Quantitative Aptitude PYQs

Shortcut Trick

LCM of given time intervals (3 mins and 4 mins) = 12 mins.

Water filled by the man in 12 mins = 4 × (12 ÷ 3) = 16 liters.

Water filled by the woman in 12 mins = 3 × (12 ÷ 4) = 9 liters.

Total water filled by both together in 12 mins = 16 + 9 = 25 liters.

Time required to fill 100 liters = (100 ÷ 25) × 12 = 48 min.

∴ The correct answer is 48 min.

Alternate Method

Given:

Man pours 4 liters every 3 minutes.

Woman pours 3 liters every 4 minutes.

Total capacity of the tank = 100 liters.

Formula Used:

Total Time = Total Work ÷ Combined Efficiency

Calculations:

⇒ Time taken by the man to independently fill 100 liters = (100 ÷ 4) × 3 = 75 min

⇒ Time taken by the woman to independently fill 100 liters = (100 ÷ 3) × 4 = 400/3 min

⇒ Combined efficiency = 4/3 + 3/4

⇒ Combined efficiency = (16 + 9) ÷ 12 = 25/12 liters/min

⇒ Total time required to fill the tank = Total Work ÷ Combined efficiency

⇒ Total time = 100 ÷ (25/12)

⇒ Total time = (100 × 12) ÷ 25

⇒ Total time = 4 × 12 = 48 min

∴ The correct answer is 48 min.

Additional Information

Pipes & Cisterns Formula

If an inlet pipe fills a tank in x hours, its 1 hour work is 1/x. For two pipes taking x and y hours independently, their combined 1 hour work is given by (1/x + 1/y).

Rate of Work Principle

Total Work = Efficiency (Rate of Work) × Time. When multiple entities work together continuously without interruption, their individual volumetric efficiencies per unit time can simply be added to find the overall combined rate.

Alternate Working (Pipes)

If two individuals or pipes work on an alternate cycle basis, calculate the total volume filled in one complete cycle (e.g., 2 minutes) to deduce how many full cycles are required before calculating the remainder of the work.

Shortcut Trick

Total sum = Average × 3 = 100 × 3 = 300.

Total ratio units = 2 + 3 + 5 = 10 units.

Value of 1 unit = 300 ÷ 10 = 30.

Difference in ratio between largest (5) and smallest (2) = 3 units.

Required difference = 3 × 30 = 90.

∴ The correct answer is 90.

Alternate Method

Given:

Ratio of the three numbers = 2 : 3 : 5

Average of the numbers = 100

Formula Used:

Sum of elements = Average × Number of elements

Total Sum = 100 × 3 = 300 (10 units)

2 units 3 units 5 units

Value = 60 Value = 90 Value = 150

Difference = 150 − 60 = 90

Calculation:

⇒ Let the three numbers be 2x, 3x, and 5x.

⇒ Sum of the numbers = 2x + 3x + 5x = 10x

⇒ Average = 10x ÷ 3 = 100

⇒ 10x = 300

⇒ x = 30

⇒ The smallest number = 2x = 2 × 30 = 60

⇒ The largest number = 5x = 5 × 30 = 150

⇒ Difference between the largest and smallest = 150 − 60 = 90

∴ The correct answer is 90.

Additional Information

Average (Mean) Concept

The total sum of all given entities in a group is always equal to the average multiplied by the number of entities: Sum = Average × N.

Ratio Distribution

To find individual values from a given ratio, you must first divide the total sum of the entities by the sum of their corresponding ratio units to determine the exact value of a single unit.

Direct Difference Shortcut

Whenever you need to find the difference or sum between quantities in a ratio, operate on their ratio unit values directly first, then multiply by the calculated base value of a single unit. This avoids the extra step of calculating both exact numerical values.

Shortcut Trick

Average CP per litre = (40 × 50 + 60 × 60) ÷ 100 = Rs. 56

Selling Price (SP) per litre = Rs. 70

Profit per litre = SP − CP = 70 − 56 = Rs. 14

Total profit on 20 litres sold = 20 × 14 = Rs. 280

∴ The correct answer is Rs. 280 profit.

