Quantitative Aptitude PYQs

Shortcut Trick

Identify the Highest Common Factor (HCF) of 72 and 96.

72 = 24 × 3 and 96 = 24 × 4.

HCF = 24.

Divide numerator and denominator by 24 → 3/4.

∴ The correct answer is 3/4.

Alternate Method

Given: Fraction = 72/96

Formula Used: Lowest form = (Numerator ÷ HCF) / (Denominator ÷ HCF)

⇒ Prime factors of 72 = 2 × 2 × 2 × 3 × 3

⇒ Prime factors of 96 = 2 × 2 × 2 × 2 × 2 × 3

⇒ HCF = 2 × 2 × 2 × 3 = 24

⇒ Lowest form = (72 ÷ 24) / (96 ÷ 24)

⇒ Lowest form = 3/4

∴ The correct answer is 3/4.

Additional Information

HCF (Highest Common Factor)

The largest positive integer that divides each of the integers without a remainder.

Simplest Form of a Fraction

A fraction is in its simplest form (or lowest terms) if its numerator and denominator are coprime, meaning their HCF is 1.

Equivalent Fractions

Fractions that have the same value even though they look different, obtained by multiplying or dividing both parts by the same non-zero number.

Shortcut Trick

For any quadrilateral, the angle formed by the bisectors of two adjacent angles is equal to half the sum of the other two angles.

∠LZK = (∠I + ∠J) ÷ 2

∠LZK = (36° + 44°) ÷ 2

∠LZK = 80° ÷ 2 = 40°

∴ The correct answer is 40°.

Alternate Method

Given:

In quadrilateral IJKL:

∠I = 36°

∠J = 44°

The bisectors of ∠K and ∠L meet at Z.

Formula Used:

1. Sum of all interior angles of a quadrilateral = 360°
2. Sum of all angles of a triangle = 180°

Calculations:

In quadrilateral IJKL:

⇒ ∠I + ∠J + ∠K + ∠L = 360°

⇒ 36° + 44° + ∠K + ∠L = 360°

⇒ 80° + ∠K + ∠L = 360°

⇒ ∠K + ∠L = 360° − 80°

⇒ ∠K + ∠L = 280°

Dividing both sides by 2:

⇒ (∠K + ∠L) ÷ 2 = 280° ÷ 2

⇒ ∠K/2 + ∠L/2 = 140°

Since KZ and LZ are angle bisectors of ∠K and ∠L respectively:

⇒ ∠ZKL = ∠K/2 and ∠ZLK = ∠L/2

⇒ ∠ZKL + ∠ZLK = 140°

In triangle LZK:

⇒ ∠LZK + ∠ZKL + ∠ZLK = 180°

⇒ ∠LZK + 140° = 180°

⇒ ∠LZK = 180° − 140°

⇒ ∠LZK = 40°

Option Analysis:

⇒ Option (1) 40° is the correct measure of ∠LZK as calculated above.

⇒ Option (2) 55° is incorrect because it does not satisfy the angle sum property of ΔLZK where the sum of the half-angles must be 140°.

⇒ Option (3) 34° is incorrect because it would require the sum of the other two angles of the quadrilateral to be 68°, which contradicts the given sum of 80°.

⇒ Option (4) 54° is incorrect because it does not match the derived relation ∠LZK = (∠I + ∠J) ÷ 2.

∴ The correct answer is 40°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is always equal to half the sum of the remaining two angles. For adjacent angles A and B, the angle between their bisectors is (∠C + ∠D) ÷ 2.

Sum of Interior Angles of a Polygon

The sum of the interior angles of any n-sided polygon is given by the formula (n − 2) × 180°. For a quadrilateral (n = 4), the sum is (4 − 2) × 180° = 360°.

Shortcut Trick

Relative Speed in opposite direction = 72 + 108 = 180 kmph.

Conversion to m/s = 180 × (5/18) = 50 m/s.

Total Distance to cover = Length₁ + Length₂ = 150 + 200 = 350 m.

Time Taken = Total Distance ÷ Relative Speed = 350 ÷ 50 = 7 sec.

