Quantitative Aptitude PYQs

Shortcut Trick

When an object is melted and recast, its total volume remains completely conserved.

Volume of Original Sphere = 8 × Volume of One Cone

4/3 × π × R³ = 8 × (1/3 × π × r² × h)

R³ = 2 × 3² × 4 = 72

R = ³√72 ≈ 4.16 cm

∴ The correct answer is 4.16 cm.

Alternate Method

Given:

Radius of each cone (r) = 3 cm

Height of each cone (h) = 4 cm

Number of cones (n) = 8

Formula Used:

Volume of a sphere = 4/3 × π × R³

Volume of a cone = 1/3 × π × r² × h

R = ? Original Sphere

Melted & Recast

h=4 r=3 8 Identical Cones

Calculations:

⇒ Volume of the original sphere = n × Volume of one cone

⇒ 4/3 × π × R³ = 8 × [1/3 × π × (3)² × 4]

⇒ 4 × R³ = 8 × 9 × 4

⇒ 4 × R³ = 288

⇒ R³ = 288 ÷ 4

⇒ R³ = 72

⇒ R = ³√72

⇒ We know that 4³ = 64 and 5³ = 125.

⇒ Since 72 is slightly greater than 64, its cube root will be slightly greater than 4.

⇒ From the given options, R ≈ 4.16 cm.

∴ The correct answer is 4.16 cm.

Additional Information

Conservation of Volume

When a three-dimensional object is melted and reshaped into another form, the total volume of the material always remains conserved. Only the surface area changes.

Volume of a Sphere

The total space occupied by a sphere of radius R is calculated using the formula V = 4/3 × π × R³.

Volume of a Cone

A cone with base radius r and vertical height h has a volume equal to V = 1/3 × π × r² × h, which is exactly one-third of the volume of a cylinder having the exact same base and height.

Shortcut Trick

A regular hexagon has 6 equal sides and 6 equal interior angles of 120°.

By joining the opposite vertices, the hexagon is symmetrically divided by its diagonals passing through the center.

The full central angle of 360° is split equally into 6 parts of 60° each.

Because the sides of the hexagon are equal and the angles are all 60°, exactly 6 perfectly equilateral triangles are formed.

∴ The correct answer is 6.

Alternate Method

O 1 2 3 4 5 6

Given:

A polygon which is a regular hexagon.

Formula Used:

Sum of interior angles of an n-sided polygon = (n − 2) × 180°

Calculations:

⇒ For a regular hexagon, the number of sides n = 6.

⇒ Each interior angle of the hexagon = [(6 − 2) × 180°] ÷ 6 = 720° ÷ 6 = 120°.

⇒ The 3 main diagonals connecting opposite vertices intersect perfectly at the geometric center (Point O).

⇒ These diagonals bisect the interior angles of the hexagon, giving base angles of 120° ÷ 2 = 60° for each formed triangle.

⇒ At the center O, the full 360° angle is divided equally into 6 parts, so the central angle for each triangle = 360° ÷ 6 = 60°.

⇒ Since all three angles of each triangle are exactly 60°, they are all equilateral triangles.

⇒ As shown in the figure above, counting them around the center results in exactly 6 equilateral triangles.

∴ The correct answer is 6.

Additional Information

Area of a Regular Hexagon

Since it is composed of 6 equilateral triangles of side length a, its total Area = 6 × (√3 ÷ 4) × a² = (3√3 ÷ 2) × a².

Perimeter of a Regular Hexagon

The sum of all six equal outer sides of length a is simply given by Perimeter = 6a.

Radius of the Circumcircle

For any regular hexagon, the side length a is exactly equal to the radius of its circumscribed circle (R = a), due to the internal equilateral triangles.

Shortcut Trick

Ratio of inner radius (fountain) to outer radius (total) = 7 : (7 + 3.5) = 7 : 10.5 = 2 : 3.

Ratio of fountain area to total area = (2)² : (3)² = 4 : 9.

Area of grass bed in ratio units = Total area − Fountain area = 9 − 4 = 5.

Ratio of grass bed area to fountain area = 5 : 4.