Alternate Method

Given:

Quantity of Can 1 = 40 litres at Rs. 50/litre

Quantity of Can 2 = 60 litres at Rs. 60/litre

Quantity sold = 20 litres at Rs. 70/litre

Formula Used:

Average CP = Total Cost ÷ Total Quantity

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

CP of Can 1 CP of Can 2

₹50

₹60

₹56 Mean CP

2 Ratio (40L)

3 Ratio (60L)

Calculations:

⇒ Total quantity of mixture = 40 + 60 = 100 litres

⇒ Total CP of 100 litres = (40 × 50) + (60 × 60) = 2000 + 3600 = Rs. 5600

⇒ Average CP per litre = 5600 ÷ 100 = Rs. 56

⇒ CP of 20 litres of mixture = 20 × 56 = Rs. 1120

⇒ SP of 20 litres = 20 × 70 = Rs. 1400

⇒ Profit = 1400 − 1120 = Rs. 280

∴ The correct answer is Rs. 280 profit.

Additional Information

Alligation Method

It is a rule that enables finding the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired average price.

Rule of Alligation

Quantity of Cheaper / Quantity of Dearer = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper)

Profit and Loss Basic Rules

Profit occurs when Selling Price > Cost Price. Profit = SP − CP. Loss occurs when Cost Price > Selling Price. Loss = CP − SP.

Shortcut Trick

The problem directly asks for the standard double-angle trigonometric identity.

According to the standard identity: cos2A = cos²A − sin²A.

Since cos²A = 1 − sin²A, substituting this directly gives:

cos2A = (1 − sin²A) − sin²A = 1 − 2sin²A

∴ The correct answer is 1−2sin²A.

Alternate Method

Given:

We need to find the equivalent formula for cos2A based on the options.

Formula Used:

Standard trigonometric identities:

cos(X + Y) = cosX × cosY − sinX × sinY

sin²X + cos²X = 1

Calculations:

⇒ Let X = A and Y = A in the cosine addition formula.

⇒ cos(A + A) = cosA × cosA − sinA × sinA

⇒ cos2A = cos²A − sin²A

⇒ From the Pythagorean identity, we know that cos²A = 1 − sin²A.

⇒ Substitute this value of cos²A into the cos2A equation:

⇒ cos2A = (1 − sin²A) − sin²A

⇒ cos2A = 1 − 2sin²A

∴ The correct answer is 1−2sin²A.

Additional Information

Double Angle Formulas for Sine and Tangent

sin2A = 2sinA × cosA = (2tanA) ÷ (1 + tan²A)

Other Forms of cos2A

cos2A = 2cos²A − 1 = (1 − tan²A) ÷ (1 + tan²A)

Pythagorean Trigonometric Identities

sin²A + cos²A = 1, sec²A − tan²A = 1, csc²A − cot²A = 1

Shortcut Trick

A straight line must pass through both given points. We can find the correct equation by substituting the given coordinates into the options.

Test point (1, 2) in the options:

⇒ Option 1 gives y = 3(1) − 1 = 2 (Valid equation)

⇒ Option 3 gives y = 4(1) − 2 = 2 (Valid equation)

Now test point (3, 8) in the valid options: Option 1 gives y = 3(3) − 1 = 8 (Valid) whereas Option 3 gives y = 4(3) − 2 = 10 (Invalid).

∴ The correct answer is y = 3x − 1.

Alternate Method

Given:

Two points on the line are (x₁, y₁) = (1, 2) and (x₂, y₂) = (3, 8).

Formula Used:

The slope (m) of a line passing through two points is: m = (y₂ − y₁) ÷ (x₂ − x₁)

The equation of a straight line passing through a point (x₁, y₁) is: y − y₁ = m(x − x₁)

X Y

1 3 2 8

A(1, 2)

B(3, 8) y = 3x − 1

Calculations:

⇒ Slope m = (8 − 2) ÷ (3 − 1)

⇒ m = 6 ÷ 2 = 3

⇒ Now, substitute m = 3 and the point (1, 2) into the point-slope formula:

⇒ y − 2 = 3 × (x − 1)

⇒ y − 2 = 3x − 3

⇒ y = 3x − 3 + 2

⇒ y = 3x − 1

∴ The correct answer is y = 3x − 1.

