Quantitative Aptitude PYQs

Shortcut Trick

Sum of x and y = 30 × 2 = 60

Sum of x, y and z = 40 × 3 = 120

Value of z = Sum of (x, y, z) − Sum of (x, y) = 120 − 60 = 60

Sum of x, y and 2z = (x + y) + 2z = 60 + (2 × 60) = 180

New Average of these 3 terms = 180 ÷ 3 = 60

∴ The correct answer is 60.

Alternate Method

Given:

⇒ Average of x and y = 30

⇒ (x + y) ÷ 2 = 30

⇒ x + y = 60

⇒ Average of x, y and z = 40

⇒ (x + y + z) ÷ 3 = 40

⇒ x + y + z = 120

x + y = 60 Sum of x, y (2 × 30) x + y = 60 z = 60 Sum of x, y, z (3 × 40) x + y = 60 z = 60 z = 60 Sum of x, y, 2z = 180 New Average = 180 ÷ 3 = 60

Calculations:

⇒ Substituting x + y = 60 into the second sum equation:

⇒ 60 + z = 120

⇒ z = 60

⇒ We are required to find the average of x, y and 2z. The terms are 3 in number.

⇒ Sum = x + y + 2z

⇒ Sum = (x + y) + 2 × z

⇒ Sum = 60 + 2 × 60

⇒ Sum = 60 + 120 = 180

⇒ Required Average = Sum ÷ 3

⇒ Required Average = 180 ÷ 3 = 60

∴ The correct answer is 60.

Additional Information

Average Basic Formula

Average = Sum of all observations ÷ Total number of observations. To find the total sum when the average is known, simply multiply the average by the number of observations.

Combined Average

If a group A has n₁ items with average A₁ and group B has n₂ items with average A₂, the Combined Average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Effect of Arithmetic Operations on Average

If a constant k is added, subtracted, multiplied, or divided to each individual term in a dataset, the new average of the dataset will correspondingly increase, decrease, be multiplied, or be divided by k.

Shortcut Trick

Cost Price (CP) of the mixture = SP ÷ 1.1 = 55 ÷ 1.1 = ₹50/kg.

Apply Alligation on CP: (60 − 50) : (50 − 40)

Ratio = 10 : 10 = 1 : 1.

∴ The correct answer is 1:1.

Alternate Method

Given:

Cost price of Variety 1 = ₹40/kg

Cost price of Variety 2 = ₹60/kg

Selling price of the mixture = ₹55/kg

Profit = 10%

Formula Used:

CP = [100 ÷ (100 + Profit%)] × SP

Calculation:

⇒ Mean CP of the mixture = [100 ÷ (100 + 10)] × 55

⇒ Mean CP = (100 ÷ 110) × 55 = ₹50/kg

₹40 Variety 1  ₹60 Variety 2  ₹50 Mean CP  10 60 − 50  10 50 − 40 Ratio = 1 : 1

⇒ By the rule of alligation, the ratio of their quantities is:

⇒ Quantity of Variety 1 : Quantity of Variety 2 = (60 − 50) : (50 − 40)

⇒ Ratio = 10 : 10 = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Rule of Alligation

It states that: Quantity of Cheaper ÷ Quantity of Dearer = (CP of Dearer − Mean Price) ÷ (Mean Price − CP of Cheaper).

Weighted Average Method

The mean price can also be verified using weights: Mean Price = (w₁ × x₁ + w₂ × x₂) ÷ (w₁ + w₂).

Cost Price from Selling Price

When profit percentage is given, always calculate the Cost Price before applying alligation, as the components being mixed are at their cost prices.

Shortcut Trick

Given sinA = P/H = 1/√10, so Perpendicular (P) = 1 and Hypotenuse (H) = √10.

Using Pythagorean theorem, Base B = √(H² − P²) = √(10 − 1) = 3.

We need to find: tanA + secA = (P/B) + (H/B) = (P + H)/B.

Directly substitute the values: (1 + √10)/3.

∴ The correct answer is (1+√10)/3.