∴ The correct answer is 5:4.

Alternate Method

r = 7 m

w = 3.5 m

R = 10.5 m

Fountain Grass Bed

Given:

Radius of the circular fountain (r) = 7 m

Width of the grass bed (w) = 3.5 m

Formula Used:

Area of a circle = πr²

Area of a circular ring = π(R² − r²), where R is the outer radius and r is the inner radius.

Calculations:

⇒ Outer radius of the grass bed (R) = r + w

⇒ R = 7 + 3.5 = 10.5 m

⇒ Area of the fountain = π × (7)² = 49π m²

⇒ Area of the grass bed = π × (R² − r²)

⇒ Area of the grass bed = π × (10.5² − 7²)

⇒ Area of the grass bed = π × (110.25 − 49) = 61.25π m²

⇒ Required Ratio = Area of grass bed ÷ Area of fountain

⇒ Required Ratio = 61.25π ÷ 49π

⇒ Required Ratio = 61.25 ÷ 49 = 6125 ÷ 4900

⇒ Simplifying the fraction, Required Ratio = 5 ÷ 4

∴ The correct answer is 5:4.

Additional Information

Area of a Circular Path

When a path of uniform width 'w' runs around a circular field of radius 'r', the area of the path can be directly calculated using the formula π × w × (2r + w).

Perimeter of a Circle

The total boundary length of a circular region is known as its circumference, which is calculated as 2πr or πd, where d is the diameter.

Ratio of Areas of Circles

The ratio of the areas of any two circles is always equal to the square of the ratio of their corresponding radii or diameters.

Shortcut Trick

For any uniform geometric prism, the volume is calculated by simply multiplying its base area by its height (or length).

Volume = Base Area × Height

Volume = 20 × 6 = 120 m³

∴ The correct answer is 120 m³.

Alternate Method

Given:

Area of the triangular base = 20 m²

Height of the tent (length of the prism) = 6 m

Formula Used:

Volume of a Prism = Base Area × Height

Base Area = 20 sq. m

Height = 6 m

Volume = Base Area × Height

Calculation:

⇒ Volume of the tent = Base Area × Height

⇒ Volume of the tent = 20 × 6

⇒ Volume of the tent = 120 m³

∴ The correct answer is 120 m³.

Additional Information

Volume of a Prism

For any prism (triangular, rectangular, etc.), the volume is calculated by multiplying the area of its base by its perpendicular height or length: V = Base Area × h.

Surface Area of a Prism

The lateral surface area of a prism is the perimeter of the base multiplied by the height: LSA = Base Perimeter × h. The total surface area is TSA = LSA + 2 × Base Area.

Volume of a Pyramid

Unlike a prism, the volume of a pyramid is one-third the product of its base area and height: V = (1 ÷ 3) × Base Area × h.

Shortcut Trick

Check the given options to find the line that has a slope of −1 and satisfies the point (2, 3).

The standard slope-intercept form is y = mx + c, where m is the slope.

Option 1: y = −x + 5 → Here, slope m = −1.

Substitute x = 2 into this equation: y = −2 + 5 = 3. This directly matches our given point (2, 3).

∴ The correct answer is y = −x + 5.

Alternate Method

Given:

Slope of the line (m) = −1

The line passes through the point (x₁, y₁) = (2, 3)

Formula Used:

The point-slope form of a linear equation is given by:

y − y₁ = m(x − x₁)

(2, 3) 5 5 2 3 0 X-axis Y-axis

y = −x + 5 Slope (m) = −1

Calculations:

⇒ Substitute the given point and slope into the formula:

⇒ y − 3 = −1(x − 2)

⇒ y − 3 = −x + 2

⇒ y = −x + 2 + 3

⇒ y = −x + 5

∴ The correct answer is y = −x + 5.

Additional Information

Slope-Intercept Form

The linear equation of a straight line can also be written directly as y = mx + c, where m is the slope and c is the y-intercept (the coordinate where the line crosses the y-axis).

Slope Formula

The slope m between any two valid points (x₁, y₁) and (x₂, y₂) on a line is calculated as m = (y₂ − y₁) ÷ (x₂ − x₁).