∴ The correct answer is 7 sec.

Alternate Method

Given:

Speed of Train 1 (S₁) = 72 km/hr

Speed of Train 2 (S₂) = 108 km/hr

Length of Train 1 (L₁) = 150 m

Length of Train 2 (L₂) = 200 m

Formula Used:

Time = (Total Length) ÷ (Relative Speed)

Relative Speed (Opposite) = S₁ + S₂

Calculations:

⇒ Total distance = 150 + 200 = 350 m

⇒ Relative Speed = 72 + 108 = 180 km/hr

⇒ Relative Speed in m/s = 180 × (5/18) = 50 m/s

⇒ Time taken = 350 ÷ 50

⇒ Time taken = 7 sec

∴ The correct answer is 7 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, relative speed is the sum of their speeds (S₁ + S₂). When they move in the same direction, relative speed is the difference (S₁ − S₂).

Speed Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Total Distance for Crossing

When a train crosses another train or a bridge, the total distance is always the sum of their lengths, regardless of the direction of movement.

Shortcut Trick

Initial Relative Speed = 80 + 60 = 140 km/hr.

New Speed of A = 80 − 10% of 80 = 72 km/hr.

New Speed of B = 60 + 10% of 60 = 66 km/hr.

New Relative Speed = 72 + 66 = 138 km/hr.

Since Time ∝ 1/Speed for constant distance: Ratio = 138 : 140 = 69 : 70.

∴ The correct answer is 69:70.

Alternate Method

Given: Distance = 9660 km, Speed A = 80 km/hr, Speed B = 60 km/hr.

Formula: Time = Distance ÷ Relative Speed (Towards each other = S₁ + S₂).

⇒ Initial Relative Speed (V₁) = 80 + 60 = 140 km/hr.

⇒ Initial Time (T₁) = 9660 ÷ 140 = 69 hours.

⇒ New Speed A = 80 × 0.9 = 72 km/hr.

⇒ New Speed B = 60 × 1.1 = 66 km/hr.

⇒ New Relative Speed (V₂) = 72 + 66 = 138 km/hr.

⇒ New Time (T₂) = 9660 ÷ 138 = 70 hours.

⇒ Ratio T₁ : T₂ = 69 : 70.

∴ The correct answer is 69:70.

Additional Information

Relative Speed Concept

When two objects move toward each other, relative speed is the sum of individual speeds. When they move in the same direction, it is the difference.

Inverse Proportion Rule

For a fixed distance, time is inversely proportional to speed (T₁/T₂ = V₂/V₁), allowing ratios to be solved without using the actual distance.

Shortcut Trick

Divide the numerator and denominator of the given expression by cos θ.

Expression = (1 − tan θ) ÷ (tan θ + 1)

Substitute tan θ = 1/2 → (1 − 1/2) ÷ (1/2 + 1) = (1/2) ÷ (3/2) = 1/3

∴ The correct answer is 1/3.

Alternate Method

Given: tan θ = 1/2 (Perpendicular/Base)

Formula Used: sin θ = P/H and cos θ = B/H; H = √(P² + B²)

⇒ Let Perpendicular (P) = 1 and Base (B) = 2

⇒ Hypotenuse (H) = √(1² + 2²) = √5

⇒ sin θ = 1/√5 and cos θ = 2/√5

⇒ Expression = ((2/√5) − (1/√5)) ÷ ((1/√5) + (2/√5))

⇒ Expression = (1/√5) ÷ (3/√5) = 1/3

∴ The correct answer is 1/3.

Additional Information

Trigonometric Ratios

tan θ = sin θ / cos θ. Always check if dividing by cos θ or sin θ simplifies the expression directly.

Pythagorean Identity

sin²θ + cos²θ = 1. This can be used to find cos θ if sin θ is known and vice versa.

Componendo and Dividendo

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). Useful for expressions involving (cos+sin)/(cos-sin).

Shortcut Trick

The difference between the outer perimeter and inner perimeter of a track with uniform width depends only on the width of the curves.

Difference = 2 × π × width = 2 × (22/7) × 18 = 792/7 ≈ 113.14 m.