Alternate Method

Given:

sinA = 1/√10

Let Perpendicular (P) = 1 and Hypotenuse (H) = √10

Formula Used:

Pythagoras Theorem: H² = P² + B²

tanA = P/B and secA = H/B

A B C H = √10 P = 1 B = 3

Calculations:

⇒ (√10)² = 1² + B²

⇒ 10 = 1 + B²

⇒ B² = 10 − 1 = 9

⇒ B = √9 = 3

⇒ tanA = P/B = 1/3

⇒ secA = H/B = √10/3

⇒ tanA + secA = 1/3 + √10/3 = (1+√10)/3

∴ The correct answer is (1+√10)/3.

Additional Information

Trigonometric Ratios

In a right-angled triangle, sine is Perpendicular ÷ Hypotenuse, cosine is Base ÷ Hypotenuse, and tangent is Perpendicular ÷ Base.

Pythagoras Theorem

It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides: H² = P² + B².

Basic Trigonometric Identities

The primary trigonometric identity connects sine and cosine as sin²θ + cos²θ = 1. Similarly, sec²θ − tan²θ = 1.

Shortcut Trick

Value of sin 90° = 1

Value of cos 60° = 1/2

Substitute these values into the expression:

⇒ (1)² + (1/2)²

⇒ 1 + 1/4 = 5/4

∴ The correct answer is 5/4.

Alternate Method

Given:

Expression: sin²90° + cos²60°

Formula Used:

sin 90° = 1

cos 60° = 1/2

Calculations:

⇒ sin²90° + cos²60°

⇒ (1)² + (1/2)²

⇒ 1 + 1/4

⇒ (4 + 1) ÷ 4

⇒ 5/4

∴ The correct answer is 5/4.

Additional Information

Standard Trigonometric Values

sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2

cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2

Trigonometric Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Complementary Angles

sin(90° − θ) = cosθ

cos(90° − θ) = sinθ

Shortcut Trick

To find the x-intercept of any linear equation, simply set y = 0.

⇒ 5x + 2(0) = 10

⇒ 5x = 10

⇒ x = 2

∴ The correct answer is 2.

Alternate Method

X Y O y-intercept (0, 5) x-intercept (2, 0) 5x + 2y = 10 1 3 4 5

Given:

Equation of the line: 5x + 2y = 10

Formula Used:

The standard intercept form of a linear equation is x/a + y/b = 1, where 'a' represents the x-intercept and 'b' represents the y-intercept.

Calculations:

Divide the entire equation by 10 to make the Right Hand Side (RHS) equal to 1:

⇒ (5x / 10) + (2y / 10) = 10 / 10

⇒ x / 2 + y / 5 = 1

Comparing this with the standard intercept form (x/a + y/b = 1):

⇒ a = 2

⇒ b = 5

The x-intercept 'a' is 2.

∴ The correct answer is 2.

Additional Information

General Form Conversion

For a linear equation in the general form ax + by + c = 0, the x-intercept is given by −c/a and the y-intercept is −c/b.

Slope-Intercept Form

The equation can also be expressed as y = mx + c, where 'm' is the slope (gradient) of the line and 'c' directly represents the y-intercept.

Intercept Properties

The x-intercept is specifically the absolute point where the line crosses the horizontal x-axis (meaning y = 0). Conversely, the y-intercept is where it crosses the vertical y-axis (meaning x = 0).

Shortcut Trick

If sin(A) = cos(B), it directly implies A + B = 90°.

If cos(A) = sin(C), it directly implies A + C = 90°.

Adding both equations gives: 2A + B + C = 180°.

We are already given that A + B + C = 90°.

Subtracting the given equation from our sum gives: A = 180° − 90° = 90°.

∴ The correct answer is 90°.

Alternate Method

Given:

sin(A) = cos(B)

cos(A) = sin(C)

A + B + C = 90°

Concept Used:

Trigonometric complementary angle identities:

cos(θ) = sin(90° − θ)

sin(θ) = cos(90° − θ)

Calculations:

⇒ sin(A) = cos(B)

⇒ sin(A) = sin(90° − B)

⇒ A = 90° − B

⇒ A + B = 90°     --- (1)

⇒ cos(A) = sin(C)

⇒ cos(A) = cos(90° − C)

⇒ A = 90° − C

⇒ A + C = 90°     --- (2)

Adding equations (1) and (2):

⇒ (A + B) + (A + C) = 90° + 90°

⇒ 2A + B + C = 180°

⇒ A + (A + B + C) = 180°

Substitute the given value of A + B + C = 90°:

⇒ A + 90° = 180°

⇒ A = 180° − 90°

⇒ A = 90°

∴ The correct answer is 90°.

