Quantitative Aptitude PYQs

Shortcut Trick

We need to convert 7 + 4√3 into a perfect square of the form (a + b)².

Compare the irrational part with 2ab: 2ab = 4√3 ⇒ ab = 2√3.

Assume a = 2 and b = √3. Checking the rational part: a² + b² = 2² + (√3)² = 4 + 3 = 7.

Since the sum matches the first term, 7 + 4√3 = (2 + √3)².

So, √(7 + 4√3) = √((2 + √3)²) = 2 + √3 = √3 + 2.

∴ The correct answer is √3 + 2.

Alternate Method

Given:

Expression = √(7 + 4√3)

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ We need to evaluate: √(7 + 4√3)

⇒ Split the number 7 as 4 + 3 to help form a perfect square.

⇒ √(4 + 3 + 4√3)

⇒ Expressing 4 as 2² and 3 as (√3)²

⇒ √(2² + (√3)² + 2 × 2 × √3)

⇒ Applying the formula (a + b)² = a² + b² + 2ab, where a = 2 and b = √3

⇒ √((2 + √3)²)

⇒ 2 + √3 = √3 + 2

∴ The correct answer is √3 + 2.

Additional Information

Square Root of Surds

To find the square root of a binomial surd like a ± √b, always express it in the form (x ± y)² by equating the rational part to x² + y² and the irrational part to 2xy.

Key Algebraic Identities

(a + b)² = a² + b² + 2ab

(a − b)² = a² + b² − 2ab

Rationalization of Surds

If a surd appears in the denominator, multiply both numerator and denominator by its conjugate. For example, the conjugate of a + √b is a − √b.

Shortcut Trick

If the entire ₹8000 was lent at the lower rate of 5%, the interest would be 5% of 8000 = ₹400.

The extra interest generated = ₹500 − ₹400 = ₹100.

This extra ₹100 is solely due to the additional 2% (7% − 5%) interest on the second part.

⇒ 2% of the sum lent at 7% = ₹100

⇒ Sum lent at 7% = ₹100 × (100 ÷ 2) = ₹5000.

∴ The correct answer is ₹5000.

Alternate Method

Given:

Total Principal = ₹8000

Rate 1 = 5% | Rate 2 = 7%

Total Simple Interest = ₹500 | Time = 1 year

Formula Used:

Overall Rate of Interest = (Total Interest ÷ Total Principal) × 100

Calculations:

⇒ Overall Rate = (500 ÷ 8000) × 100 = 6.25%

Applying the rule of alligation to find the ratio of the invested amounts:

Part 1 5% Part 2 7% Overall Rate 6.25% 0.75 1.25 Ratio = 3 : 5 Total Parts = 8 units → ₹8000

⇒ Ratio of Sum at 5% to Sum at 7% = (7 − 6.25) : (6.25 − 5)

⇒ Ratio = 0.75 : 1.25 = 3 : 5

⇒ Total units = 3 + 5 = 8 units

⇒ 8 units = ₹8000 ⇒ 1 unit = ₹1000

⇒ Sum lent at 7% = 5 units = 5 × 1000 = ₹5000

∴ The correct answer is ₹5000.

Additional Information

Alligation in Interest Problems

When a principal is divided into two parts given at different rates, the alligation method quickly yields the ratio of the principal amounts using the overall effective interest rate.

Shift Method Concept

By assuming the entire amount is lent at the lower interest rate, you calculate a base interest. The difference between the actual interest and this base interest is entirely produced by the extra percentage rate on the second part.

Simple Interest Rule

Unlike Compound Interest, Simple Interest remains constant for every year. It is calculated solely on the original principal using the formula I = (P × R × T) ÷ 100.

Shortcut Trick

Let the total distance be the LCM of (4 × 20), (4 × 10), and (2 × 70) = LCM(80, 40, 140) = 560 km.

The distances for the three parts are 140 km, 140 km, and 280 km respectively.

Total time taken = (140 ÷ 20) + (140 ÷ 10) + (280 ÷ 70) = 7 + 14 + 4 = 25 hours.

