Quantitative Aptitude PYQs

Shortcut Trick

Area of a circular path outside a circle = π × w × (2r + w)

Area = (22 ÷ 7) × 1.5 × (2 × 14 + 1.5)

Area = (33 ÷ 7) × 29.5 = 973.5 ÷ 7 ≈ 139.07 m²

∴ The correct answer is 139 m².

Alternate Method

O

14 m

1.5 m

R = 15.5 m

Given:

⇒ Radius of the circular park, r = 14 m

⇒ Width of the path built around it, w = 1.5 m

Formula Used:

⇒ Area of the path = Area of the outer circle − Area of the inner circle

⇒ Area of a circle = πR²

Calculations:

⇒ Outer radius, R = r + w

⇒ R = 14 + 1.5 = 15.5 m

⇒ Area of the path = πR² − πr²

⇒ Area = π × (15.5² − 14²)

⇒ Area = (22 ÷ 7) × (15.5 + 14) × (15.5 − 14)

⇒ Area = (22 ÷ 7) × 29.5 × 1.5

⇒ Area = (33 ÷ 7) × 29.5

⇒ Area = 973.5 ÷ 7 ≈ 139.07 m²

⇒ The approximate area of the path is 139 m².

∴ The correct answer is 139 m².

Additional Information

Area of Circular Path Outside

When a path of width 'w' is constructed outside a circle of radius 'r', the Area is calculated as πw(2r + w).

Area of Circular Path Inside

When a path of width 'w' is constructed inside a circle of radius 'r', the Area is calculated as πw(2r − w).

Area of a Circle

The total space occupied by a circular region is denoted by πr².

Shortcut Trick

To convert an angle from radians to degrees quickly, directly substitute π = 180°.

Given angle = 5π ÷ 4

⇒ Angle = (5 × 180°) ÷ 4

⇒ Angle = 5 × 45°

∴ The correct answer is 225°.

Alternate Method

Given:

Angle in radians = 5π ÷ 4

Formula Used:

Angle in degrees = Angle in radians × (180° ÷ π)

0°90°180°270°5π/4 = 225°225°

Calculations:

⇒ Angle in degrees = (5π ÷ 4) × (180° ÷ π)

⇒ Angle in degrees = (5 × 180°) ÷ 4

⇒ Angle in degrees = 900° ÷ 4

⇒ Angle in degrees = 225°

∴ The correct answer is 225°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply the given angle by (π ÷ 180°).

Radian Measure

A radian is the angle subtended at the center of a circle by an arc whose length is exactly equal to the radius of the circle.

Common Angle Conversions

π ÷ 2 = 90°, π ÷ 3 = 60°, π ÷ 4 = 45°, and π ÷ 6 = 30°.

Shortcut Trick

Rewrite the equation y = -x + 4 as x + y = 4.

For a point to lie on the line, the sum of its coordinates must be 4.

Check the options: (5, 4) → 9; (1, 2) → 3; (3, 6) → 9; (2, 2) → 4.

Only the point (2, 2) gives a sum of 4.

∴ The correct answer is (2, 2).

Alternate Method

Given:

Equation of the line: y = -x + 4

X Y O

y = -x + 4

(2, 2)

(5, 4)

(1, 2)

(3, 6)

Calculations:

⇒ Let us substitute the given coordinates (x, y) into the equation x + y = 4.

⇒ Option 1: For (2, 2), LHS = 2 + 2 = 4. This satisfies the equation.

⇒ Option 2: For (5, 4), LHS = 5 + 4 = 9 ≠ 4. This does not satisfy.

⇒ Option 3: For (1, 2), LHS = 1 + 2 = 3 ≠ 4. This does not satisfy.

⇒ Option 4: For (3, 6), LHS = 3 + 6 = 9 ≠ 4. This does not satisfy.

∴ The correct answer is (2, 2).

Additional Information

Slope-Intercept Form

The equation of a straight line is often written as y = mx + c, where m is the slope of the line and c is the y-intercept. For y = -x + 4, the slope is -1 and the y-intercept is 4.

Checking a Point on a Line

A coordinate point (x₁, y₁) lies on a specific line if and only if its coordinates satisfy the equation of that line.