Outer Perimeter = Inner Perimeter + Difference = 504 + 113.14 = 617.14 m.

Rounding 617.14 m to the nearest tens gives 620 m.

∴ The correct answer is 620 m.

Alternate Method

Given: Straight segment length (L) = 120 m, Track width (w) = 18 m, Inner perimeter = 504 m, π = 22/7.

The inner perimeter of the track consists of two straight segments and two semicircular arcs:

⇒ Inner Perimeter = 2 × L + 2 × π × r

⇒ 504 = 2(120) + 2 × (22/7) × r

⇒ 504 = 240 + (44/7)r

⇒ 264 = (44/7)r

⇒ r = (264 × 7) ÷ 44 = 6 × 7 = 42 m

The outer radius (R) is the sum of the inner radius and the track width:

⇒ R = r + width = 42 + 18 = 60 m

Now, calculate the outer perimeter using the outer radius:

⇒ Outer Perimeter = 2 × L + 2 × π × R

⇒ Outer Perimeter = 2(120) + 2 × (22/7) × 60

⇒ Outer Perimeter = 240 + 2640 ÷ 7

⇒ Outer Perimeter = 240 + 377.14 ≈ 617.14 m

Rounding to the nearest tens:

⇒ 617.14 m → 620 m

∴ The correct answer is 620 m.

Additional Information

Track Perimeter Formula

For a stadium-shaped track, Perimeter = 2 × Length + 2 × π × Radius. The distance between the straight lines is the diameter (2r).

Area of a Track

Area = (Outer Area − Inner Area) = 2 × L × w + π(R² − r²), where L is the straight length and w is the width.

Nearest Tens Rounding Rule

If the last digit is 5 or greater, round up to the next ten (e.g., 617 becomes 620). If less than 5, round down.

Shortcut Trick

Assume a base value for the Mean, let it be 60.

Calculate deviations from 60: (+34, −1, −14, −26, −45, −5, +29, +38, +8).

Net sum of deviations = 34 − 1 − 14 − 26 − 45 − 5 + 29 + 38 + 8 = +18.

Correct Mean = Assumed Mean + (Net Deviation ÷ Total Observations) = 60 + (18 ÷ 9) = 60 + 2 = 62.

∴ The correct answer is 62.

Alternate Method

Given: Observations = 94, 59, 46, 34, 15, 55, 89, 98, 68

Formula Used: Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Total sum = 94 + 59 + 46 + 34 + 15 + 55 + 89 + 98 + 68

⇒ Total sum = 558

⇒ Total number of observations (n) = 9

⇒ Arithmetic Mean = 558 ÷ 9

⇒ Mean = 62

∴ The correct answer is 62.

Additional Information

Properties of Arithmetic Mean

The sum of deviations of items from their arithmetic mean is always zero. ∑(x − x̄) = 0.

Weighted Arithmetic Mean

When different weights (w) are assigned to observations (x), the mean is calculated as ∑wx ÷ ∑w.

Effect of Change of Origin and Scale

If each observation is increased or decreased by a constant 'k', the mean also increases or decreases by 'k'. If multiplied by 'k', the mean is also multiplied by 'k'.

Shortcut Trick

Volume of a sphere = 4/3 × π × r³

Since the formula uses π (22/7), the volume must be a multiple of 11.

Check 38,808: (3 + 8 + 8) − (8 + 0) = 19 − 8 = 11 (Divisible by 11).

Also, since r = 21, the answer must be a multiple of 9.

Sum of digits: 3 + 8 + 8 + 0 + 8 = 27 (Divisible by 9).

∴ The correct answer is 38,808 m³.

Alternate Method

Given:

Radius of the spherical tank (r) = 21 m

Value of π = 22/7

Formula Used:

Volume of a sphere = (4/3) × π × r³

Calculations:

⇒ Volume = (4/3) × (22/7) × 21 × 21 × 21

⇒ Volume = (4/3) × (22/7) × 9261

⇒ Volume = (88/21) × 9261

⇒ Volume = 88 × 441

⇒ Volume = 38,808 m³

∴ The correct answer is 38,808 m³.