Average speed = Total Distance ÷ Total Time = 560 ÷ 25 = 22.4 km/h.

∴ The correct answer is 22.4 km/h.

Alternate Method

Given:

First part of the journey = D/4 at a speed of 20 km/h.

Second part of the journey = D/4 at a speed of 10 km/h.

Remaining distance = D − D/4 − D/4 = D/2 at a speed of 70 km/h.

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

     1/4 of D 1/4 of D 1/2 of D  20 km/h 10 km/h 70 km/h   Total Distance = D

Calculations:

⇒ Total Time (T) = t₁ + t₂ + t₃

⇒ T = (D/4 ÷ 20) + (D/4 ÷ 10) + (D/2 ÷ 70)

⇒ T = D/80 + D/40 + D/140

⇒ Taking LCM of 80, 40, and 140, which is 560:

⇒ T = (7D + 14D + 4D) ÷ 560

⇒ T = 25D ÷ 560

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = D ÷ (25D ÷ 560)

⇒ Average Speed = 560 ÷ 25

⇒ Average Speed = 22.4 km/h

∴ The correct answer is 22.4 km/h.

Additional Information

Average Speed for Two Equal Distances

When a person travels two equal halves of a journey at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

Average Speed for Three Equal Distances

When travelling three equal parts of a journey at speeds x, y, and z, Average Speed = (3 × x × y × z) ÷ (xy + yz + zx).

Average Speed vs Average of Speeds

Average speed must always be calculated as Total Distance ÷ Total Time. It is never the simple arithmetic mean of the individual speeds.

Shortcut Trick

When the angle of elevation of an object of height 'h' changes from θ₁ to θ₂ upon moving distance 'd' towards it, the relation is:

d = h(cot θ₁ − cot θ₂)

⇒ 15 = h(cot 45° − cot 60°) = h(1 − 1/√3)

⇒ h = (15√3) / (√3 − 1)

⇒ Rationalising the denominator: h = 15(3 + √3) / 2

∴ The correct answer is 15(3+√3)/2 m.

Alternate Method

    45°  60°  A B C D     15 m    x     h

Given:

Initial angle of elevation (∠DAC) = 45°

Distance walked towards the pole (AB) = 15 m

Final angle of elevation (∠DBC) = 60°

Formula Used:

tan θ = Perpendicular / Base

Calculations:

Let the height of the pole (DC) be 'h' and distance BC be 'x'.

In right-angled ΔDBC:

⇒ tan 60° = DC / BC

⇒ √3 = h / x

⇒ x = h / √3 ..... (Equation 1)

In right-angled ΔDAC:

⇒ tan 45° = DC / AC

⇒ 1 = h / (15 + x)

⇒ 15 + x = h

Substitute the value of x from Equation 1:

⇒ 15 + (h / √3) = h

⇒ h − (h / √3) = 15

⇒ h(√3 − 1) / √3 = 15

⇒ h = 15√3 / (√3 − 1)

Multiply numerator and denominator by (√3 + 1) to rationalise the denominator:

⇒ h = [15√3 × (√3 + 1)] / [(√3 − 1)(√3 + 1)]

⇒ h = 15(3 + √3) / (3 − 1)

⇒ h = 15(3 + √3) / 2

∴ The correct answer is 15(3+√3)/2 m.

Additional Information

Angle of Elevation & Depression

The angle of elevation is formed when looking up at an object, while the angle of depression is formed when looking down. Both are measured from the horizontal line of sight.

Direct Formula for Heights & Distances

When moving a distance 'd' directly towards or away from an object of height 'h', changing the angle of elevation from θ₁ to θ₂, the generalized relation used is: d = h(cot θ₁ − cot θ₂).

Key Trigonometric Values

Memorize these essential values: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3. Since cot θ = 1/tan θ, their inverse values are cot 30° = √3, cot 45° = 1, and cot 60° = 1/√3.

Shortcut Trick

The diameter of the largest possible sphere equals the smallest dimension of the rectangular tank.