Additional Information

Slope-Intercept Form

The equation of a straight line can be written directly as y = mx + c, where 'm' represents the slope and 'c' represents the y-intercept.

Point-Slope Form

The linear equation for a line with slope 'm' passing through a specific coordinate (x₁, y₁) is given by y − y₁ = m(x − x₁).

Collinear Points

Three or more points are considered collinear if they lie on the same straight line, which implies that the slope between any two pairs of these points is exactly equal.

Intercept Form

The equation of a line making intercepts 'a' and 'b' on the x-axis and y-axis respectively is written as (x ÷ a) + (y ÷ b) = 1.

Shortcut Trick

If sin A = cos B, then A + B = 90°.

⇒ (3x + 10°) + (2x − 5°) = 90°

⇒ 5x + 5° = 90°

⇒ 5x = 85°

⇒ x = 17°

∴ The correct answer is 17°.

Alternate Method

A B C

3x+10°

2x−5° sin A = BC/AC = cos C

Given:

⇒ sin(3x + 10°) = cos(2x − 5°)

Formula Used:

⇒ cos θ = sin(90° − θ)

Calculations:

⇒ sin(3x + 10°) = sin(90° − (2x − 5°))

Comparing the angles on both sides:

⇒ 3x + 10° = 90° − 2x + 5°

⇒ 3x + 10° = 95° − 2x

⇒ 3x + 2x = 95° − 10°

⇒ 5x = 85°

⇒ x = 17°

∴ The correct answer is 17°.

Additional Information

Complementary Angle Relations

Two angles are complementary if their sum is 90°. The relations are: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Identities

The fundamental identity is sin²θ + cos²θ = 1. Others include sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1.

Equality of Sine and Cosine

For acute angles A and B, if sin A = cos B, then A + B = 90°. This geometric property stems from the two non-right angles in a right-angled triangle.

Shortcut Trick

For any convex polygon with 'n' sides, the number of triangles formed by drawing all possible diagonals from a single vertex is given directly by the formula (n − 2).

An octagon has 8 sides, so n = 8.

Number of triangles = 8 − 2 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

A polygon with 8 sides (Octagon).

Concept Used:

When drawing diagonals from one vertex of a polygon to all other possible non-adjacent vertices, the polygon gets divided completely into non-overlapping triangles.

Vertex

1 2 3 4 5 6

n = 8 sides 5 diagonals 6 Triangles

Calculations:

⇒ Total vertices in an octagon = 8

⇒ If we choose one specific vertex, we cannot draw a diagonal to itself or to the 2 vertices adjacent to it.

⇒ Number of valid diagonals from one vertex = 8 − 3 = 5 diagonals.

⇒ Each newly drawn diagonal divides the remaining area, essentially adding one more triangle to the total count.

⇒ Total number of triangles formed = Number of diagonals + 1

⇒ Total number of triangles = 5 + 1 = 6

∴ The correct answer is 6.

Additional Information

Total Number of Diagonals

The total number of diagonals that can be drawn in a convex polygon with n sides from all vertices is n(n − 3) ÷ 2.

Sum of Interior Angles

The sum of the interior angles of any n-sided polygon is calculated using the formula (n − 2) × 180°.

Sum of Exterior Angles

The sum of the exterior angles (taken one at each vertex) of any convex polygon is always exactly 360°.

Interior Angle of a Regular Polygon

For a regular polygon with equal sides and equal angles, each interior angle measures [(n − 2) × 180°] ÷ n.

Shortcut Trick

In a right-angled triangle, the circumcenter always lies exactly at the midpoint of the hypotenuse.

The given points P(0, 0), Q(8, 0), and R(0, 6) form a right triangle at the origin P(0, 0).

The hypotenuse is the line segment joining the other two vertices Q(8, 0) and R(0, 6).

Midpoint of hypotenuse = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2) = ((8 + 0) ÷ 2, (0 + 6) ÷ 2) = (4, 3).

∴ The correct answer is (4, 3).