Intercepts of a Line

The x-intercept is calculated by setting y = 0, giving (4, 0). The y-intercept is calculated by setting x = 0, giving (0, 4). These are the points where the line crosses the respective axes.

Shortcut Trick

The relationship between arc length (l), radius (r), and central angle (θ) in radians is directly calculated using the formula:

θ = l ÷ r

⇒ θ = 10 ÷ 5 = 2 radians

∴ The correct answer is 2 radians.

Alternate Method

θ = ? r = 5 cm r = 5 cm Arc length (l) = 10 cm

Center O

Given:

Arc length of the sector (l) = 10 cm

Radius of the circle (r) = 5 cm

Formula Used:

The central angle in radians subtended by an arc at the center of a circle is the ratio of the arc length to its radius.

Angle (θ) = Arc length (l) ÷ Radius (r)

Calculations:

⇒ θ = 10 cm ÷ 5 cm

⇒ θ = 2

Since the formula directly yields the angle in radians, the central angle is 2 radians.

∴ The correct answer is 2 radians.

Additional Information

Conversion between Radians and Degrees

To convert an angle from radians to degrees, multiply by (180° ÷ π). Similarly, to convert degrees to radians, multiply by (π ÷ 180°). Thus, π radians = 180°.

Area of a Sector

The area of a sector with radius r and central angle θ (in radians) is determined by the formula A = (1 ÷ 2) × r² × θ. It can also be found using the arc length as A = (1 ÷ 2) × l × r.

Full Circle Arc Length

The total arc length encompassing a complete circle is its circumference, calculated as C = 2 × π × r. This aligns with a full rotation having a central angle of 2π radians.

Shortcut Trick

Given: tan(90° − A) = √3

We know from standard trigonometric values that tan 60° = √3.

Comparing both sides, we get: 90° − A = 60°

⇒ A = 30°

Therefore, sin A = sin 30° = 1/2.

∴ The correct answer is 1/2.

Alternate Method

A Base = √3 Perp = 1 Hyp = 2

Given:

tan(90° − A) = √3

Formula Used:

tan(90° − θ) = cot θ

cot θ = Base ÷ Perpendicular

sin θ = Perpendicular ÷ Hypotenuse

Calculations:

⇒ cot A = √3

⇒ Base ÷ Perpendicular = √3 ÷ 1

⇒ Let Base = √3 and Perpendicular = 1

⇒ By Pythagoras Theorem, Hypotenuse = √(Base² + Perpendicular²)

⇒ Hypotenuse = √((√3)² + 1²) = √(3 + 1) = √4 = 2

⇒ sin A = Perpendicular ÷ Hypotenuse = 1/2

∴ The correct answer is 1/2.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Trigonometric ratios for complementary angles satisfy relations like sin(90°−θ) = cos θ and tan(90°−θ) = cot θ.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its base and perpendicular: H² = P² + B².

Standard Trigonometric Values

Common values frequently used in exams include sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and tan 60° = √3.

Shortcut Trick

Substitute x = √7 directly into the expression x + 1/x.

Calculate 1/x = 1/√7 and rationalize it by multiplying the numerator and denominator by √7 to get √7/7.

Add the terms: √7 + √7/7 = (7√7 + √7) / 7.

Simplify the numerator to get 8√7/7.

∴ The correct answer is 8√7/7.

Alternate Method

Given:

x = √7

Formula Used:

For rationalization: 1/√a = √a / a

Calculations:

⇒ Substitute the given value of x in the expression x + 1/x

⇒ x + 1/x = √7 + 1/√7

⇒ Taking the common denominator as √7 for addition:

⇒ x + 1/x = (√7 × √7 + 1) / √7

⇒ x + 1/x = (7 + 1) / √7

⇒ x + 1/x = 8 / √7

⇒ Multiply the numerator and denominator by √7 to rationalize the denominator:

⇒ x + 1/x = (8 × √7) / (√7 × √7)

⇒ x + 1/x = 8√7 / 7

∴ The correct answer is 8√7/7.

Additional Information

Rationalizing the Denominator

When a fraction has an irrational number like √a in the denominator, multiplying both the numerator and denominator by √a removes the radical from the denominator. Formula: 1/√a = √a / a.

