! × k₂! × …) |
| Number of subsets of an n-element set | 2ⁿ |
Quick values: 0! = 1; 1! = 1; 2! = 2; 3! = 6; 4! = 24; 5! = 120; 6! = 720; 7! = 5,040.
How many ways to arrange the letters of the word “MATH”?
4 distinct letters: 4! = 24 ways.
From a group of 8 people, how many ways to choose a committee of 3?
8C3 = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56 ways.
20.10 Probability
Probability of event E = P(E) = (Number of favourable outcomes) / (Total number of outcomes).
- 0 ≤ P(E) ≤ 1.
- P(certain event) = 1; P(impossible event) = 0.
- P(not E) = 1 − P(E).
- Mutually exclusive events: P(A ∪ B) = P(A) + P(B).
- Independent events: P(A ∩ B) = P(A) × P(B).
- General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
What is the probability of getting at least one head when two coins are tossed together?
Total outcomes = 2² = 4: HH, HT, TH, TT. Favourable (at least one H) = 3.
P = 3/4.
A die is rolled. Probability of getting an even number?
Favourable: 2, 4, 6 → 3 outcomes. Total: 6.
P = 3/6 = 1/2.
20.11 Practice Questions
A salary is increased by 20 % and then decreased by 20 %. What is the net change?
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A book is bought for ₹160 and sold for ₹200. What is the profit percentage?
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₹8,000 is invested at 6 % per annum simple interest for 5 years. The interest earned is:
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₹10,000 invested at 10 % per annum compounded annually for 2 years. The compound interest is:
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A train 180 m long crosses a stationary man in 12 seconds. The speed of the train in km/h is:
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A can do a job in 10 days; B can do it in 15 days. How many days will they take working together?
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From a group of 8 people, how many committees of 3 can be formed?
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Two coins are tossed together. What is the probability of getting at least one head?
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- Percentage net effect: a − b − ab/100.
- Profit % = (Profit / CP) × 100.
- SI = PRT/100; CI: A = P(1 + R/100)ᵀ.
- D = ST; km/h → m/s = ×5/18.
- Combined time (A + B) = ab/(a + b).
- nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!).
- P(E) = favourable / total; P(not E) = 1 − P(E).
- Average of first n natural numbers = (n+1)/2.