Additional Information

Surface Area of Sphere

The total surface area of a sphere is calculated using the formula 4 × π × r².

Volume of Hemisphere

The volume of a hemisphere is exactly half that of a sphere: (2/3) × π × r³.

Total Surface Area of Hemisphere

For a solid hemisphere, the total surface area (TSA) is 3 × π × r².

Shortcut Trick

Successive reduction formula: Present Population = Initial Population × (100 − 40)% × (100 − 35)%

Initial : Present = 100 : (60/100 × 65)

Ratio = 100 : (0.6 × 65) = 100 : 39

∴ The correct answer is 100:39.

Alternate Method

Given: First reduction = 40%, Second reduction = 35% of remaining, Present Population = 2340.

⇒ Let Initial Population be 100 units.

⇒ After 40% left, Remaining population = 100 − 40 = 60 units.

⇒ 35% of 60 left for jobs = 0.35 × 60 = 21 units.

⇒ Present Population = 60 − 21 = 39 units.

⇒ Ratio of Initial to Present = 100 : 39.

⇒ Check with given value: 39 units = 2340 ⇒ 1 unit = 60. Initial = 100 × 60 = 6000. Ratio = 6000 : 2340 = 100 : 39.

∴ The correct answer is 100:39.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the net change is [x + y + (xy/100)]%.

Population Depreciation Formula

Final Value = Initial Value × (1 − r₁/100) × (1 − r₂/100), where r is the rate of decrease.

Shortcut Trick

Use the Empirical Relationship: 2 × Mean = 3 × Median − Mode.

Substitute the ratio parts directly: Median = 9 and Mode = 11.

2 × Mean = 3(9) − 11 ⇒ 2 × Mean = 27 − 11 = 16.

Mean = 8.

The required ratio Mean : Median = 8 : 9.

∴ The correct answer is 8:9.

Alternate Method

Given:

Ratio of Mode and Median = 11 : 9

Formula Used:

Empirical Formula: Mode = 3 × Median − 2 × Mean

Calculations:

⇒ Let Mode = 11x and Median = 9x.

⇒ According to the formula: 11x = 3(9x) − 2(Mean)

⇒ 11x = 27x − 2(Mean)

⇒ 2(Mean) = 27x − 11x

⇒ 2(Mean) = 16x

⇒ Mean = 8x

⇒ Ratio of Mean and Median = 8x : 9x

⇒ Mean : Median = 8 : 9

∴ The correct answer is 8:9.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is given by: Mean − Mode = 3 × (Mean − Median).

Symmetrical Distribution

In a perfectly symmetrical distribution (Normal Distribution), the Mean, Median, and Mode are all equal (Mean = Median = Mode).

Central Tendency Usage

Mean is sensitive to outliers; Median is the middle value used for skewed data; Mode represents the most frequent value.

Shortcut Trick

Given $x = \sqrt{10} + 3$, then its reciprocal $1/x = \sqrt{10} − 3$.

Calculate the difference: $k = x − 1/x = (\sqrt{10} + 3) − (\sqrt{10} − 3) = 6$.

Use the shortcut formula: $x^3 − 1/x^3 = k^3 + 3k$.

Substitute $k = 6$: $x^3 − 1/x^3 = 6^3 + 3(6) = 216 + 18 = 234$.

∴ The correct answer is 234.

Alternate Method

Given: $x = \sqrt{10} + 3$

Formula Used: $x^3 − 1/x^3 = (x − 1/x)^3 + 3(x − 1/x)$

⇒ $x = \sqrt{10} + 3$

⇒ $1/x = 1 / (\sqrt{10} + 3)$

⇒ Rationalizing the denominator: $1/x = (\sqrt{10} − 3) / [(\sqrt{10} + 3)(\sqrt{10} − 3)]$

⇒ $1/x = (\sqrt{10} − 3) / (10 − 9) = \sqrt{10} − 3$

⇒ $x − 1/x = (\sqrt{10} + 3) − (\sqrt{10} − 3) = 6$

⇒ $x^3 − 1/x^3 = (x − 1/x)^3 + 3 \times x \times (1/x) \times (x − 1/x)$

⇒ $x^3 − 1/x^3 = 6^3 + 3 \times 6$

⇒ $x^3 − 1/x^3 = 216 + 18 = 234$

∴ The correct answer is 234.