Operations with Surds

When multiplying square roots of the same base number, the square root symbol is eliminated: √a × √a = a. This basic property simplifies addition and multiplication of irrational fractions.

Standard Algebraic Identities

If x + 1/x = k, then x² + 1/x² = k² − 2. This rule is highly useful for finding higher powers of algebraic expressions quickly in quantitative aptitude exams.

Shortcut Trick

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Ratio of areas = (3 ÷ 5)²

Ratio of areas = 9 / 25

∴ The correct answer is 9:25.

Alternate Method

Side = 3 units Area 1

Side = 5 units Area 2

Given:

Ratio of corresponding sides of two similar triangles = 3:5

Formula Used:

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Area₁ ÷ Area₂ = (Side₁ ÷ Side₂)²

Calculations:

⇒ Area₁ ÷ Area₂ = (3 ÷ 5)²

⇒ Area₁ ÷ Area₂ = (3² ÷ 5²)

⇒ Area₁ ÷ Area₂ = 9 ÷ 25

⇒ The ratio of their areas = 9:25

∴ The correct answer is 9:25.

Additional Information

Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. Similarity preserves the shape but not necessarily the size of the figure.

Ratio of Perimeters

The ratio of the perimeters of two similar triangles is identical to the ratio of their corresponding sides.

Ratio of Volumes

For similar three-dimensional objects, the ratio of their volumes is equal to the cube of the ratio of their corresponding sides (V₁ ÷ V₂ = s₁³ ÷ s₂³).

Shortcut Trick

Rationalize the denominator inside the root to find x.

x = √[(2 + √3)² ÷ (4 − 3)] = 2 + √3

Substitute x into the expression: x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

⇒ 7 + 4√3 + 2 + √3 − 9 = 5√3

∴ The correct answer is 5√3.

Alternate Method

Given:

x = √[(2 + √3) ÷ (2 − √3)]

Calculations:

Rationalizing the denominator inside the square root:

⇒ x = √[((2 + √3) × (2 + √3)) ÷ ((2 − √3) × (2 + √3))]

⇒ x = √[(2 + √3)² ÷ (2² − (√3)²)]

⇒ x = √[(2 + √3)² ÷ (4 − 3)]

⇒ x = √[(2 + √3)² ÷ 1]

⇒ x = 2 + √3

Now, substitute the value of x in the given expression:

⇒ x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

Using the algebraic formula (a + b)² = a² + 2ab + b²:

⇒ (2² + 2 × 2 × √3 + (√3)²) + 2 + √3 − 9

⇒ (4 + 4√3 + 3) + 2 + √3 − 9

⇒ 7 + 4√3 + 2 + √3 − 9

⇒ 9 + 5√3 − 9

⇒ 5√3

∴ The correct answer is 5√3.

Additional Information

Rationalization of Surds

To eliminate a square root from the denominator, multiply both the numerator and the denominator by its conjugate. The conjugate of a − √b is a + √b.

Algebraic Identity for Squares

The square of a binomial sum is calculated using the identity (a + b)² = a² + 2ab + b².

Difference of Squares

The product of the sum and difference of the same two terms yields the difference of their squares: (a − b)(a + b) = a² − b².

Shortcut Trick

The sum of opposite angles in a cyclic quadrilateral is exactly 180°.

The given ratio of angles is 1 : 2 : 3 : 4. Let the angles be x, 2x, 3x, and 4x.

Since the sum of opposite angle pairs must be equal, we must pair them as (x, 4x) and (2x, 3x) because x + 4x = 5x and 2x + 3x = 5x.

Equating the sum to 180°: 5x = 180° ⇒ x = 36°.

The smallest angle in the ratio is x, which is 36°.

∴ The correct answer is 36°.

Alternate Method

Given:

The angles of a cyclic quadrilateral are in the ratio of 1 : 2 : 3 : 4.

Formula Used:

The sum of the opposite angles of a cyclic quadrilateral is always 180°.

O x 2x 3x 4x

Calculations:

⇒ Let the four angles of the cyclic quadrilateral be x, 2x, 3x, and 4x.

⇒ In a valid cyclic quadrilateral, the pairs of opposite angles must sum to the same value (180°).