Additional Information

Complementary Angles

Two angles are complementary if their sum is exactly 90°. In right-angled triangles, the two acute angles are always complementary.

Trigonometric Co-functions

Sine and cosine are co-functions, meaning sin(θ) = cos(90° − θ) and cos(θ) = sin(90° − θ).

Tangent and Cotangent

Similarly, tangent and cotangent are co-functions where tan(θ) = cot(90° − θ) and cot(θ) = tan(90° − θ).

Shortcut Trick

For any polygon with n sides, the sum of its interior angles is simply calculated as (n − 2) × 180°.

Here, the polygon has 20 sides, so n = 20.

⇒ Sum = (20 − 2) × 180° = 18 × 180° = 3240°.

∴ The correct answer is 3240°.

Alternate Method

Given:

Number of sides of the polygon, n = 20

Formula Used:

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

  Generic Polygon (Triangles = n − 2)  Sum of Interior Angles = (n − 2) × 180° For 20-sided: n = 20

Calculations:

⇒ Sum of interior angles = (20 − 2) × 180°

⇒ Sum of interior angles = 18 × 180°

⇒ Sum of interior angles = 3240°

∴ The correct answer is 3240°.

Additional Information

Sum of Exterior Angles

The sum of all exterior angles of any convex polygon is always exactly 360°, regardless of how many sides the polygon has.

Each Interior Angle

If the polygon is regular (all sides and angles are equal), each interior angle can be found by dividing the total sum by n: [(n − 2) × 180°] ÷ n.

Number of Diagonals

The total number of diagonals that can be drawn in an n-sided polygon is given by the direct formula: n × (n − 3) ÷ 2.

Shortcut Trick

The given points A(0, 0), B(4, 0), and C(0, 3) lie on the axes, forming a right-angled triangle with the right angle at the origin A(0, 0).

A point equidistant from all three vertices of a triangle is known as its circumcenter.

For any right-angled triangle, the circumcenter always lies exactly at the midpoint of its hypotenuse.

The hypotenuse connects the points B(4, 0) and C(0, 3).

Coordinates of P = ((4 + 0) ÷ 2, (0 + 3) ÷ 2) = (2, 1.5).

∴ The correct answer is (2, 1.5).

Alternate Method

Given:

Vertices of the triangle are A(0, 0), B(4, 0), and C(0, 3).

Let the coordinates of point P be (x, y).

Point P is equidistant from all three vertices, meaning PA = PB = PC.

Formula Used:

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

      A(0, 0)  B(4, 0)  C(0, 3)  P(2, 1.5)

Calculation:

⇒ Equating PA² = PB²

⇒ (x − 0)² + (y − 0)² = (x − 4)² + (y − 0)²

⇒ x² + y² = x² − 8x + 16 + y²

⇒ 8x = 16

⇒ x = 2

⇒ Similarly, evaluating PA² = PC²

⇒ (x − 0)² + (y − 0)² = (x − 0)² + (y − 3)²

⇒ x² + y² = x² + y² − 6y + 9

⇒ 6y = 9

⇒ y = 1.5

⇒ The coordinates of point P are (2, 1.5).

∴ The correct answer is (2, 1.5).

Additional Information

Circumcenter of a Right-Angled Triangle

In any right-angled triangle, the circumcenter is precisely the midpoint of its hypotenuse. The length of the circumradius is equal to half the length of the hypotenuse.

Centroid of a Triangle

The centroid is the point of intersection of the three medians. For vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the coordinates are derived as ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3).

Circumcenter Position based on Triangle Type

The location of the circumcenter varies distinctively: it lies strictly inside an acute triangle, exactly on the midpoint of the hypotenuse for a right-angled triangle, and entirely outside an obtuse triangle.