Alternate Method

P(0, 0) Q(8, 0) R(0, 6) C(4, 3) 8 units 6 units 10 units

Given:

Vertices of the triangle are P(0, 0), Q(8, 0), and R(0, 6).

Formula Used:

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint formula: M(x, y) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Calculations:

⇒ Length of PQ = √((8 − 0)² + (0 − 0)²) = √(64) = 8

⇒ Length of PR = √((0 − 0)² + (6 − 0)²) = √(36) = 6

⇒ Length of QR = √((0 − 8)² + (6 − 0)²) = √(64 + 36) = √100 = 10

⇒ Here, PQ² + PR² = 8² + 6² = 64 + 36 = 100.

⇒ Since PQ² + PR² = QR², triangle PQR satisfies Pythagoras' theorem and is right-angled at P.

⇒ For a right-angled triangle, the circumcenter is located exactly at the midpoint of its hypotenuse (QR).

⇒ Circumcenter coordinates = Midpoint of QR = ((8 + 0) ÷ 2, (0 + 6) ÷ 2)

⇒ Circumcenter = (4, 3)

∴ The correct answer is (4, 3).

Additional Information

Circumcenter of Different Triangles

For an acute triangle, the circumcenter lies strictly inside the triangle. For an obtuse triangle, it lies outside. For a right-angled triangle, it lies exactly at the midpoint of the hypotenuse.

Orthocenter in Right Triangles

The orthocenter (intersection of altitudes) of a right-angled triangle is exactly at the vertex containing the right angle. In this question, the orthocenter is directly at P(0, 0).

Circumradius

The radius of the circumcircle (circumradius) for a right-angled triangle is exactly half the length of the hypotenuse. Here, Circumradius = 10 ÷ 2 = 5 units.

Shortcut Trick

Factorize the numbers inside the roots into perfect squares:

√12 = √(4 × 3) = 2√3

√75 = √(25 × 3) = 5√3

Add the similar surds: 2√3 + 5√3 = 7√3

∴ The correct answer is 7√3.

Alternate Method

Given:

Expression = √12 + √75

Concept Used:

√(a × b) = √a × √b (Identify the largest perfect square factor)

Calculations:

⇒ √12 + √75

⇒ √(4 × 3) + √(25 × 3)

⇒ √4 × √3 + √25 × √3

⇒ 2×√3 + 5×√3

⇒ (2 + 5)√3

⇒ 7√3

∴ The correct answer is 7√3.

Additional Information

Properties of Surds

Multiplication: √(ab) = √a × √b | Division: √(a/b) = √a ÷ √b.

Like Surds

Surds that have the exact same irrational factor are known as like surds (e.g., 2√3 and 5√3). Only like surds can be directly added or subtracted together.

BODMAS in Simplification

Always simplify the contents under roots or brackets before performing addition or subtraction when solving surd equations.

Shortcut Trick

At the same time of day, the ratio of height to shadow length remains constant due to direct proportion.

Height of tree / Shadow of tree = Height of person / Shadow of person

Height / 20 = 5 / 4

Height = 20 × (5 / 4) = 25

∴ The correct answer is 25 feet.

Alternate Method

Given:

Height of the person = 5 feet

Length of the person's shadow = 4 feet

Length of the tree's shadow = 20 feet

Formula Used:

At a specific time, objects and their shadows form similar triangles, meaning their corresponding sides are proportional:

H₁ / S₁ = H₂ / S₂

20 ft x ft Tree

4 ft 5 ft Person

Similar Triangles Formed by Sun Rays

Calculations:

Let the height of the tree be x feet.

⇒ x / 20 = 5 / 4

⇒ x = (5 × 20) / 4

⇒ x = 100 / 4

⇒ x = 25

∴ The correct answer is 25 feet.

Additional Information

Similar Triangles Rule

When two triangles are similar, their corresponding angles are equal, and the ratio of their corresponding sides is proportional: a/x = b/y = c/z.

Direct Proportion

Two values are in direct proportion if an increase in one leads to a constant proportional increase in the other. In this case, taller objects cast proportionally longer shadows.