Additional Information

Reciprocal Conjugate Rule

If $x = \sqrt{a} + \sqrt{b}$ and $a − b = 1$, then $1/x = \sqrt{a} − \sqrt{b}$. This simplifies calculation of $x \pm 1/x$.

Cube Algebraic Identities

$x^3 − 1/x^3 = (x − 1/x)^3 + 3(x − 1/x)$ and $x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x)$.

Square Relation

$x^2 + 1/x^2 = (x − 1/x)^2 + 2 = (x + 1/x)^2 − 2$.

Shortcut Trick

Total ratio parts = 11 + 16 + 6 = 33.

Profit = 70% of Total Investment = ₹69,300.

Total Investment = (69,300 ÷ 70) × 100 = ₹99,000.

Unitary value = 99,000 ÷ 33 = ₹3,000 per ratio part.

Maya's investment = 16 ratio parts × ₹3,000 = ₹48,000.

∴ The correct answer is ₹48,000.

Alternate Method

Given → Investment ratio = 11 : 16 : 6 | Total Profit = ₹69,300 | Profit % = 70% of Investment.
Formulas: Investment = (Profit ÷ Profit %) × 100 | Individual Investment = (Individual Ratio ÷ Total Ratio) × Total Investment.

⇒ Let investments be 11x, 16x, and 6x. Total investment = 33x.
⇒ 70% of 33x = 69,300.
⇒ 0.70 × 33x = 69,300.
⇒ 23.1x = 69,300.
⇒ x = 69,300 ÷ 23.1 = 3,000.
⇒ Maya investment = 16x = 16 × 3,000 = 48,000.

∴ The correct answer is ₹48,000.

Additional Information

Partnership Profit Distribution

In a partnership, if the time period is constant, the profit is distributed in the direct ratio of the capital invested.

Capital Calculation from Profit %

If profit is given as a percentage of investment, then Total Investment = (Profit Amount × 100) ÷ Profit %.

Shortcut Trick

Sum of ages 4 years ago = 40 years

Present sum = 40 + (2 × 4) = 48 years

Ratio of Mohan : Suresh = 2 : 1

Suresh's present age = 48 × (1 / 3) = 16 years

∴ The correct answer is 16 years.

Alternate Method

Given:

Present age of Suresh = x

Present age of Mohan = 2x

Calculations:

⇒ Age of Suresh 4 years ago = x − 4

⇒ Age of Mohan 4 years ago = 2x − 4

⇒ (2x − 4) + (x − 4) = 40

⇒ 3x − 8 = 40

⇒ 3x = 48

⇒ x = 16

∴ The correct answer is 16 years.

Additional Information

Age Ratio Problems

When comparing ages at different points in time, the difference between ages of two people always remains constant regardless of the number of years passed.

Sum of Ages Shift

If there are 'n' people, the sum of their ages increases by n × t after 't' years and decreases by the same amount for 't' years ago.

Shortcut Trick

Incomes ratio = 5 : 8 | Expenditure ratio = 1 : 3

Since savings are equal (₹84,000), make the difference between ratio parts equal.

Difference (5, 8) = 3 and (1, 3) = 2. Multiply Income by 2 and Expenditure by 3.

New Income ratio = 10 : 16 | New Expenditure ratio = 3 : 9

Savings (Income − Exp) = 10 − 3 = 7 units. Given 7 units = ₹84,000 → 1 unit = ₹12,000.

Rupesh income = 10 units = 10 × ₹12,000 = ₹1,20,000.

∴ The correct answer is ₹1,20,000.

Alternate Method

Given: Ratio of income of Rupesh and Manjit = 5 : 8. Monthly saving of each = ₹84,000. Ratio of expenditure = 1 : 3.

Formula: Income − Savings = Expenditure

⇒ Let the incomes be 5x and 8x.