Shortcut Trick

Combine the numerator and denominator under a single square root: √( (5 + 2√6) / (5 − 2√6) )

Rationalize the fraction inside the root by multiplying both numerator and denominator by the conjugate (5 + 2√6).

The denominator becomes 5² − (2√6)² = 25 − 24 = 1.

The numerator becomes (5 + 2√6)².

The expression directly simplifies to √( (5 + 2√6)² / 1 ) = 5 + 2√6.

∴ The correct answer is 5 + 2√6.

Alternate Method

Given:

Expression: √(5 + 2√6) / √(5 − 2√6)

Formula Used:

(a + b)² = a² + b² + 2ab

(a − b)² = a² + b² − 2ab

(a + b)(a − b) = a² − b²

Calculations:

⇒ First, express the terms inside the square roots as perfect squares.

⇒ 5 + 2√6 = 3 + 2 + 2√(3 × 2) = (√3)² + (√2)² + 2(√3)(√2) = (√3 + √2)²

⇒ 5 − 2√6 = 3 + 2 − 2√(3 × 2) = (√3)² + (√2)² − 2(√3)(√2) = (√3 − √2)²

⇒ Substitute these back into the original expression:

⇒ √(√3 + √2)² / √(√3 − √2)²

⇒ Simplify the square roots:

⇒ (√3 + √2) / (√3 − √2)

⇒ Rationalize the denominator by multiplying the numerator and denominator by the conjugate (√3 + √2):

⇒ [(√3 + √2) × (√3 + √2)] / [(√3 − √2) × (√3 + √2)]

⇒ Numerator: (√3 + √2)² = (√3)² + (√2)² + 2(√3)(√2) = 3 + 2 + 2√6 = 5 + 2√6

⇒ Denominator: (√3)² − (√2)² = 3 − 2 = 1

⇒ Final value = (5 + 2√6) / 1 = 5 + 2√6

∴ The correct answer is 5 + 2√6.

Additional Information

Rationalization of Surds

Rationalization is the process of removing surds from a denominator by multiplying both the numerator and denominator by its conjugate. The conjugate of a + √b is a − √b.

Perfect Squares Involving Surds

Expressions of the form a + 2√b can frequently be rewritten as a perfect square (√x + √y)² if two numbers x and y can be found such that x + y = a and xy = b.

Quotient Property of Radicals

The quotient property states that √x ÷ √y = √(x/y). This allows combining separate roots in a fraction into a single root for easier algebraic simplification.

Shortcut Trick

In a right-angled triangle, the altitude drawn to the hypotenuse is given by the direct formula: Altitude = (Perpendicular × Base) ÷ Hypotenuse.

Hypotenuse AC = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Altitude BD = (AB × BC) ÷ AC = (6 × 8) ÷ 10 = 4.8 cm.

∴ The correct answer is 4.8 cm.

Alternate Method

Given:

Legs of right-angled triangle AB = 6 cm and BC = 8 cm.

 A B C D 6 cm 8 cm h = ?     10 cm

Formula Used:

Area of a triangle = 1/2 × Base × Height

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

Calculations:

⇒ By Pythagoras theorem, AC² = AB² + BC²

⇒ AC = √(6² + 8²) = √(36 + 64) = √100 = 10 cm

⇒ Area of ∆ABC using legs as base and height = 1/2 × AB × BC

⇒ Area = 1/2 × 6 × 8 = 24 cm²

⇒ We can also express Area of ∆ABC using the hypotenuse as the base: Area = 1/2 × AC × BD

⇒ 24 = 1/2 × 10 × BD

⇒ 24 = 5 × BD

⇒ BD = 24 ÷ 5 = 4.8 cm

∴ The correct answer is 4.8 cm.

Additional Information

Altitude to Hypotenuse Area Rule

The product of the legs of a right triangle is always equal to the product of the hypotenuse and the altitude drawn to it (Leg₁ × Leg₂ = Hypotenuse × Altitude).

Similar Triangles Theorem

Drawing an altitude to the hypotenuse of a right triangle divides it into two smaller triangles that are mathematically similar to each other and to the original parent triangle.