Sun Angle Constant

At any specific moment, the angle of elevation of the sun is identical for all objects in the same area. This identical angle causes objects and their shadows to naturally form similar right-angled triangles.

Shortcut Trick

Apply the Componendo and Dividendo rule twice for quick results:

⇒ √(7+x) / √(7−x) = (4 + 1) / (4 − 1) = 5 / 3

Squaring both sides gives: (7 + x) / (7 − x) = 25 / 9

Applying the rule again: 7 / x = (25 + 9) / (25 − 9) = 34 / 16 = 17 / 8

⇒ x = (7 × 8) / 17 = 56/17

∴ The correct answer is 56/17.

Alternate Method

Given:

[ √(7+x) + √(7−x) ] / [ √(7+x) − √(7−x) ] = 4

Formula Used:

Componendo and Dividendo: If a / b = c / d, then (a + b) / (a − b) = (c + d) / (c − d)

Calculations:

Applying Componendo and Dividendo to the given equation:

⇒ [ (√(7+x) + √(7−x)) + (√(7+x) − √(7−x)) ] / [ (√(7+x) + √(7−x)) − (√(7+x) − √(7−x)) ] = (4 + 1) / (4 − 1)

⇒ 2√(7+x) / 2√(7−x) = 5 / 3

⇒ √(7+x) / √(7−x) = 5 / 3

Squaring both sides:

⇒ (7 + x) / (7 − x) = 25 / 9

Cross-multiplying to solve for x:

⇒ 9 × (7 + x) = 25 × (7 − x)

⇒ 63 + 9x = 175 − 25x

⇒ 9x + 25x = 175 − 63

⇒ 34x = 112

⇒ x = 112 / 34

⇒ x = 56 / 17

∴ The correct answer is 56/17.

Additional Information

Componendo and Dividendo

If a / b = c / d, then (a + b) / (a − b) = (c + d) / (c − d). This property is extremely useful for simplifying rational expressions involving conjugate pairs.

Invertendo Property

If two ratios are equal, i.e., a / b = c / d, then their inverse ratios are also equal: b / a = d / c.

Alternendo Property

If a / b = c / d, then by alternating the consequent of the first and the antecedent of the second, a / c = b / d.

Shortcut Trick

For quarterly compounding, Rate = 20% ÷ 4 = 5%, and Time = 9 months = 3 quarters.

Using the 3-period multiplying ratio method (3 : 3 : 1) for compound interest:

Interest = 3 × (5% of 16000) + 3 × (5% of 800) + 1 × (5% of 40)

Interest = 3 × 800 + 3 × 40 + 1 × 2

Interest = 2400 + 120 + 2 = 2522

∴ The correct answer is ₹2522.

Alternate Method

Given:

Principal (P) = ₹16000

Rate (R) = 20% per annum

Time (T) = 9 months

Formula Used:

When interest is compounded quarterly:

Amount = P(1 + r/100)ⁿ

Where, r = R/4 and n = T × 4 (in years) or total number of quarters.

P = ₹16000 5% 5% 5% Quarter 1 Quarter 2 Quarter 3 800 800 800 5% 40 5% 80 5% 2 Total CI = 2400 + 120 + 2 = ₹2522

Calculations:

⇒ Effective rate per quarter (r) = 20% ÷ 4 = 5%

⇒ Number of quarters (n) = 9 months ÷ 3 months = 3

⇒ Amount = 16000 × (1 + 5/100)³

⇒ Amount = 16000 × (105/100)³ = 16000 × (21/20)³

⇒ Amount = 16000 × (9261 / 8000)

⇒ Amount = 2 × 9261 = ₹18522

⇒ Compound Interest = Amount − Principal

⇒ Compound Interest = 18522 − 16000 = ₹2522

∴ The correct answer is ₹2522.

Additional Information

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2T in years).

Effective Interest Rate

The effective annual rate of interest for a nominal rate R compounded n times a year is calculated using the formula: (1 + R/n)ⁿ − 1.

Simple vs Compound Interest

For the first period, SI and CI are strictly equal. From the second period onwards, CI is always greater than SI due to the interest earned on accumulated past interest.