Parallel and Perpendicular Lines

Parallel lines always share the exact same slope value (m₁ = m₂). Conversely, two lines are perpendicular if their slopes are negative reciprocals of each other (m₁ × m₂ = −1).

Shortcut Trick

Express each term as a multiple of the simplest surd form.

√45 = √(9 × 5) = 3√5

√20 = √(4 × 5) = 2√5

Now, substitute the simplified terms back into the expression:

3√5 + 2√5 − √5 = (3 + 2 − 1)√5 = 4√5

∴ The correct answer is 4√5.

Alternate Method

Given:

Expression = √45 + √20 − √5

Method Used:

Factorize the numbers inside the square roots to extract the perfect square factors.

Calculations:

⇒ Expression = √(3 × 3 × 5) + √(2 × 2 × 5) − √5

⇒ Expression = 3√5 + 2√5 − 1√5

⇒ Taking √5 common from all the terms

⇒ Expression = (3 + 2 − 1)√5

⇒ Expression = 4√5

∴ The correct answer is 4√5.

Additional Information

Rules of Surds

√(a × b) = √a × √b. This property is strictly used to simplify square roots by factoring out perfect squares.

Addition and Subtraction of Surds

Only like surds (surds having the exact same irrational part) can be directly added or subtracted. For example: x√a + y√a = (x + y)√a.

BODMAS Rule

Always follow the correct order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction for successful simplification.

Shortcut Trick

The maximum number of distinct pairs that can be formed from 3 non-overlapping, non-touching circles is 3.

For each distinct pair of separate circles, a maximum of 4 common tangents (2 direct and 2 transverse) can be drawn.

To maximize the total number of distinct tangents, we assume no single line acts as a tangent to all three circles simultaneously.

Maximum distinct common tangents = 3 pairs × 4 tangents/pair = 12.

∴ The correct answer is 12 tangents.

Alternate Method

Given:

Three non-overlapping, non-touching circles are drawn in a plane.

Formula Used:

Maximum common tangents for 2 disjoint circles = 4

Number of pairs from n items = n(n − 1) ÷ 2

C1C2C34 distinct common tangents shown between C1 and C2Total for all 3 pairs = 3 × 4 = 12 distinct tangents

Calculations:

⇒ Number of given circles = 3

⇒ Number of possible distinct pairs of circles = 3 × 2 ÷ 2 = 3 pairs

⇒ Let the 3 circles be C₁, C₂, and C₃. The respective pairs are (C₁, C₂), (C₂, C₃), and (C₃, C₁).

⇒ For each pair of distinctly separated circles, we can draw exactly 4 common tangents (2 Direct Common Tangents + 2 Transverse Common Tangents).

⇒ To achieve the maximum total distinct tangents, we place the circles such that no tangent is shared between all three circles simultaneously.

⇒ Maximum distinct common tangents = 3 pairs × 4 tangents/pair = 12.

∴ The correct answer is 12 tangents.

Additional Information

Direct Common Tangent (DCT)

It is a common tangent that does not intersect the line segment joining the centers of the two circles. The formula for the length of a DCT is √(d² − (r₁ − r₂)²).

Transverse Common Tangent (TCT)

It is a common tangent that intersects the line segment joining the centers of the two circles. The formula for the length of a TCT is √(d² − (r₁ + r₂)²).

Number of Tangents Based on Circle Positions

Two completely separated circles have exactly 4 common tangents. Externally touching circles have 3. Intersecting circles have 2. Internally touching circles have 1. If one circle is completely inside the other, there are 0 common tangents.

Shortcut Trick

By Intersecting Chords Theorem, the product of the segments of intersecting chords are equal.

⇒ (2/5 × x) × (3/5 × x) = (5/7 × y) × (2/7 × y)

⇒ 6x² / 25 = 10y² / 49

⇒ x² / y² = (10 × 25) / (6 × 49)

∴ The correct answer is x²/y² = 125/147.

Alternate Method

Given:

AE : EB = 2 : 3 and AB = x

CE : ED = 5 : 2 and CD = y

Formula Used:

Intersecting Chords Theorem: If two chords AB and CD intersect at E, then AE × EB = CE × ED.