Geometric Mean Theorem

The altitude drawn to the hypotenuse acts as the geometric mean of the lengths of the two smaller segments it divides the hypotenuse into: h² = Segment₁ × Segment₂.

Shortcut Trick

If a principal becomes n times in t years at Compound Interest, the rate is R = (n^1/t − 1) × 100%.

Given n = 3 and t = 3 years.

R = (3^1/3 − 1) × 100%

Knowing 3^1/3 ≈ 1.4422

R = (1.4422 − 1) × 100% = 44.22% ≈ 44.2%

∴ The correct answer is 44.2%.

Alternate Method

Given:

Amount (A) = 3P

Time (t) = 3 years

Formula Used:

A = P × (1 + R/100)^t

       Start Year 1 Year 2 Year 3 P 3P  Time = 3 Years, CI Rate = R% A = P × (1 + R/100)³ = 3P

Calculations:

⇒ 3P = P × (1 + R/100)³

⇒ 3 = (1 + R/100)³

⇒ 1 + R/100 = 3^1/3

⇒ 1 + R/100 ≈ 1.4422

⇒ R/100 = 1.4422 − 1

⇒ R/100 = 0.4422

⇒ R = 0.4422 × 100 = 44.22%

⇒ R ≈ 44.2%

∴ The correct answer is 44.2%.

Additional Information

Compound Interest Formula

Amount (A) = P × (1 + R/100)^t, where P is the Principal, R is the Rate of interest per compounding period, and t is the Time period.

SI vs CI for 2 Years

The exact difference between compound interest and simple interest for 2 years on the same principal is computed as: Difference = P × (R/100)².

Rule of 72

To quickly estimate the number of years required to double a sum of money at compound interest, divide 72 by the annual interest rate (t ≈ 72 ÷ R).

Shortcut Trick

A fraction in its lowest terms represents a terminating decimal if its denominator has only 2 and/or 5 as prime factors.

Checking the options:

Denominator of 3/8 is 8, and its prime factorization is 8 = 2³.

Since the only prime factor is 2, the fraction 3/8 will have a terminating decimal representation.

∴ The correct answer is 3/8.

Alternate Method

Given:

Fractions provided in the options: 3/8, 1/6, 2/7, 4/9

Method Used:

We can manually divide the numerator by the denominator to observe the decimal sequence.

Calculations:

⇒ 3 ÷ 8 = 0.375 (This stops after three decimal places, meaning it terminates)

⇒ 1 ÷ 6 = 0.16666... (The digit 6 repeats endlessly)

⇒ 2 ÷ 7 = 0.285714285714... (The sequence 285714 repeats endlessly)

⇒ 4 ÷ 9 = 0.44444... (The digit 4 repeats endlessly)

⇒ Out of all options, only 3/8 gives a finite and exact decimal output.

∴ The correct answer is 3/8.

Additional Information

Terminating Decimal

A decimal number that contains a finite number of digits after the decimal point. This happens precisely when the fraction's denominator (in simplest form) is of the form 2^m × 5ⁿ, where m and n are non-negative integers.

Non-Terminating Repeating Decimal

A decimal that continues endlessly but features a repeating pattern of digits (e.g., 1/3 = 0.333...). This happens when the denominator has prime factors other than 2 or 5 (like 3, 7, 11, etc.).

Irrational Numbers

Numbers that have non-terminating and non-repeating decimal expansions (e.g., √2, √3, π). These cannot be expressed as a simple fraction of two integers.

Shortcut Trick

When two objects move in the same direction, their relative speed is the difference of their individual speeds.

Relative speed = 5 − 4 = 1 km/h.

Distance gap after 2 hours = Relative speed × Time = 1 × 2 = 2 km.

∴ The correct answer is 2 km.

Alternate Method

      Start    Ajay (4 km/h) → 8 km     Vijay (5 km/h) → 10 km     Gap = 2 km

Given:

Ajay's speed = 4 km/h

Vijay's speed = 5 km/h

Time duration = 2 hours

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Distance covered by Ajay in 2 hours = 4 × 2 = 8 km

⇒ Distance covered by Vijay in 2 hours = 5 × 2 = 10 km

⇒ Since they are walking in the same direction, the distance between them is the difference of distances covered.