Difference Between CI and SI

The absolute difference between CI and SI for 2 time periods is P(R/100)² and for 3 time periods it is P(R/100)²(3 + R/100).

Shortcut Trick

Total work remains constant, so we apply the MDH concept: M₁ × D₁ = M₂ × D₂

⇒ 10 Men × 12 days = 20 Women × 16 days

⇒ 120M = 320W

⇒ M / W = 320 / 120 = 8 / 3

∴ The correct answer is 8:3.

Alternate Method

Given:

20 women complete the work in 16 days.

10 men complete the work in 12 days.

Formula Used:

Total Work = Total Days × Efficiency

Time taken by 1 person = Number of persons × Days taken

Calculations:

⇒ Time taken by 1 woman to complete the work = 20 × 16 = 320 days

⇒ Time taken by 1 man to complete the work = 10 × 12 = 120 days

⇒ Efficiency of 1 Man = 8 units/day

⇒ Efficiency of 1 Woman = 3 units/day

⇒ Ratio of efficiency of a Man to a Woman = 8 : 3

∴ The correct answer is 8:3.

Additional Information

MDH Rule

It states that (M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂, where M = Men, D = Days, H = Hours, and W = Work.

Work and Efficiency

Work done is the product of time and efficiency. For a constant amount of work, the time taken is always inversely proportional to the efficiency.

Pipes & Cisterns

Follows the exact same principles as Time & Work, but an outlet pipe or leak is considered to have negative efficiency.

Shortcut Trick

Integers include positive numbers, negative numbers, and zero (..., −2, −1, 0, 1, 2, ...).

Whole numbers are all non-negative integers (0, 1, 2, 3, ...).

Every whole number is automatically an integer.

From the given options, only 1 is a non-negative integer.

∴ The correct answer is 1.

Alternate Method

Given:

Options: −5, 1, 3/5, √2

Definition Used:

⇒ Integers (Z) = {..., −2, −1, 0, 1, 2, ...}

⇒ Whole Numbers (W) = {0, 1, 2, 3, ...}

Calculations:

⇒ Option 1: −5 is a negative integer, hence it is not a whole number.

⇒ Option 2: 1 is an integer and it is non-negative, hence it is also a whole number.

⇒ Option 3: 3/5 is a rational fraction, so it is neither an integer nor a whole number.

⇒ Option 4: √2 is an irrational number, so it is neither an integer nor a whole number.

∴ The correct answer is 1.

Additional Information

Natural Numbers

These are counting numbers starting from 1 (1, 2, 3, ...). Zero is not included in this set.

Rational Numbers

Numbers that can be expressed as p/q, where p and q are integers and q is not zero.

Irrational Numbers

Numbers that cannot be expressed as simple fractions. Their decimal expansions are non-terminating and non-repeating (e.g., √2, π).

Real Numbers

The complete set of numbers formed by combining both rational and irrational numbers.

Shortcut Trick

When equal distances are covered at two different speeds x and y, Average Speed = (2 × x × y) ÷ (x + y)

Given, x = 12 km/h and y = 3 km/h.

Average Speed = (2 × 12 × 3) ÷ (12 + 3)

Average Speed = 72 ÷ 15 = 4.8 km/h

∴ The correct answer is 4.8 km/h.

Alternate Method

Given:

Speed for the first half of the journey (v₁) = 12 km/h

Speed for the second half of the journey (v₂) = 3 km/h

Let the total distance be 2D, so each half of the journey is D.

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

    Distance = D Distance = D Speed = 12 km/h Speed = 3 km/h Total Distance = 2D

Calculations:

⇒ Time taken for the first half (t₁) = D ÷ 12

⇒ Time taken for the second half (t₂) = D ÷ 3

⇒ Total Time = (D ÷ 12) + (D ÷ 3)

⇒ Total Time = (D + 4D) ÷ 12 = 5D ÷ 12

⇒ Average Speed = 2D ÷ (5D ÷ 12)

⇒ Average Speed = (2D × 12) ÷ 5D

⇒ Average Speed = 24 ÷ 5 = 4.8 km/h

∴ The correct answer is 4.8 km/h.