A B C D E

2x/5 3x/5 5y/7 2y/7

Calculation:

⇒ Let AE = 2k and EB = 3k.

⇒ Total length AB = 2k + 3k = 5k.

⇒ Given AB = x, so 5k = x ⇒ k = x/5.

⇒ Thus, AE = 2x/5 and EB = 3x/5.

⇒ Similarly, let CE = 5m and ED = 2m.

⇒ Total length CD = 5m + 2m = 7m.

⇒ Given CD = y, so 7m = y ⇒ m = y/7.

⇒ Thus, CE = 5y/7 and ED = 2y/7.

⇒ Applying the Intersecting Chords Theorem:

⇒ AE × EB = CE × ED

⇒ (2x/5) × (3x/5) = (5y/7) × (2y/7)

⇒ 6x² / 25 = 10y² / 49

⇒ x² / y² = (10 / 49) × (25 / 6)

⇒ x² / y² = 250 / 294

⇒ x² / y² = 125 / 147

∴ The correct answer is x²/y² = 125/147.

Additional Information

Intersecting Chords Theorem

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other. Formula: AE × EB = CE × ED

Secant-Secant Theorem

If two secant lines intersect outside a circle, the product of the outer segment and the whole secant is equal for both lines. Formula: PA × PB = PC × PD

Tangent-Secant Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the tangent length equals the product of the secant and its outer segment. Formula: PT² = PA × PB

Shortcut Trick

Area of minor segment = Area of a Quadrant − Area of the Right-Angled Triangle

⇒ Area = (πr² ÷ 4) − (r² ÷ 2)

⇒ Area = (π × 100 ÷ 4) − (100 ÷ 2)

⇒ Area = 25π − 50

∴ The correct answer is (25π−50) sq. cm.

Alternate Method

OAB10 cm10 cm90°Minor Segment

Given:

Radius of the circle (r) = 10 cm

Angle subtended at the center (θ) = 90°

Formula Used:

Area of minor segment = Area of sector − Area of triangle

Area of sector = (θ ÷ 360°) × πr²

Area of a right-angled triangle = (1/2) × base × height

Calculations:

⇒ Area of sector OAB = (90° ÷ 360°) × π × (10)²

⇒ Area of sector OAB = (1 ÷ 4) × 100π = 25π sq. cm

⇒ Area of triangle OAB = (1 ÷ 2) × 10 × 10

⇒ Area of triangle OAB = 50 sq. cm

⇒ Area of minor segment = (25π − 50) sq. cm

∴ The correct answer is (25π−50) sq. cm.

Additional Information

Area of a Major Segment

The Area of the Major Segment is calculated as Area of the Circle − Area of the Minor Segment.

Length of an Arc

The length of an arc subtending angle θ at the center is determined by (θ ÷ 360°) × 2πr.

Area of Triangle in a Circle

For any chord subtending angle θ at the center, the area of the triangle formed with the center is (1/2)r²sin(θ).

Shortcut Trick

Observe the relation between the numerator and denominator terms.

Each term in the denominator is exactly 5 times the corresponding term in the numerator.

Denominator = (5 × 0.05)³ + (5 × 0.01)³ = 5³ × (0.05³ + 0.01³)

Expression = 1 ÷ 5³ = 1 ÷ 125 = 0.008

∴ The correct answer is 0.008.

Alternate Method

Given:

Expression: (0.05³ + 0.01³) ÷ (0.25³ + 0.05³)

Calculations:

⇒ Numerator = 0.05³ + 0.01³

⇒ Denominator = 0.25³ + 0.05³

⇒ Rewrite the denominator by extracting 5 as a common multiple:

⇒ Denominator = (5 × 0.05)³ + (5 × 0.01)³

⇒ Denominator = 5³ × 0.05³ + 5³ × 0.01³

⇒ Denominator = 5³ × (0.05³ + 0.01³)

⇒ Substitute this back into the expression:

⇒ (0.05³ + 0.01³) ÷ [125 × (0.05³ + 0.01³)]

⇒ Cancel out the common terms from the numerator and denominator:

⇒ Expression = 1 ÷ 125

⇒ 1 ÷ 125 = 0.008

∴ The correct answer is 0.008.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction, dictating the correct sequence for evaluating mathematical expressions.