⇒ Grouping the angles to find equal sums: (x + 4x) = 5x and (2x + 3x) = 5x.

⇒ Therefore, the opposite angle pairs are (x, 4x) and (2x, 3x).

⇒ Summing either opposite pair gives: 5x = 180°

⇒ x = 180° ÷ 5 = 36°

⇒ The measures of the four angles are 36°, 72°, 108°, and 144°.

⇒ Smallest angle = x = 36°

∴ The correct answer is 36°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all four of its vertices lie on the circumference of a circle. The sum of either pair of opposite angles is exactly 180°.

Exterior Angle Property

If any side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the interior opposite angle.

Ptolemy's Theorem

For any cyclic quadrilateral with side lengths a, b, c, d and diagonals p, q, the product of the diagonals equals the sum of the products of opposite sides: p × q = (a × c) + (b × d).

Brahmagupta's Formula

The area of a cyclic quadrilateral is given by Area = √(s − a)(s − b)(s − c)(s − d), where s is the semi-perimeter calculated as s = (a + b + c + d) ÷ 2.

Shortcut Trick

We know that sec A = Hypotenuse ÷ Adjacent = 13/5.

This directly forms the standard Pythagorean triplet (5, 12, 13).

Thus, the Opposite side to angle A is 12.

Since sin A = Opposite ÷ Hypotenuse, we get 12/13.

∴ The correct answer is 12/13.

Alternate Method

Given:

sec A = 13/5

Formula Used:

Pythagoras Theorem: Hypotenuse² = Opposite² + Adjacent²

sin A = Opposite ÷ Hypotenuse

A B C Adjacent = 5 Opposite = 12 Hypotenuse = 13

Let the sides of the right-angled triangle be:

⇒ Hypotenuse = 13k

⇒ Adjacent = 5k

Applying the Pythagoras theorem:

⇒ (13k)² = Opposite² + (5k)²

⇒ 169k² = Opposite² + 25k²

⇒ Opposite² = 169k² − 25k²

⇒ Opposite² = 144k²

⇒ Opposite = 12k

Now, substitute the values to find sin A:

⇒ sin A = Opposite / Hypotenuse

⇒ sin A = 12k / 13k

⇒ sin A = 12 / 13

∴ The correct answer is 12/13.

Additional Information

Trigonometric Ratios

In a right-angled triangle, sin A = Opposite ÷ Hypotenuse, cos A = Adjacent ÷ Hypotenuse, and tan A = Opposite ÷ Adjacent.

Pythagorean Triplets

A set of three positive integers a, b, and c that satisfy a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Trigonometric Identities

The fundamental identities are: sin²A + cos²A = 1, sec²A − tan²A = 1, and cosec²A − cot²A = 1.

Shortcut Trick

Given: 60% of A = 0.3 of B = 1/6 of C

⇒ (3/5)A = (3/10)B = (1/6)C

Take the LCM of the numerators (3, 3, 1) = 3.

Divide the entire equation by 3 to get direct denominators:

⇒ A / 5 = B / 10 = C / 18

∴ The correct answer is 5 : 10 : 18.

Alternate Method

Given:

60% of A = 0.3 of B = 1/6 of C

5 A

10 B

18 C

Ratio Comparison of A, B, and C

Calculations:

⇒ 60% of A = 0.3 of B = 1/6 of C

⇒ (60/100) × A = (3/10) × B = (1/6) × C

⇒ (3/5) × A = (3/10) × B = (1/6) × C

Let (3/5)A = (3/10)B = (1/6)C = k

⇒ A = 5k / 3

⇒ B = 10k / 3

⇒ C = 6k = 18k / 3

Now, finding the ratio A : B : C by substituting the values:

⇒ A : B : C = (5k / 3) : (10k / 3) : (18k / 3)

⇒ A : B : C = 5 : 10 : 18

∴ The correct answer is 5 : 10 : 18.

Additional Information

Ratio of Equalities

If pA = qB = rC, then the ratio A : B : C is calculated as (1/p) : (1/q) : (1/r) or by dividing each term by the LCM of numerators p, q, and r.

Compound Ratio

The compound ratio of two individual ratios a:b and c:d is generated by multiplying the antecedents and consequents respectively, given by (a × c) : (b × d).