⇒ Distance between them = 10 − 8 = 2 km

∴ The correct answer is 2 km.

Additional Information

Relative Speed in Same Direction

When two bodies move in the same direction, their relative speed is calculated as the difference between their individual speeds. Relative speed = S₁ − S₂.

Relative Speed in Opposite Direction

When two bodies move in opposite directions, their relative speed is the sum of their individual speeds. Relative speed = S₁ + S₂.

Distance Formula

Distance is defined as the product of speed and time. The formula used is D = S × T, maintaining corresponding units like km/h and hours.

Shortcut Trick

The surface area of a sphere is proportional to the square of its radius (r²).

When a dimension increases by x%, the percentage increase in area is found using the successive percentage formula: x + x + (x × x) ÷ 100

Given the increase in radius, x = 10%

Percentage increase in surface area = 10 + 10 + (10 × 10) ÷ 100 = 20 + 1 = 21%

∴ The correct answer is 21%.

Alternate Method

Given:

Increase in the radius of the spherical balloon = 10%

Formula Used:

Surface Area of a sphere = 4πr²

r Initial Area = 4πr²    1.1r New Area = 1.21 × 4πr²   +10% Radius Surface Area Increase = 21%

Calculations:

⇒ Let the initial radius of the sphere be r.

⇒ Initial Surface Area (A₁) = 4πr²

⇒ New radius after 10% increase = r + (10 ÷ 100)r = 1.1r

⇒ New Surface Area (A₂) = 4π(1.1r)²

⇒ A₂ = 4π(1.21r²) = 1.21(4πr²)

⇒ Increase in Area = A₂ − A₁ = 1.21(4πr²) − 1(4πr²) = 0.21(4πr²)

⇒ Percentage increase = (Increase in Area ÷ Initial Area) × 100

⇒ Percentage increase = [0.21(4πr²) ÷ 4πr²] × 100

⇒ Percentage increase = 0.21 × 100 = 21%

∴ The correct answer is 21%.

Additional Information

Volume Percentage Change

Since the volume of a sphere depends on the cube of its radius (r³), the percentage change in volume is calculated by applying the percentage change successively three times, or by evaluating [(1 + x/100)³ − 1] × 100.

Successive Percentage Formula

For two consecutive changes of a% and b%, the net percentage change is given by (a + b + ab ÷ 100)%. This is particularly useful for 2D mensuration problems involving area.

1D Dimension Change

Any one-dimensional measurement like circumference, diameter, or perimeter changes by the exact same percentage as the radius. Therefore, a 10% increase in radius will yield exactly a 10% increase in the circumference.

Shortcut Trick

Volume of 1 cubic meter = 1000 liters.

Bucket capacity expressed as a fraction: 6.25 = 25/4 liters.

Number of buckets needed = Total Volume ÷ Bucket capacity

Number of buckets = 1000 ÷ (25 / 4) = 1000 × (4 / 25) = 160.

∴ The correct answer is 160 buckets.

Alternate Method

Given:

Capacity of a single bucket = 6.25 liters

Volume of the tank = 1 m³

Formula Used:

1 m³ = 1000 liters

Number of buckets = Total volume of the tank ÷ Capacity of one bucket

 1 m³ Tank (1000 Liters)    × N buckets     Bucket 6.25 L  N = 1000 ÷ 6.25

Calculation:

⇒ Total volume of the tank = 1000 liters

⇒ Number of buckets = 1000 ÷ 6.25

⇒ Number of buckets = 1000 ÷ (625 ÷ 100)

⇒ Number of buckets = (1000 × 100) ÷ 625

⇒ Number of buckets = 100000 ÷ 625

⇒ Number of buckets = 160

∴ The correct answer is 160 buckets.

Additional Information

Volume Conversions

1 m³ = 1000 liters, and 1 liter = 1000 cm³. These standard unit conversions are critical for solving 3D mensuration and capacity problems effectively.

Capacity vs Volume

Volume measures the amount of 3-dimensional space occupied by an object, while capacity refers to the amount of liquid a hollow container can hold. Capacity is generally measured in liters (L) or milliliters (mL).