Laws of Exponents

When calculating the power of a product, distribute the exponent to each individual factor: (a × b)ⁿ = aⁿ × bⁿ.

Decimal Conversion Trick

To convert 1 ÷ 125 to a decimal quickly, multiply both the numerator and denominator by 8: (1 × 8) ÷ (125 × 8) = 8 ÷ 1000 = 0.008.

Shortcut Trick

Break the given decimal into its integer and fractional parts:

3.375 = 3 + 0.375

Recall the standard decimal conversion for eighths: 0.375 = 3/8

⇒ 3.375 = 3 + 3/8 = (3 × 8 + 3) ÷ 8

⇒ 3.375 = 27/8

∴ The correct answer is 27/8.

Alternate Method

Given:

Decimal value = 3.375

Method Used:

To convert a decimal to a fraction, write the entire number without the decimal point over 1 followed by as many zeros as there are decimal places, and then simplify.

Calculations:

⇒ Since 3.375 has 3 decimal places, we write it over 1000.

⇒ 3.375 = 3375 / 1000

⇒ To simplify, divide both the numerator and denominator by 25:

⇒ Numerator: 3375 ÷ 25 = 135

⇒ Denominator: 1000 ÷ 25 = 40

⇒ The fraction becomes 135 / 40.

⇒ Now, divide both by 5 to reduce it further:

⇒ Numerator: 135 ÷ 5 = 27

⇒ Denominator: 40 ÷ 5 = 8

⇒ The fraction in its simplest form is 27/8.

∴ The correct answer is 27/8.

Additional Information

Common Decimal Equivalents

Memorizing common base-8 fractions speeds up calculations in exams: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Proper vs Improper Fractions

A proper fraction has a numerator smaller than its denominator (e.g., 3/8). An improper fraction has a numerator greater than or equal to its denominator (e.g., 27/8). Mixed numbers can be formed directly from improper fractions.

Converting Repeating Decimals

For non-terminating but repeating decimals, place the repeating digits over an equal number of 9s instead of 10s. Example: 0.444... = 4/9, and 0.3636... = 36/99 = 4/11.

Shortcut Trick

Given, 15% of a = 25% of b

Directly equate the percentage values: 15 × a = 25 × b

Rearranging the terms, we get the ratio: a ÷ b = 25 ÷ 15

Simplifying the fraction gives: a : b = 5 : 3

∴ The correct answer is 5 : 3.

Alternate Method

Given:

15% of a = 25% of b

Visualising the Ratio

Value of a (5 Parts)

Value of b (3 Parts)

Ratio a : b = 5 : 3

Calculations:

⇒ (15 ÷ 100) × a = (25 ÷ 100) × b

⇒ 15 × a = 25 × b

⇒ a ÷ b = 25 ÷ 15

⇒ a ÷ b = 5 ÷ 3

⇒ a : b = 5 : 3

∴ The correct answer is 5 : 3.

Additional Information

Ratio and Proportion Basics

A ratio indicates how many times one number contains another. The ratio of x to y is mathematically represented as x : y or x ÷ y.

Inverse Relationship

If p × A = q × B, then the ratio of A to B is simply q : p. This happens because crossing over the terms yields A ÷ B = q ÷ p.

Percentage to Fraction

To convert a percentage into a fraction, divide the percentage value by 100. For example, 25% is equivalent to 25 ÷ 100, which simplifies to 1 ÷ 4.

Shortcut Trick

To find the number x to be subtracted from a, b, c, and d to make them proportional, use the direct formula:

x = (ad − bc) ÷ ((a + d) − (b + c))

Here, a = 7, b = 9, c = 8, d = 11.

x = (7 × 11 − 9 × 8) ÷ ((7 + 11) − (9 + 8))

x = (77 − 72) ÷ (18 − 17) = 5 ÷ 1 = 5

∴ The correct answer is 5.