Additional Information

Average Speed for Unequal Distances

When different distances are covered at different speeds, Average Speed = Total Distance ÷ Total Time.

Average Speed for 3 Equal Distances

When three equal distances are covered at speeds x, y, and z, Average Speed = 3xyz ÷ (xy + yz + zx).

Relative Speed Concept

For two bodies moving in the same direction at speeds a and b (where a > b), Relative Speed = a − b. For bodies moving in opposite directions, Relative Speed = a + b.

Shortcut Trick

When a hemisphere is inscribed in a cube, its diameter is equal to the cube's side length.

Radius of hemisphere (r) = 6 ÷ 2 = 3 cm

Volume = (2/3) × π × (3)³ = 18 × 3.1416 = 56.55 cm³

∴ The correct answer is 56.55 cm³.

Alternate Method

Given:

Side of the cube (a) = 6 cm

Formula Used:

Radius of inscribed hemisphere (r) = a ÷ 2

Volume of hemisphere = (2/3)πr³

r = 3 cm

r = 3 cm

Side = 6 cm Hemisphere inscribed in a Cube

Calculations:

⇒ Diameter of the hemisphere = Side of the cube = 6 cm

⇒ Radius (r) = 6 ÷ 2 = 3 cm

⇒ Volume of hemisphere = (2/3) × π × (3)³

⇒ Volume = (2/3) × 3.14159 × 27

⇒ Volume = 18 × 3.14159

⇒ Volume ≈ 56.55 cm³

∴ The correct answer is 56.55 cm³.

Additional Information

Hemisphere Formulas

Curved Surface Area (CSA) = 2πr² | Total Surface Area (TSA) = 3πr² | Volume = (2/3)πr³

Sphere Formulas

Surface Area = 4πr² | Volume = (4/3)πr³

Cube Formulas

Total Surface Area = 6a² | Volume = a³ | Diagonal = a√3

Shortcut Trick

Mileage = Total distance traveled ÷ Total fuel consumed

Mileage = 13.5 ÷ 1.8

Multiply numerator and denominator by 10 to remove decimals:

Mileage = 135 ÷ 18 = 7.5 km/liter

∴ The correct answer is 7.5 km.

Alternate Method

Given:

Total distance traveled = 13.5 km

Total fuel used = 1.8 liters

Formula Used:

Distance traveled per liter = Total distance ÷ Total fuel used

1.8 Liters travels 13.5 km 1 Liter travels ? km

Calculation:

⇒ Distance per liter = 13.5 ÷ 1.8

⇒ Distance per liter = 135 ÷ 18

⇒ Distance per liter = 15 ÷ 2

⇒ Distance per liter = 7.5 km

∴ The correct answer is 7.5 km.

Additional Information

Unitary Method

It is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Fuel Economy / Mileage

Fuel economy is the relationship between the distance traveled and the amount of fuel consumed by the vehicle, expressed as kilometers per liter (km/L).

Converting Decimals to Fractions

When dividing two numbers with the same number of decimal places, the decimals can be removed by multiplying both the numerator and the denominator by multiples of 10. For example, 13.5 ÷ 1.8 = 135 ÷ 18.

Shortcut Trick

Remaining profit after C's 20% share = 80% of Rs. 2,50,000 = Rs. 2,00,000.

Ratio of capital invested by A and B = 40% : 60% = 2 : 3.

B's share in profit = (3 ÷ 5) × 2,00,000 = Rs. 1,20,000.

∴ The correct answer is Rs. 1,20,000.

Alternate Method

Total Profit = Rs. 2,50,000  C (20%) Rs. 50,000  A (40% Capital) Rs. 80,000  B (60% Capital) Rs. 1,20,000 B's Share = Rs. 1,20,000

Given:

⇒ Total profit = Rs. 2,50,000

⇒ C's profit share percentage = 20%

⇒ A's capital contribution = 40%

⇒ B's capital contribution = 100% − 40% = 60%

Formula Used:

When the time period is the same, Profit Ratio = Capital Ratio

Calculations:

⇒ C's profit share = 20% of 2,50,000 = (20 ÷ 100) × 2,50,000 = Rs. 50,000

⇒ Remaining profit available for A and B = 2,50,000 − 50,000 = Rs. 2,00,000

⇒ Ratio of capitals of A and B = 40 : 60 = 2 : 3

⇒ B's share = (3 ÷ (2 + 3)) × Remaining profit

⇒ B's share = (3 ÷ 5) × 2,00,000

⇒ B's share = 3 × 40,000 = Rs. 1,20,000

∴ The correct answer is Rs. 1,20,000.