⇒ According to the question: (5x − 84,000) ÷ (8x − 84,000) = 1 ÷ 3

⇒ 3 × (5x − 84,000) = 1 × (8x − 84,000)

⇒ 15x − 2,52,000 = 8x − 84,000

⇒ 15x − 8x = 2,52,000 − 84,000

⇒ 7x = 1,68,000

⇒ x = 24,000

⇒ Monthly income of Rupesh = 5x = 5 × 24,000 = ₹1,20,000

∴ The correct answer is ₹1,20,000.

Additional Information

Income-Expenditure Relation

Income = Expenditure + Savings. This is the fundamental equation for all financial ratio problems.

Proportionality in Ratios

When a constant value is added to or subtracted from both terms of a ratio, the new terms are found by equalizing the difference between the ratio parts.

Cross-Multiplication Method

For a : b and c : d with a constant change k, use |(a × d) − (b × c)| units = |k × (c − d)| to solve for 1 unit directly.

Shortcut Trick

Savings % = 100% − Expenditure % = 100% − 65% = 35%.

As per the question, 35% of Income = ₹42,000.

Total Income (100%) = (42,000 ÷ 35) × 100.

⇒ 1,200 × 100 = ₹1,20,000.

∴ The correct answer is ₹1,20,000.

Alternate Method

Given:

Expenditure = 65% of Income

Total Savings = ₹42,000

Formula Used:

Income = Expenditure + Savings

Calculations:

Let the total income be 100 units.

⇒ Expenditure = 65 units

⇒ Savings = 100 − 65 = 35 units

⇒ 35 units = ₹42,000

⇒ 1 unit = 42,000 ÷ 35 = ₹1,200

⇒ 100 units = 1,200 × 100 = ₹1,20,000

∴ The correct answer is ₹1,20,000.

Additional Information

Percentage Change in Income

If income increases by x% and expenditure increases by y%, the change in savings depends on the initial ratio of income to expenditure.

Base Value Concept

In problems involving income and expenditure, Total Income is always considered as the base value (100%) unless specified otherwise.

Savings Ratio

Savings to Income ratio = (100 − Expenditure %)/100. For this case, the ratio is 35/100 or 7/20.

Shortcut Trick

Net percentage change for 10% compound interest for 2 years = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = (100 + 21)% of Principal.

Amount = 121% of 8,800 = 1.21 × 8,800 = ₹10,648.

∴ The correct answer is ₹10,648.

Alternate Method

Given:

Principal (P) = ₹8,800

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 8,800 × (1 + 10/100)²

⇒ A = 8,800 × (110/100) × (110/100)

⇒ A = 8,800 × (1.1) × (1.1)

⇒ A = 8,800 × 1.21

⇒ A = 88 × 121

⇒ A = ₹10,648

∴ The correct answer is ₹10,648.

Additional Information

Compound Interest Calculation

Compound Interest is calculated as CI = Amount − Principal. In this case, CI = 10,648 − 8,800 = ₹1,848.

Successive Percentage Rule

For two successive changes of a% and b%, the effective change is [a + b + (ab/100)]%.

Compounding Frequency

If compounded half-yearly, the rate becomes R/2 and the time becomes 2n. If quarterly, the rate becomes R/4 and time becomes 4n.

Shortcut Trick

Total deduction = 71 + 38 + 20 = ₹129.

Remaining Total Sum = 6,129 − 129 = ₹6,000.

Sum of Ratio parts = 14 + 3 + 8 = 25 units.

Difference between Q and R shares (after deduction) = 8 − 3 = 5 units.

Difference in original shares = (6,000 ÷ 25) × 5 + (Difference of deductions) = 240 × 5 + (38 − 20) = 1,200 − 18 = ₹1,182.

∴ The correct answer is ₹1,182.

Alternate Method

Given: Total sum = ₹6,129; Deductions for P, Q, R are ₹71, ₹38, ₹20; Ratio after deduction = 14 : 3 : 8.

⇒ Total Amount after deduction = 6,129 − (71 + 38 + 20) = ₹6,000

⇒ Let shares be 14x, 3x, and 8x.