Invertendo

If a : b = c : d, then applying invertendo property gives b : a = d : c.

Shortcut Trick

Convert the total time into minutes: 1 hour 45 minutes = 105 minutes.

Since the distance covered in 1 minute is 0.875 km, directly multiply the distance by the total time.

Total Distance = 0.875 × 105 = 91.875 km.

∴ The correct answer is 91.875 km.

Alternate Method

Given:

Distance covered in 1 minute = 0.875 km

Total time traveled = 1 hour 45 minutes

Formula Used:

Distance = Speed × Time

0 min 1 min 105 min

0 km 0.875 km ? km

1 min gap

× 105 multiplier

Calculations:

⇒ First, convert the total time strictly into minutes.

⇒ Total Time = 1 hour + 45 minutes

⇒ Total Time = 60 minutes + 45 minutes = 105 minutes

⇒ The car's speed is effectively 0.875 km per minute.

⇒ Distance in 105 minutes = 0.875 × 105

⇒ Distance = 0.875 × (100 + 5)

⇒ Distance = 87.5 + 4.375

⇒ Distance = 91.875 km

∴ The correct answer is 91.875 km.

Additional Information

Unitary Method

The unitary method involves finding the value of a single base unit and then multiplying it by the necessary number of units to determine the final total value.

Speed & Distance Formula

Distance is always calculated as the product of Speed and Time (Distance = Speed × Time). Here, the car's speed acts directly as the rate per minute.

Unit Matching

Before executing a multiplication, ensure all parameters share the same base dimension. In this case, converting hours to minutes allowed a direct multiplication with the given rate.

Shortcut Trick

Combine the first two fractional terms directly by cross-multiplying their denominators.

Sum = [(7 − √2) + (7 + √2)] ÷ [(7 + √2) × (7 − √2)]

Sum = 14 ÷ (49 − 2)

Notice that the third term is exactly identical to this sum.

Expression = [14 ÷ (49 − 2)] − [14 ÷ (49 − 2)] = 0

∴ The correct answer is 0.

Alternate Method

Given:

Expression = [1 / (7 + √2)] + [1 / (7 − √2)] − [14 / (49 − 2)]

Formula Used:

(a + b)(a − b) = a² − b²

Calculations:

⇒ First, add the first two fractions by taking the LCM of their denominators.

⇒ LCM = (7 + √2)(7 − √2) = 7² − (√2)² = 49 − 2 = 47

⇒ Numerator = (7 − √2) + (7 + √2) = 14

⇒ Sum of first two terms = 14 / 47

⇒ Now, evaluate the third term individually:

⇒ Third term = 14 / (49 − 2) = 14 / 47

⇒ Substitute these simplified values back into the overall expression:

⇒ Result = 14 / 47 − 14 / 47

⇒ Result = 0

∴ The correct answer is 0.

Additional Information

Rationalization of Surds

To rationalize a denominator of the form a + √b, multiply both the numerator and denominator by its conjugate a − √b. This simplifies the surd expression into standard integers.

Algebraic Identities

The difference of squares formula (a + b)(a − b) = a² − b² is the primary mathematical tool used to eliminate square roots from denominators.

Simplification Rules

When simplifying complex fractional expressions, always check if combining certain elements directly reduces the equation to a common base value, which often results in zero or one.

Shortcut Trick

Using the weighted average method for net percentage change:

Net % Change = [ (2 × −10%) + (4 × 5%) + (4 × −15%) ] ÷ (2 + 4 + 4)

Net % Change = ( −20% + 20% − 60% ) ÷ 10

Net % Change = −60% ÷ 10 = −6%

The negative sign represents a decrease.

∴ The correct answer is 6% decrease.

Alternate Method

Given:

Ratio of expenditure on Food, Clothing, and Entertainment = 2 : 4 : 4

Let actual initial expenditures be a convenient multiple of the ratio, such as 200, 400, and 400.