Additional Information

Simple Partnership

When all partners invest their capital for the same time period, the profit is distributed exactly in the ratio of their invested capitals. Profit Ratio = C₁ : C₂.

Compound Partnership

When partners invest for different time periods, the profit is distributed in the ratio of the product of their capital and time. Profit Ratio = C₁×T₁ : C₂×T₂.

Active vs Sleeping Partner

An active partner manages the business operations and is often entitled to a fixed salary or a fixed percentage of the total profit, after which the remaining profit is shared according to capital contribution.

Shortcut Trick

Using the cross-multiplication method for addition/subtraction in ratios:

Difference in cross-products of ratios = (4 × 1) − (3 × 1) = 1 unit.

Difference in cross-products of changes = (1 × 6) − (1 × −8) = 14.

⇒ 1 unit = 14.

Total initial fruits = (4 + 3) units = 7 × 14 = 98.

∴ The correct answer is 98 fruits.

Alternate Method

Given:

Initial ratio of apples and oranges = 4 : 3

Apples removed = 8, Oranges added = 6

Final ratio = 1 : 1

Initial Fruits Apples: 4x | Oranges: 3x −8 App, +6 Org Final Fruits 4x − 8 = 3x + 6 x = 14 → 7x = 98

Calculation:

Let the initial number of apples be 4x and oranges be 3x.

⇒ Total fruits initially = 4x + 3x = 7x

⇒ New number of apples = 4x − 8

⇒ New number of oranges = 3x + 6

⇒ As per the new ratio given, the quantities become equal:

⇒ (4x − 8) / (3x + 6) = 1 / 1

⇒ 4x − 8 = 3x + 6

⇒ 4x − 3x = 6 + 8

⇒ x = 14

⇒ Total original fruits = 7 × 14 = 98

∴ The correct answer is 98 fruits.

Additional Information

Cross-Multiplication Method

When an initial ratio a:b changes to c:d after quantities x and y are added/subtracted, the base unit is found using: (a×d − b×c) units = (c×y − d×x). Pay strict attention to the signs of x and y based on addition or removal.

Means and Extremes Property

For any standard proportion a:b :: c:d, the fundamental rule states that the product of the means equals the product of the extremes. Formally, b × c = a × d.

Discrete Quantity Ratios

When components of a system are given in an integer ratio a:b, the total original quantity must always be a multiple of the sum of the ratio parts (a + b) since items like fruits cannot exist in fractions.

Shortcut Trick

Area of triangle = 45% of 200 = 90 m².

Height = (2 × Area) ÷ Base

⇒ h = (2 × 90) ÷ 10

⇒ h = 180 ÷ 10 = 18 m.

∴ The correct answer is 18 m.

Alternate Method

Given:

Area of square window = 200 m²

Base of triangular window (b) = 10 m

Area of triangle = 45% of square window's area

Formula Used:

Area of triangle = ½ × base × height

    Square Area = 200 m²   45%   Area = 90 m²    h = ?     Base = 10 m

Calculations:

⇒ Area of triangular window = (45 ÷ 100) × 200

⇒ Area = 90 m²

Substituting the values in the formula:

⇒ ½ × 10 × h = 90

⇒ 5 × h = 90

⇒ h = 90 ÷ 5

⇒ h = 18 m

∴ The correct answer is 18 m.

Additional Information

Area of Equilateral Triangle

Area = (√3 ÷ 4) × side². In an equilateral triangle, all sides and internal angles are equal.

Area of Square

Area = side². Alternatively, it can be computed using the diagonal as Area = ½ × diagonal².

Right-Angled Triangle

Area = ½ × base × perpendicular. It strictly follows Pythagoras Theorem: hypotenuse² = base² + perpendicular².