⇒ Diameter = 6 cm ⇒ Radius = 3 cm

⇒ Unoccupied Space = Volume of Tank − Volume of Sphere

⇒ Unoccupied Space = (12 × 10 × 6) − (4/3 × π × 3³)

⇒ Unoccupied Space = (720 − 36π) cm³

∴ The correct answer is (720-36π) cm³.

Alternate Method

Given:

Length of the rectangular tank (L) = 12 cm

Width of the rectangular tank (W) = 10 cm

Height of the rectangular tank (H) = 6 cm

Formula Used:

Volume of a Cuboid = L × W × H

Volume of a Sphere = 4/3 × π × r³

Length = 12 cm  Height = 6 cm   r = 3 cm Front View of Tank and Sphere

Calculations:

⇒ The largest spherical ball that can fit entirely inside the tank must have a diameter equal to the smallest dimension of the tank to avoid overlapping the boundaries.

⇒ Smallest dimension among 12 cm, 10 cm, and 6 cm is 6 cm.

⇒ Diameter of the sphere = 6 cm

⇒ Radius of the sphere (r) = 6 ÷ 2 = 3 cm

⇒ Volume of the rectangular tank = L × W × H

⇒ Volume of the rectangular tank = 12 × 10 × 6 = 720 cm³

⇒ Volume of the spherical ball = 4/3 × π × r³

⇒ Volume of the spherical ball = 4/3 × π × (3)³ = 4/3 × π × 27 = 36π cm³

⇒ Volume of the space not occupied = Volume of the tank − Volume of the spherical ball

⇒ Unoccupied volume = (720 − 36π) cm³

∴ The correct answer is (720-36π) cm³.

Additional Information

Volume of a Cone

The volume of a cone with base radius r and vertical height h is given by V = 1/3 × π × r² × h.

Volume of a Cylinder

The volume of a right circular cylinder with base radius r and height h is given by V = π × r² × h.

Surface Area of a Sphere

The total surface area of a sphere with radius r is given by A = 4 × π × r².

Largest Sphere inside a Cube

For a cube of side a, the largest sphere that can be placed inside it has a diameter exactly equal to the side length a, meaning its radius = a/2.

Shortcut Trick

Ratio of profit of A to B = 7 : 5

5 units = Rs. 15,000

1 unit = Rs. 3,000

A's profit = 7 units = 7 × 3,000 = Rs. 21,000

∴ The correct answer is Rs. 21,000.

Alternate Method

Given:

Profit ratio of A and B = 7 : 5

B's share of profit = Rs. 15,000

7 parts A A = ? 5 parts B Rs. 15,000

Calculations:

Let the profits of A and B be 7x and 5x respectively.

⇒ B's profit = 5x = 15000

⇒ x = 15000 ÷ 5

⇒ x = 3000

⇒ A's profit = 7x

⇒ A's profit = 7 × 3000 = 21000

∴ The correct answer is Rs. 21,000.

Additional Information

Partnership Profit Ratio

When the time period of investment is the same for all partners, the profit is divided exactly in the ratio of their invested capital.

Compound Partnership

If partners invest capital for different time periods, Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Sleeping vs Active Partner

An active partner manages the business and usually receives a fixed percentage or salary from the total profit before the remainder is distributed among all partners based on their capital ratio.

Shortcut Trick

Each interior angle of a regular polygon is given by 180° − (360° ÷ n), where n is the number of sides.

Interior angle of a regular hexagon (n = 6) = 180° − 60° = 120°.

Interior angle of a regular 12-sided polygon (n = 12) = 180° − 30° = 150°.

Required Ratio = 120° : 150° = 4 : 5.

∴ The correct answer is 4 : 5.

Alternate Method

Given:

Number of sides of regular hexagon (n₁) = 6

Number of sides of regular dodecagon (n₂) = 12

Formula Used:

Interior angle of a regular n-sided polygon = [(n − 2) × 180°] ÷ n

Hexagon 120° Dodecagon 150° Interior Angle Ratio = 120° : 150° = 4 : 5

Calculations:

⇒ Interior angle of a regular hexagon = [(6 − 2) × 180°] ÷ 6

⇒ Interior angle of a regular hexagon = [4 × 180°] ÷ 6 = 120°

⇒ Interior angle of a regular 12-sided polygon = [(12 − 2) × 180°] ÷ 12

⇒ Interior angle of a regular 12-sided polygon = [10 × 180°] ÷ 12 = 150°

⇒ Ratio of their interior angles = 120° : 150°

⇒ Ratio = 12 : 15 = 4 : 5

∴ The correct answer is 4 : 5.