Food200−10%Clothing400+5%Entertainment400−15%−20+20−60Net Change = −60 out of Total 1000 = −6%

Calculations:

⇒ Total initial expenditure = 200 + 400 + 400 = 1000

⇒ New expenditure on Food = 200 − (10% of 200) = 200 − 20 = 180

⇒ New expenditure on Clothing = 400 + (5% of 400) = 400 + 20 = 420

⇒ New expenditure on Entertainment = 400 − (15% of 400) = 400 − 60 = 340

⇒ Total new expenditure = 180 + 420 + 340 = 940

⇒ Net change in expenditure = 940 − 1000 = −60

⇒ Percentage change = (Net change ÷ Total initial expenditure) × 100

⇒ Percentage change = (−60 ÷ 1000) × 100 = −6%

The negative value implies a decrease of 6%.

∴ The correct answer is 6% decrease.

Additional Information

Weighted Average in Percentages

When a quantity is divided into specific parts, the net percentage change is the sum of (part ratio × % change) divided by the total ratio sum.

Base Shifting Technique

Assuming a convenient base value like 100 or a multiple of 10 for abstract ratio amounts makes calculating percentage increases or decreases purely integer-based and minimizes error.

Net Percentage Change

A negative final percentage signifies an overall decrement (loss/fall), while a positive percentage denotes an increment (gain/rise).

Shortcut Trick

Profit ratio A : B = (1,20,000 × 10) : (1,60,000 × 7) = 15 : 14.

Total profit units = 15 + 14 = 29 units.

Given B's share (14 units) = ₹16,800.

Value of 1 unit = 16,800 ÷ 14 = ₹1,200.

Total profit (29 units) = 29 × 1,200 = ₹34,800.

∴ The correct answer is ₹34,800.

Alternate Method

Given:

Investment by A = ₹1,20,000

Investment by B = ₹1,60,000

Time period for A = 10 months

Time period for B = 7 months

B's profit share = ₹16,800

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Partner A1.2L × 10 monthsProfit: 15 UnitsPartner B1.6L × 7 monthsProfit: 14 Units14 Units = ₹16,800 → 29 Units = ₹34,800

Calculations:

⇒ Profit Ratio of A : B = (1,20,000 × 10) : (1,60,000 × 7)

⇒ Profit Ratio = 12,00,000 : 11,20,000

⇒ By simplifying, we get Profit Ratio = 120 : 112 = 15 : 14

⇒ Let A's profit be 15x and B's profit be 14x.

⇒ Total profit = 15x + 14x = 29x

⇒ We are given that B's profit share is ₹16,800.

⇒ 14x = 16,800

⇒ x = 16,800 ÷ 14 = 1,200

⇒ Total profit = 29 × 1,200 = 34,800

∴ The correct answer is ₹34,800.

Additional Information

Partnership Profit Ratio

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time. P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

A sleeping partner only invests capital, while an active partner also manages the business. An active partner may receive a fixed salary or percentage from total profit before the remainder is split.

Joining Mid-way

If a partner joins x months after the start of a year, their investment time period will be calculated as (12 − x) months for the annual profit distribution.

Shortcut Trick

Convert all animals to equivalent cows to find the total grazing units.

1 horse = 5/3 cows and 1 sheep = 3/5 cows.

A's total grazing = (21 × 5/3 × 6) + (15 × 2) = 210 + 30 = 240 cow-months.

B's total grazing = (15 × 6) + (40 × 3/5 × 7.5) = 90 + 180 = 270 cow-months.

A's rent portion = 240 ÷ (240 + 270) = 240 ÷ 510 = 8/17.

∴ The correct answer is 8/17.

Alternate Method

Given:

A places 21 horses for 6 months and 15 cows for 2 months.

B places 15 cows for 6 months and 40 sheep for 7.5 months.

Food equality: 3 horses = 5 cows and 6 cows = 10 sheep.

Formula Used:

Total Usage = Equivalent Number of Animals × Time Period

240 Units Ratio: 8 270 Units Ratio: 9 A's Rent Share B's Rent Share

Calculations:

⇒ First, convert all animals into equivalent cow units for calculation consistency.

⇒ 3 horses = 5 cows ⇒ 1 horse = 5/3 cows

⇒ 6 cows = 10 sheep ⇒ 1 sheep = 6/10 = 3/5 cows

⇒ Total grazing consumption for A = (21 × 5/3 × 6) + (15 × 2)

⇒ Total equivalent units for A = 210 + 30 = 240 cow-months

⇒ Total grazing consumption for B = (15 × 6) + (40 × 3/5 × 7.5)

⇒ Total equivalent units for B = 90 + 180 = 270 cow-months

⇒ The total rent will be distributed in the ratio of their total grazing consumption.