⇒ 14x + 3x + 8x = 6,000 ⇒ 25x = 6,000

⇒ x = 6,000 ÷ 25 = 240

⇒ Original share of Q = 3x + 38 = 3(240) + 38 = 720 + 38 = ₹758

⇒ Original share of R = 8x + 20 = 8(240) + 20 = 1,920 + 20 = ₹1,940

⇒ Difference = 1,940 − 758 = ₹1,182

∴ The correct answer is ₹1,182.

Additional Information

Ratio and Proportion Basics

If a sum S is divided in ratio a:b:c, the share of b is [b ÷ (a+b+c)] × S.

Deduction Principle

When fixed amounts are subtracted from various shares, the total sum is reduced by the sum of those subtractions before applying the ratio.

Difference calculation

Difference between (A+k1) and (B+k2) is (A−B) + (k1−k2).

Shortcut Trick

Assume all 20 coins are ₹5 coins: Total value = 20 × 5 = ₹100.

Difference from actual value = 100 − 70 = ₹30.

Difference between ₹5 and ₹2 coin values = 5 − 2 = ₹3.

Number of ₹2 coins = Total difference ÷ Value difference = 30 ÷ 3 = 10.

∴ The correct answer is 10 coins.

Alternate Method

Given: Total coins = 20, Total value = ₹70.

Formula: Let number of ₹2 coins be x and ₹5 coins be (20 − x).

⇒ 2x + 5(20 − x) = 70

⇒ 2x + 100 − 5x = 70

⇒ −3x = 70 − 100

⇒ −3x = −30

⇒ x = 10

∴ The correct answer is 10 coins.

Additional Information

Alligation Cross Method

Used to find the ratio of two quantities when their individual values and average value are known. Ratio = (High − Mean) : (Mean − Low).

Linear Equations in Two Variables

Problems involving two types of items can be solved using x + y = Total and ax + by = Value.

Assumption Method

By assuming all items belong to one category, the discrepancy in the total value directly helps find the quantity of the other category.

Shortcut Trick

Check odd numbers between 472 and 486: 473, 475, 477, 479, 481, 483, 485.

Eliminate non-primes: 473 (11×43), 475 (ends in 5), 477 (sum 18, div by 9), 481 (13×37), 483 (sum 15, div by 3), 485 (ends in 5).

Only 479 is not divisible by any prime up to √486 ≈ 22.

∴ The correct answer is 1.

Alternate Method

Given: Range of numbers from 472 to 486.

Method: Listing and checking divisibility for numbers in the range.

⇒ Even numbers (474, 476, 478, 480, 482, 484) are not prime.

⇒ 473 ÷ 11 = 43 (Not prime)

⇒ 475 is divisible by 5 (Not prime)

⇒ 477: Sum of digits = 4+7+7 = 18, divisible by 3 and 9 (Not prime)

⇒ 479: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 (Prime)

⇒ 481 ÷ 13 = 37 (Not prime)

⇒ 483: Sum of digits = 4+8+3 = 15, divisible by 3 (Not prime)

⇒ 485 is divisible by 5 (Not prime)

⇒ Total prime numbers found = 1 (which is 479).

∴ The correct answer is 1.

Additional Information

Prime Number Definition

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Primality Test

To check if n is prime, test divisibility by all primes less than or equal to √n.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and even places is 0 or a multiple of 11.

Shortcut Trick

SP = MP × (100 − Discount%) ÷ 100

SP = 8,460 × (100 − 30) ÷ 100

SP = 8,460 × 0.70 = 5,922

∴ The correct answer is ₹5,922.

Alternate Method

Given: Marked Price (MP) = ₹8,460 and Discount Percentage = 30%.
Formulas: Discount = (Discount% ÷ 100) × MP | SP = MP − Discount.

⇒ Calculate Discount amount = (30 ÷ 100) × 8,460.
⇒ Discount = 0.3 × 8,460 = ₹2,538.
⇒ Calculate Selling Price (SP) = 8,460 − 2,538.
⇒ SP = ₹5,922.

∴ The correct answer is ₹5,922.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP to reach the Selling Price.

Net Selling Price

If the discount is d%, then SP = MP × (100 − d)/100.