Additional Information

Sum of Interior Angles

The sum of all interior angles of an n-sided polygon is given by (n − 2) × 180°.

Exterior Angle of Regular Polygon

Each exterior angle of a regular n-sided polygon is given by 360° ÷ n. The sum of all exterior angles is always 360°.

Number of Diagonals

The total number of diagonals in an n-sided polygon can be calculated using the formula [n × (n − 3)] ÷ 2.

Shortcut Trick

Side of the square = √Area = √121 = 11 m.

Distance walked in one round = Perimeter of the square.

Perimeter = 4 × Side = 4 × 11 = 44 m.

∴ The correct answer is 44 m.

Alternate Method

Given:

Area of the square plot = 121 m²

Formula Used:

Area of a square = (Side)²

Perimeter of a square = 4 × Side

Area = 121 m2 Side = 11 m 11 m 11 m 11 m 11 m Distance Walked = Perimeter

Calculations:

⇒ (Side)² = 121

⇒ Side = √121

⇒ Side = 11 m

The man walks once around the boundary, which is exactly equal to the perimeter of the square plot.

⇒ Perimeter = 4 × Side

⇒ Perimeter = 4 × 11

⇒ Perimeter = 44 m

∴ The correct answer is 44 m.

Additional Information

Area of a Square

The region enclosed by the boundaries of a square is its area. It is given by the square of its side: Area = a², or by using the diagonal length (d): Area = d² ÷ 2.

Perimeter of a Square

The total length of the outer boundary of the square is its perimeter. It is calculated by adding all four equal sides together: Perimeter = 4a.

Diagonal of a Square

The straight line segment connecting two opposite corners of a square. Its length is calculated using the Pythagorean theorem as: Diagonal = a√2.

Shortcut Trick

Equalize the gap between the ratio parts of the two numbers.

Initial ratio difference = 7 − 5 = 2. Final ratio difference = 11 − 8 = 3.

Multiply initial ratio by 3 ⇒ 15 : 21

Multiply final ratio by 2 ⇒ 16 : 22

Increase in ratio parts = 16 − 15 = 1 unit. Given increase = 9.

∴ The correct answer is 135 and 189.

Alternate Method

Given:

Initial ratio of the two numbers = 5 : 7

Number added to each = 9

Final ratio becomes = 8 : 11

     Initial Ratio Final Ratio  5 : 7 Gap = 2  8 : 11 Gap = 3   × 3 × 2  15 : 21  16 : 22  +1 unit = 9 Numbers: 15 × 9 = 135 | 21 × 9 = 189

Calculations:

⇒ Let the original numbers be 5x and 7x.

⇒ According to the question conditions:

⇒ (5x + 9) / (7x + 9) = 8 / 11

⇒ 11 × (5x + 9) = 8 × (7x + 9)

⇒ 55x + 99 = 56x + 72

⇒ 56x − 55x = 99 − 72

⇒ x = 27

⇒ First number = 5x = 5 × 27 = 135

⇒ Second number = 7x = 7 × 27 = 189

∴ The correct answer is 135 and 189.

Additional Information

Equating Ratios / Equalizing Gaps

When a constant number is added or subtracted from two terms in a ratio, you can equate the horizontal differences of the ratios. Multiply each ratio by the other's difference to standardize the unit value without variables.

Cross Multiplication Method

If the initial ratio is a:b, final ratio is c:d, and the changes are x and y respectively, then 1 ratio part = (cx − ay) ÷ (ad − bc). This acts as a direct algebraic shortcut.

Proportion Fundamentals

Four numbers a, b, c, d are in proportion if a : b = c : d, meaning ad = bc. The outer terms a and d are called extremes, while the inner terms b and c are called means.