⇒ Ratio of rent share (A : B) = 240 : 270 = 8 : 9

⇒ Total ratio parts = 8 + 9 = 17 parts

⇒ Portion of the rent A should cover = 8 / 17

∴ The correct answer is 8/17.

Additional Information

Partnership / Joint Rent Ratio

When multiple partners invest in a business or rent a shared resource, the profit or expense is distributed in the exact ratio of the product of their investment (or resource usage) and the time period. Ratio = (Usage₁ × Time₁) : (Usage₂ × Time₂).

Proportional Division

To divide a total quantity representing fractions of a whole into a given ratio a:b, the individual fractional parts are calculated directly as [a / (a+b)] and [b / (a+b)] respectively.

Shortcut Trick

Convert cows to equivalent sheep: 4 cows = 8 sheep ⇒ 1 cow = 2 sheep.

A's equivalent sheep-months = (16 × 2 × 3) + (20 × 4) = 96 + 80 = 176.

B's equivalent sheep-months = 30 × 6 = 180.

Ratio of A's rent to B's rent = 176 : 180 = 44 : 45.

A's portion out of the total rent = 44 ÷ (44 + 45) = 44/89.

∴ The correct answer is 44/89.

Alternate Method

Given:

A uses 16 cows for 3 months and 20 sheep for 4 months.

B uses 30 sheep for 6 months.

Equivalence: 4 cows = 8 sheep.

Formula Used:

Rent Share Ratio = Total equivalent animal-months of A : Total equivalent animal-months of B

Rent Sharing Proportion

A: 176 units

B: 180 units Ratio = 44 Ratio = 45

Calculations:

⇒ First, find the sheep equivalent for A's cows.

⇒ 4 cows = 8 sheep → 1 cow = 2 sheep.

⇒ A's 16 cows = 16 × 2 = 32 sheep.

⇒ Total equivalent sheep-months for A = (32 sheep × 3 months) + (20 sheep × 4 months).

⇒ Total for A = 96 + 80 = 176 sheep-months.

⇒ Total equivalent sheep-months for B = 30 sheep × 6 months.

⇒ Total for B = 180 sheep-months.

⇒ Rent Ratio (A : B) = 176 : 180 = 44 : 45.

⇒ Total proportion parts = 44 + 45 = 89.

⇒ A's portion of the rent = 44/89.

∴ The correct answer is 44/89.

Additional Information

Partnership Basic Rule

When resources are used for different time periods, the profit or rent sharing ratio is directly proportional to the product of capital (or equivalent resource) and time period. Formula: Ratio = C₁×T₁ : C₂×T₂.

Compound Partnership

When individuals invest different amounts in different phases, their effective capital is calculated by multiplying each invested amount by its respective time period and summing them up.

Equivalent Units Principle

In scenarios involving different entities (like cows, sheep, or men, women), a common unit ratio must first be established to calculate the total effective capital or work completed before determining any proportions.

Shortcut Trick

Assume the overall average is 20 years (the middle value).

Calculate net deviation: (18 − 20)×2 + (20 − 20)×3 + (22 − 20)×5

⇒ Net deviation = (−4) + 0 + 10 = +6

Average deviation per student = +6 ÷ (2 + 3 + 5) = +0.6

Overall average = Assumed Average + Average Deviation = 20 + 0.6 = 20.6

∴ The correct answer is 20.6 years.

Alternate Method

Given:

Average ages of three groups = 18, 20, and 22

Ratio of number of students in the groups = 2 : 3 : 5

Formula Used:

Weighted Average = (N₁A₁ + N₂A₂ + N₃A₃) ÷ (N₁ + N₂ + N₃)

Ratio: 2 Avg: 18

Ratio: 3 Avg: 20

Ratio: 5 Avg: 22

Total Ratio Sum = 10 Combined Average = 20.6

Calculations:

Let the exact number of students in the three groups be 2x, 3x, and 5x respectively.