Alternate Method

Given:

The numbers are 7, 9, 8, 11.

Formula Used:

If four numbers A, B, C, D are in proportion, then A ÷ B = C ÷ D.

Cross Multiplication for Proportion

7 − x

9 − x

8 − x

11 − x

=

(7 − x)(11 − x) = (9 − x)(8 − x)

Calculations:

Let the number to be subtracted be x.

⇒ (7 − x) ÷ (9 − x) = (8 − x) ÷ (11 − x)

⇒ (7 − x)(11 − x) = (9 − x)(8 − x)

⇒ 77 − 11x − 7x + x² = 72 − 9x − 8x + x²

⇒ 77 − 18x = 72 − 17x

⇒ 77 − 72 = 18x − 17x

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Continued Proportion

If three numbers a, b, c are in continued proportion, then a ÷ b = b ÷ c, which means b² = ac. Here, b is known as the mean proportional.

Componendo and Dividendo

If a ÷ b = c ÷ d, then according to this rule: (a + b) ÷ (a − b) = (c + d) ÷ (c − d). This is a highly useful tool for simplifying complex ratio equations.

Fourth Proportional

For any three given numbers a, b, and c, their fourth proportional is a number d such that a ÷ b = c ÷ d, giving the result d = (b × c) ÷ a.

Shortcut Trick

Convert the fractional part to a decimal directly: 5/8 = 0.625.

So, 2 ⁵/₈ = 2.625.

Given expression: 6.4 − (2.625 + 1.75)

⇒ 6.4 − 4.375 = 2.025

∴ The correct answer is 2.025.

Alternate Method

Given:

6.4 − (2 ⁵/₈ + 1.75)

Concept Used:

Follow the BODMAS rule to solve the expression in the brackets first.

Calculations:

⇒ Convert 2 ⁵/₈ to decimal: (2 × 8 + 5) ÷ 8 = 21 ÷ 8 = 2.625

⇒ Substitute the decimal value into the given expression:

⇒ 6.4 − (2.625 + 1.75)

⇒ Solve the addition inside the bracket:

⇒ 2.625 + 1.75 = 4.375

⇒ Subtract the calculated sum from 6.4:

⇒ 6.400 − 4.375 = 2.025

∴ The correct answer is 2.025.

Additional Information

BODMAS Rule

It determines the correct sequence of operations in any mathematical expression: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Common Fraction to Decimal Conversions

Memorizing eighth-based fractions saves time in calculations: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Decimal Addition & Subtraction

Always align the decimal points vertically before calculating. You can append trailing zeros to numbers (e.g., writing 6.4 as 6.400) to balance the number of decimal places without changing the value.

Shortcut Trick

Profit is distributed in the ratio of (Capital × Time).

Alok's total capital = (90,000 × 6) + (1,20,000 × 6) = 12,60,000

Deepak's time in business = 12 − 3 (joined late) − 3 (left early) = 6 months.

Deepak's total capital = 1,20,000 × 6 = 7,20,000

Ratio = 12,60,000 : 7,20,000 = 7 : 4

∴ The correct answer is 7:4.

Alternate Method

0m 3m 6m 9m 12m Alok 90k × 6m 120k × 6m Deepak 120k × 6m

Given:

Alok's initial investment = ₹90,000

Alok's increased investment after 6 months = ₹90,000 + ₹30,000 = ₹1,20,000

Deepak's investment = ₹1,20,000

Total time for business = 12 months

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Calculations:

⇒ Alok invested ₹90,000 for the first 6 months.

⇒ Alok invested ₹1,20,000 for the remaining 6 months.

⇒ Equivalent capital for Alok = (90,000 × 6) + (1,20,000 × 6)

⇒ Equivalent capital for Alok = 5,40,000 + 7,20,000 = 12,60,000

⇒ Deepak joined after 3 months and left 3 months before the end.

⇒ Deepak's total investment time = 12 − 3 − 3 = 6 months.

⇒ Equivalent capital for Deepak = 1,20,000 × 6 = 7,20,000

⇒ Profit sharing ratio (Alok : Deepak) = 12,60,000 : 7,20,000

⇒ Alok : Deepak = 126 : 72 = 7 : 4

∴ The correct answer is 7:4.