Compound Ratio

The compound ratio of two sets of ratios a : b and c : d is defined as ac : bd. It is found by multiplying the corresponding antecedent and consequent terms together.

Shortcut Trick

To find the ratio of the first and last terms directly, multiply all intermediate ratios.

A/D = (A/B) × (B/C) × (C/D)

A/D = (3/4) × (6/7) × (14/15)

A/D = 252 / 420 = 3 / 5

∴ The correct answer is 3:5.

Alternate Method

Given:

A:B = 3:4

B:C = 6:7

C:D = 14:15

Formula Used:

Equalizing the common terms or using the adjacent filling method to find the combined ratio.

A B C D 3 4 4 4 6 6 7 7 14 14 14 15 252 336 392 420 A : B : C : D → 9 : 12 : 14 : 15

Calculations:

⇒ Using the adjacent filling method, we fill empty spaces with the nearest adjacent value in that row.

⇒ Multiply terms vertically: A = 3 × 6 × 14 = 252

⇒ D = 4 × 7 × 15 = 420

⇒ Combined ratio A:B:C:D = 252 : 336 : 392 : 420 ⇒ 9 : 12 : 14 : 15

⇒ Extracting A:D = 9 : 15 = 3 : 5

∴ The correct answer is 3:5.

Additional Information

Compound Ratio

When two or more ratios are multiplied term-wise, it is called a compound ratio. For a:b and c:d, the compound ratio is ac:bd.

Duplicate Ratio

The duplicate ratio of a:b is its square, i.e., a²:b². The sub-duplicate ratio is √a:√b.

Invertendo Property

If a:b = c:d, then by the invertendo property, b:a = d:c.

Shortcut Trick

Let the total number of balls in the bag be 100%.

Percentage of blue balls = 100% − (20% + 30%) = 50%

We are given that 50% represents 80 balls.

Total balls (100%) = 80 × (100 ÷ 50) = 160

∴ The correct answer is 160 balls.

Alternate Method

Given:

Percentage of red balls = 20%

Percentage of green balls = 30%

Number of blue balls = 80

20% 30% 50% (80 balls) Red Green Blue    Total Balls = 100%

Calculations:

⇒ Let the total number of balls in the bag be x.

⇒ Total percentage of red and green balls = 20% + 30% = 50%

⇒ Percentage of blue balls = 100% − 50% = 50%

⇒ 50% of x = 80

⇒ (50 ÷ 100) × x = 80

⇒ (1 ÷ 2) × x = 80

⇒ x = 80 × 2 = 160

∴ The correct answer is 160 balls.

Additional Information

Percentage Composition

When parts of a mixture or collection are expressed as a percentage of the same total, their individual percentages can be directly added or subtracted. The sum of all component percentages must exactly equal 100%.

Finding the Base Quantity

If p% of a total quantity is given as X, then the total quantity (which represents 100%) can be directly calculated by multiplying the partial value X with (100 ÷ p).

Shortcut Trick

Total right = Correct from attempted + Grace from unattempted

⇒ Total right = (80% of 75% of 200) + (10% of 25% of 200)

⇒ Total right = (0.8 × 150) + (0.1 × 50)

⇒ Total right = 120 + 5 = 125

∴ The correct answer is 125 questions.

Alternate Method

Given:

Total number of questions = 200

Percentage of attempted questions = 75%

Percentage of correct attempted questions = 80%

Percentage of unattempted questions awarded grace marks = 10%

Total: 200 Attempted: 150 Unattempted: 50 Correct: 120 Grace Marks: 5 Total Right = 120 + 5 = 125

Calculations:

⇒ Number of attempted questions = 75% of 200 = 150

⇒ Number of unattempted questions = 200 − 150 = 50

⇒ Number of correct attempted questions = 80% of 150 = 120

⇒ Number of unattempted questions awarded grace marks = 10% of 50 = 5

⇒ Total questions he got right = 120 + 5 = 125

∴ The correct answer is 125 questions.

Additional Information

Percentage Change

Percentage Change = (New Value − Old Value) ÷ Old Value × 100.

Successive Percentage

If a value is successively changed by a% and b%, the effective percentage change is given by (a + b + a×b ÷ 100)%.