Fraction Shortcut for Decimals

Converting decimals to fractions often speeds up arithmetic division operations. For example, knowing that 0.25 is 1/4 means 6.25 can be written as 25/4. Dividing by a fraction is the same as multiplying by its reciprocal.

Shortcut Trick

Profit Ratio = (Capital × Time)

A : B : C = (75 × 12) : (45 × 12) : (60 × 9)

A : B : C = 900 : 540 : 540 = 5 : 3 : 3

Total profit ratio parts = 5 + 3 + 3 = 11

C's share = (3 ÷ 11) × 40000 = 120000 ÷ 11 ≈ 10909

∴ The correct answer is Rs. 10,909.

Alternate Method

Given:

A's investment = Rs. 75,000 for 12 months

B's investment = Rs. 45,000 for 12 months

C's investment = Rs. 60,000 for (12 − 3) = 9 months

Total profit = Rs. 40,000

Partner A Rs. 75k × 12 months Partner B Rs. 45k × 12 months Partner C Rs. 60k × 9 months Profit Ratio: A : B : C = 5 : 3 : 3 0m 3m 12m

Formula Used:

Profit Ratio = (Capital_A × Time_A) : (Capital_B × Time_B) : (Capital_C × Time_C)

Calculations:

⇒ Profit Ratio = (75000 × 12) : (45000 × 12) : (60000 × 9)

⇒ Profit Ratio = 900000 : 540000 : 540000

⇒ Simplifying the ratio by dividing with 180000,

⇒ Profit Ratio = 5 : 3 : 3

⇒ Total ratio parts = 5 + 3 + 3 = 11 parts

⇒ C's share = (3 ÷ 11) × 40000

⇒ C's share = 120000 ÷ 11

⇒ C's share ≈ Rs. 10909.09

∴ The correct answer is Rs. 10,909.

Additional Information

Compound Partnership

When partners invest capital for different time durations, the profit is divided in the ratio of the product of their capital and the respective time period.

Working and Sleeping Partners

A working partner actively manages the business and usually gets a fixed salary from the profit. A sleeping partner only invests capital and receives profit proportional to their investment.

Joining Mid-way

If a partner joins the business after 't' months in an annual partnership, their active investment duration is calculated as (12 − t) months.

Shortcut Trick

Difference in ratio parts = 6 − 5 = 1 part.

Given that the difference is 11, so 1 part = 11.

First number = 6 parts = 6 × 11 = 66.

Second number = 5 parts = 5 × 11 = 55.

∴ The correct answer is 66 and 55.

Alternate Method

Given:

Ratio of two numbers = 6:5

Difference between the numbers = 11

First Number (6 parts) Second Number (5 parts) Diff = 11 (1 part) 6 parts = 66 | 5 parts = 55

Calculation:

Let the two numbers be 6x and 5x.

⇒ Difference between the numbers = 6x − 5x = x

⇒ According to the problem, x = 11

⇒ First number = 6x = 6 × 11 = 66

⇒ Second number = 5x = 5 × 11 = 55

∴ The correct answer is 66 and 55.

Additional Information

Ratio & Proportion

A ratio indicates how many times one number contains another. For two numbers a and b, the ratio is a:b, which can also be expressed as a fraction a/b.

Duplicate Ratio

The duplicate ratio of two numbers a and b is the ratio of their squares, which is written as a² : b².

Compound Ratio

When two or more ratios are multiplied together term by term, the resulting ratio is known as a compound ratio. For a:b and c:d, it is ac:bd.

Shortcut Trick

Area of the rectangular sheet = 72 × 50 = 3600 m².

Since the area of the square is equal to the rectangle, Side of the square = √3600 = 60 m.

Perimeter of the square plot = 4 × Side = 4 × 60 = 240 m.

∴ The correct answer is 240 m.