Additional Information

Joining Mid-way

If a partner joins the business after t months, their investment is considered active for (12 − t) months in an annual profit cycle.

Compound Ratio in Partnership

The profit is always distributed in the compound ratio of individual investments and time periods: (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Active and Sleeping Partner

An active partner is paid a fixed amount or a specific percentage of the profit for managing the operations, and the remaining profit is distributed based on their capital and time ratios.

Shortcut Trick

Teacher's Age = New Average + (Old Members × Increase in Average)

Old Members = 19 students

Increase in Average = 18 − 17 = 1 year

Teacher's Age = 18 + (19 × 1) = 37 years

∴ The correct answer is 37 years.

Alternate Method

Given:

Average age of 19 students = 17 years

New average age after teacher joins = 18 years

Formula Used:

Sum of values = Average × Number of values

323 years

19 students (19 × 17)

19 students Teacher

20 members total = 20 × 18 = 360 years

360 − 323 = 37

Calculations:

⇒ Total age of 19 students = 19 × 17 = 323 years

⇒ Number of members after teacher joins = 19 + 1 = 20 members

⇒ Total age of 20 members = 20 × 18 = 360 years

⇒ Teacher's age = Total age of 20 members − Total age of 19 students

⇒ Teacher's age = 360 − 323 = 37 years

∴ The correct answer is 37 years.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ with averages A₁ and A₂ are mixed, Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Exclusion of a Member

If a person leaves the group, Age of leaving person = Old Average − (New Total Members × Change in Average). Keep the positive or negative sign of the change intact.

Replacement of a Member

When a new person replaces an old one, Age of new person = Age of removed person + (Total Members × Increase in Average).

Shortcut Trick

Deviation of 30 employees from overall average = ₹72,000 − ₹62,000 = +₹10,000 each.

Total extra deviation for these 30 employees = 30 × 10,000 = +₹3,00,000.

To maintain the overall average, the remaining 50 employees must compensate by −₹3,00,000.

Average deviation for remaining employees = −₹3,00,000 ÷ 50 = −₹6,000.

Required average = ₹62,000 − ₹6,000 = ₹56,000.

∴ The correct answer is ₹56,000.

Alternate Method

Given:

Total number of employees = 80

Overall average salary = ₹62,000

Number of employees in the first group = 30

Average salary of the first group = ₹72,000

Total: 80 employees | Avg = ₹62,000

30 employees Avg: ₹72,000

50 employees Avg: ?

Formula Used:

Total Sum = Average × Number of Observations

Average of Remaining = Sum of Remaining ÷ Number of Remaining Observations

Calculations:

⇒ Total salary of 80 employees = 80 × 62000 = ₹49,60,000

⇒ Total salary of 30 employees = 30 × 72000 = ₹21,60,000

⇒ Remaining employees = 80 − 30 = 50

⇒ Total salary of remaining 50 employees = 49,60,000 − 21,60,000 = ₹28,00,000

⇒ Average salary of remaining employees = 28,00,000 ÷ 50 = ₹56,000

∴ The correct answer is ₹56,000.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, the combined average is given by: A = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Net Deviation Method

When checking against a mean average, the sum of deviations of all elements from that mean is always zero. This principle is extremely helpful for calculating missing values quickly without doing large multiplications.

Weighted Average Trick

Instead of using raw numbers, you can use the ratio of the group sizes (e.g., 30 : 50 = 3 : 5) in your formulas to further simplify and speed up calculation steps.

Shortcut Trick

In Compound Interest, if a sum becomes n times in t years, it becomes n^x times in xt years.

The sum becomes 4 times in 6 years.

We need to find the time taken for it to become 16 times.

Since 16 = 4² times, we multiply the time by 2.

Required time = 6 × 2 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

P 4P 16P

6 years

6 years

× 4

× 4

Total: 12 years

Given:

A principal sum P becomes 4P in 6 years at Compound Interest.

Formula Used:

A = P(1 + R/100)^T

Where A is the Amount, P is the Principal, R is the Rate of interest, and T is the Time in years.