Base Multiplication Rule

To find x% of y% of a quantity Z, use the direct calculation: (x ÷ 100) × (y ÷ 100) × Z.

Shortcut Trick

The direct relation between Cost Price (CP) and Marked Price (MP) is:

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

⇒ CP / MP = (100 − 20) / (100 + 10) = 80 / 110 = 8 / 11

⇒ Markup % = [(MP − CP) / CP] × 100 = [(11 − 8) / 8] × 100

⇒ Markup % = (3 / 8) × 100 = 37.5%

∴ The correct answer is 37.5%.

Alternate Method

Given:

Discount on Marked Price = 20%

Profit on Cost Price = 10%

Formula Used:

Selling Price (SP) = CP × (100 + Profit%) / 100

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

Markup % = [(MP − CP) / CP] × 100

           CP 100 SP 110 MP 137.5    Markup = +37.5% +10% Profit −20% Discount

Calculations:

Let the Cost Price (CP) be 100 units.

⇒ Selling Price (SP) = 100 × (100 + 10) / 100 = 110 units

⇒ 110 = MP × (100 − 20) / 100

⇒ 110 = MP × 80 / 100

⇒ MP = (110 × 100) / 80 = 137.5 units

Now, finding the percentage by which the Marked Price is above Cost Price:

⇒ Markup = MP − CP = 137.5 − 100 = 37.5 units

⇒ Markup % = (37.5 / 100) × 100 = 37.5%

∴ The correct answer is 37.5%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP) given by the retailer. The reduced value is the Selling Price (SP) = MP − Discount.

Successive Discounts

For two successive discounts of a% and b%, the effective net discount is equivalent to (a + b − ab/100)%.

Dishonest Dealer

When a dealer sells goods at CP but uses a false weight, Profit % = [Error ÷ (True Value − Error)] × 100.

Shortcut Trick

The score in the 9^th inning is 80, which is 40 more than the previous average of 40.

These extra 40 runs will be distributed equally among all the 9 innings.

Increase in average = 40 ÷ 9 ≈ 4.44

New average = Old average + Increase = 40 + 4.44 = 44.44

∴ The correct answer is 44.44.

Alternate Method

Avg: 40 8 Innings  Score: 80 9<sup>th</sup> Inning  New Avg: 44.44

Given:

Average score of the first 8 innings = 40

Score in the 9^th inning = 80

Formula Used:

Average = Total Sum of Values ÷ Number of Values

Calculations:

⇒ Total score of first 8 innings = 8 × 40 = 320

⇒ Score in the 9^th inning = 80

⇒ Total score after the 9^th inning = 320 + 80 = 400

⇒ Total number of innings played = 8 + 1 = 9

⇒ New average = 400 ÷ 9

⇒ New average ≈ 44.44

∴ The correct answer is 44.44.

Additional Information

Batting Average

A cricketer's batting average is calculated by dividing their total runs scored by the number of times they have been out. In problems where outs are not mentioned, it is taken as total runs divided by total innings played.

Deviation Method for Averages

When a new element is added to a group, the new average can be directly computed without finding total sums: New Average = Old Average + (Difference between new value and old average ÷ New total number of elements).

Combined Average

When combining two different sets of values, the combined average is calculated as A = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick

The sum of all 4 interior angles of a quadrilateral is 360°.

Sum of the remaining angles A, B, and C = 360° − 120° = 240°.

The angles A, B, and C are divided in the ratio 1 : 2 : 3 (Total = 6 parts).

Value of 1 part = 240° ÷ 6 = 40°.

Angle A corresponds to 1 part, which is 40°.

∴ The correct answer is 40°.

Alternate Method

Given:

Ratio of ∠A : ∠B : ∠C = 1 : 2 : 3

∠D = 120°

Formula Used:

Sum of interior angles of a quadrilateral = 360°

  x 2x 3x 120° A B C D Quadrilateral ABCD

Calculations:

⇒ Let the angles ∠A, ∠B, and ∠C be x, 2x, and 3x respectively.