⇒ Total sum of ages = (18 × 2x) + (20 × 3x) + (22 × 5x)

⇒ Total sum = 36x + 60x + 110x = 206x

⇒ Total number of students = 2x + 3x + 5x = 10x

⇒ Overall average age = 206x ÷ 10x

⇒ Overall average age = 20.6 years

∴ The correct answer is 20.6 years.

Additional Information

Weighted Average

It is used when different subsets of data have different frequencies or sample sizes. Formula: A_w = (N₁A₁ + N₂A₂ + ... + N_nA_n) ÷ (N₁ + N₂ + ... + N_n).

Net Deviation Method

This shortcut simplifies heavy calculations by selecting an assumed central mean. The true average is obtained using the relation: True Average = Assumed Mean + (Sum of Net Deviations ÷ Total Frequency).

Key Property of Averages

If every observation in a specific dataset is increased, decreased, multiplied, or divided by a constant 'k', the overall average of that dataset will automatically reflect the exact same mathematical operation.

Shortcut Trick

Using the deviation method from the total average (68):

Total deviation = 5 × (63 − 68) + 5 × (72 − 68)

Total deviation = −25 + 20 = −5

To balance the sum to zero, the 6^th number must offset this deviation: 68 − (−5) = 73.

∴ The correct answer is 73.

Alternate Method

Given:

Average of 11 numbers = 68

Average of the first 5 numbers = 63

Average of the last 5 numbers = 72

First 5 Avg = 63

Last 5 Avg = 72

6^th ?

Total 11 Numbers (Avg = 68)

Formula Used:

Sum of items = Average × Number of items

6^th Number = Total Sum − (Sum of first 5 + Sum of last 5)

Calculations:

⇒ Total sum of 11 numbers = 11 × 68 = 748

⇒ Sum of the first 5 numbers = 5 × 63 = 315

⇒ Sum of the last 5 numbers = 5 × 72 = 360

⇒ Sum of these 10 numbers = 315 + 360 = 675

⇒ 6^th Number = 748 − 675

⇒ 6^th Number = 73

∴ The correct answer is 73.

Additional Information

Weighted Average

When combining groups with different averages, the overall average is (N₁A₁ + N₂A₂) ÷ (N₁ + N₂).

Deviation Method

The sum of deviations of all observations from their true average is exactly zero. This principle provides a powerful shortcut to compute an unknown item dynamically.

Overlapping Groups

If the given groups overlap on a central item (e.g., first 6 and last 6 out of 11), that central item can be found directly using: Overlapping Item = (Sum of Group 1 + Sum of Group 2) − Total Sum.

Shortcut Trick The numbers exactly divisible by 13 form an Arithmetic Progression (AP). For any AP, the average is simply the midpoint of the first and last terms. First multiple of 13 after 300 = 24 × 13 = 312. Last multiple of 13 before 450 = 34 × 13 = 442. Average = (312 + 442) ÷ 2 = 754 ÷ 2 = 377. ∴ The correct answer is 377.

Alternate Method Given: Range = 300 to 450 Divisor = 13 Formula Used: Average of an Arithmetic Progression (AP) = (First term + Last term) ÷ 2

300 450

312 442 377 Average (Midpoint)

Avg = (312 + 442) ÷ 2

Calculations: ⇒ First integer > 300 divisible by 13: 300 ÷ 13 ≈ 23.07 → 24 × 13 = 312 ⇒ Last integer < 450 divisible by 13: 450 ÷ 13 ≈ 34.61 → 34 × 13 = 442 ⇒ The sequence of numbers forms an AP: 312, 325, 338, ..., 442 ⇒ First term (a) = 312 ⇒ Last term (l) = 442 ⇒ Average = (312 + 442) ÷ 2 ⇒ Average = 754 ÷ 2 ⇒ Average = 377 ∴ The correct answer is 377.

Additional Information Arithmetic Progression (AP) An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. Example: 312, 325, 338, ... with a common difference of 13. Average of an Arithmetic Progression The average of an equally spaced series (AP) is strictly the midpoint, which equals (First Term + Last Term) ÷ 2. Finding the Number of Terms To find the total terms 'n' in an AP, use the formula: n = [(Last term − First term) ÷ Common difference] + 1.