Alternate Method

Given:

Length of the rectangular sheet (L) = 72 m

Breadth of the rectangular sheet (B) = 50 m

Area of square plot = Area of rectangular sheet

Formula Used:

Area of rectangle = L × B

Area of square = a² (where a is the side length)

Perimeter of square = 4a

   72 m 50 m Area = 3600 m²  a = 60 m a = 60 m Area = 3600 m² Perimeter = 240 m  Equal Area

Calculations:

⇒ Area of rectangular sheet = 72 × 50 = 3600 m²

⇒ Area of square plot = Area of rectangular sheet = 3600 m²

⇒ a² = 3600

⇒ a = √3600 = 60 m

⇒ Perimeter of the square plot = 4 × 60 = 240 m

∴ The correct answer is 240 m.

Additional Information

Properties of a Rectangle

Opposite sides are parallel and equal, and each angle measures 90°. Its diagonals are equal in length and bisect each other. Perimeter = 2(L + B).

Properties of a Square

All four sides are equal, and each interior angle is 90°. The diagonals are equal, perpendicular to each other, and bisect opposite angles. Diagonal length = a√2.

Area Formulas for Quadrilaterals

Area of Rectangle = Length × Breadth. Area of Square = Side² or (Diagonal)² ÷ 2. Area of Rhombus = (d₁ × d₂) ÷ 2.

Shortcut Trick

For any 2D geometrical figure, Area ∝ (Side)².

Since the area increases by 44%, the new area becomes 144% of the original, which corresponds to a multiplying factor of 1.44.

The multiplying factor for the side length will be the square root of the area factor: √1.44 = 1.2.

A multiplier of 1.2 means the side length has increased by 0.2, which is exactly 20%.

∴ The correct answer is 20%.

Alternate Method

Area = A Side = a Area = 1.44A Side = 1.2a Area +44% Side +20%

Given:

The area of a regular (equilateral) triangle increases by 44%.

Formula Used:

Area of an equilateral triangle = √3 ÷ 4 × a², where a is the side length.

Calculations:

⇒ Let the initial side length of the triangle be a.

⇒ Initial Area (A₁) = √3 ÷ 4 × a²

⇒ Final Area (A₂) = A₁ + 44% of A₁ = 1.44 × A₁

⇒ Let the final side length of the triangle be a'.

⇒ √3 ÷ 4 × (a')² = 1.44 × (√3 ÷ 4 × a²)

⇒ By cancelling out the common terms (√3 ÷ 4) on both sides:

⇒ (a')² = 1.44 × a²

⇒ Taking the square root on both sides:

⇒ a' = 1.2 × a

⇒ Increase in side length = a' − a = 1.2a − a = 0.2a

⇒ Percentage increase in side length = (0.2a ÷ a) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Successive Percentage Change

When a 2D figure has its dimensions increased uniformly by x%, the total percentage change in its area is given by the formula: Area % Change = x + x + (x² ÷ 100). For x = 20%, Area Change = 20 + 20 + 400÷100 = 44%.

Area of Similar Figures

The ratio of the areas of any two similar regular geometric figures is directly proportional to the square of the ratio of their corresponding side lengths.

Properties of a Regular Triangle

A regular triangle is an equilateral triangle where all three sides are equal, all interior angles are 60°, and its height is determined by the formula: h = √3 ÷ 2 × side.

Shortcut Trick

Since the percentages are taken on the same base number, we can subtract them directly.

Difference in percentage = 20% − 10% = 10%

This 10% of the number corresponds to the value 120.

Number = (120 ÷ 10) × 100 = 1200

∴ The correct answer is 1200.

Alternate Method

    0% 10% 20% 100%  10% = 120  Total Number = ?

Given:

⇒ The difference between 20% of a number and 10% of the same number is 120.

Calculations:

⇒ Let the required number be x.

⇒ 20% of x − 10% of x = 120

⇒ (20 ÷ 100)x − (10 ÷ 100)x = 120

⇒ (10 ÷ 100)x = 120

⇒ 0.1x = 120

⇒ x = 120 × 10

⇒ x = 1200

∴ The correct answer is 1200.

Additional Information

Percentage Change

Percentage change is evaluated as: ((New Value − Old Value) ÷ Old Value) × 100.

Base of Percentage

Percentages can be added or subtracted directly only if they are calculated on the exact same base number.

Successive Percentage

The net percentage change after two successive changes of a% and b% is: (a + b + (a × b) ÷ 100)%.