Calculations:

Let the original sum be P.

According to the given condition, the amount becomes 4 times in 6 years:

⇒ 4P = P(1 + R/100)⁶

⇒ 4 = (1 + R/100)⁶

We need to find the time T when the amount becomes 16P:

⇒ 16P = P(1 + R/100)^T

⇒ 16 = (1 + R/100)^T

We can write 16 as 4²:

⇒ 4² = (1 + R/100)^T

Substitute the value of 4 from the first equation:

⇒ ((1 + R/100)⁶)² = (1 + R/100)^T

⇒ (1 + R/100)¹² = (1 + R/100)^T

By equating the powers on both sides, we get:

⇒ T = 12 years

∴ The correct answer is 12 years.

Additional Information

Compound Interest Multiplier Concept

In Compound Interest, the amount grows geometrically over equal periods. If a sum becomes n times in t years, it will become n² times in 2t years, n³ times in 3t years, and so on.

Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate is halved (R/2) and the total number of periods is doubled (2T) in the CI formula.

Difference Between SI and CI

The difference between SI and CI on a given sum P at a rate R% for 2 years is directly calculated as: Difference = P(R/100)².

Shortcut Trick

Given the ratio of Loss to Cost Price (CP) = 1 : 10.

Let the Loss be 1 unit and the Cost Price be 10 units.

Selling Price (SP) = Cost Price − Loss = 10 − 1 = 9 units.

The ratio of Selling Price to Cost Price = SP : CP = 9 : 10.

∴ The correct answer is 9:10.

Alternate Method

Given:

Ratio of Loss to Cost Price (CP) = 1 : 10

Formula Used:

Selling Price (SP) = Cost Price (CP) − Loss

SP = 9x Loss = x

Cost Price (CP) = 10x

Ratio of SP to CP = 9 : 10

Calculations:

⇒ Let the loss incurred be x and the Cost Price (CP) be 10x.

⇒ SP = CP − Loss

⇒ SP = 10x − x = 9x

⇒ Required Ratio = SP : CP

⇒ SP : CP = 9x : 10x

⇒ SP : CP = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Cost Price and Selling Price

Cost Price (CP) is the price at which the item is purchased by a seller, and Selling Price (SP) is the price at which it is sold to the customer.

Profit and Loss Logic

When Selling Price > Cost Price, there is a Profit calculated as (SP − CP). When Cost Price > Selling Price, there is a Loss calculated as (CP − SP).

Percentage Calculation

Both profit and loss are always evaluated against the base Cost Price (CP). Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100.

Shortcut Trick

The ratio of Cost Price (CP) to Marked Price (MP) can be directly calculated by the formula:

CP / MP = (100 − Discount%) / (100 + Profit%)

⇒ CP / 150 = (100 − 10) / (100 + 25)

⇒ CP / 150 = 90 / 125 = 18 / 25

⇒ CP = (18 / 25) × 150 = 18 × 6 = ₹108

∴ The correct answer is ₹108.

Alternate Method

Given:

Marked Price (MP) = ₹150, Discount = 10%, Profit = 25%

Formula Used:

Selling Price (SP) = MP × (100 − Discount%) ÷ 100

Cost Price (CP) = SP × 100 ÷ (100 + Profit%)

CP ₹108

SP ₹135

MP ₹150

+25% Profit

-10% Discount

Cost Price = ₹108

Calculations:

⇒ SP = 150 × (100 − 10) ÷ 100

⇒ SP = 150 × 90 ÷ 100 = ₹135

⇒ CP = 135 × 100 ÷ (100 + 25)

⇒ CP = 135 × 100 ÷ 125

⇒ CP = 135 × 4 ÷ 5

⇒ CP = 27 × 4 = ₹108

∴ The correct answer is ₹108.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). The resulting value is the Selling Price: SP = MP − Discount.

Profit and Loss Calculation

Profit or Loss is always calculated on the Cost Price (CP). Thus, SP = CP × (100 + Profit%) ÷ 100.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is (d₁ + d₂ − d₁×d₂÷100)%.