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ x + 2x + 3x + 120° = 360°

⇒ 6x + 120° = 360°

⇒ 6x = 360° − 120°

⇒ 6x = 240°

⇒ x = 240° ÷ 6

⇒ x = 40°

⇒ ∠A = x = 40°

∴ The correct answer is 40°.

Additional Information

Polygon Angle Sum Property

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

Exterior Angles

The sum of exterior angles of any convex polygon, taken one at each vertex, is always 360° regardless of the total number of sides.

Special Quadrilaterals

In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°). In a parallelogram, opposite angles are equal, and adjacent angles sum to 180°.

Shortcut Trick

Total amount paid for 2 shirts = 2 × 1200 = Rs. 2400

Total shirts received = 2 + 1 = 3

Effective price per shirt = 2400 ÷ 3 = Rs. 800

∴ The correct answer is Rs. 800.

Alternate Method

Given:

Price of a single shirt = Rs. 1200

Combo offer = Buy 2 shirts, get a third shirt free

Shirt 1 Rs. 1200 Shirt 2 Rs. 1200 Shirt 3 FREE Total Cost = Rs. 2400 for 3 Shirts

Formula Used:

Effective Price = Total Amount Paid ÷ Total Quantity Received

Calculations:

⇒ Total amount paid for 2 shirts = 2 × 1200 = Rs. 2400

⇒ Total number of shirts received = 2 (paid) + 1 (free) = 3

⇒ Effective price per shirt = 2400 ÷ 3 = Rs. 800

∴ The correct answer is Rs. 800.

Additional Information

Buy X Get Y Free Discount Percentage

The effective discount percentage in such combo offers is calculated as: Discount % = [Y ÷ (X + Y)] × 100.

Effective Discount of Successive Discounts

For two successive discounts d₁ and d₂, the net effective discount is: d₁ + d₂ − (d₁ × d₂) ÷ 100.

Marked Price and Selling Price Relationship

Selling Price (SP) = Marked Price (MP) × (1 − Discount % ÷ 100).

Shortcut Trick

Total capacity of the tub = LCM(6, 4, 3) = 12 units.

Efficiency of Tap 1 (fill) = +2 units/min.

Efficiency of Tap 2 (fill) = +3 units/min.

Efficiency of Pipe (empty) = −4 units/min.

Net efficiency = 2 + 3 − 4 = +1 unit/min.

Time to fill the tub = Total capacity ÷ Net efficiency = 12 ÷ 1 = 12 min.

∴ The correct answer is 12 min.

Alternate Method

Given:

Tap 1 can fill the tub in 6 minutes.

Tap 2 can fill the tub in 4 minutes.

Pipe 3 can empty the tub in 3 minutes.

Calculations:

⇒ Net efficiency of all three operating together = (+2) + (+3) + (−4)

⇒ Net efficiency = 5 − 4 = +1 unit/min.

⇒ Time taken to fill the tub = Total Capacity ÷ Net efficiency

⇒ Time taken = 12 ÷ 1 = 12 minutes.

∴ The correct answer is 12 min.

Additional Information

Inlet vs Outlet Pipe

An inlet pipe fills the tank, hence its work or efficiency is taken as positive. An outlet pipe empties the tank, so its work or efficiency is taken as negative.

Fraction Method for Pipes

If a pipe fills a tank in x minutes, part filled in 1 minute is 1/x. If another pipe empties it in y minutes, part emptied in 1 minute is 1/y. Net part filled in 1 minute = (1/x) − (1/y).

Net Rate & State Reversal

If the net efficiency turns out to be negative when all pipes are open, it implies the system will eventually be completely emptied instead of being filled.

Shortcut Trick

Using the assumed average (net deviation) method:

Assume base average = Rs. 30,000.

Excess salary of 4 seniors = 4 × (60,000 − 30,000) = Rs. 120,000.

Net increment per employee = 120,000 ÷ 9 ≈ Rs. 13,333.33.

Combined Average = 30,000 + 13,333.33 = Rs. 43,333.33.

∴ The correct answer is Rs. 43,333.33.

Alternate Method

Given:

Average salary of 5 junior employees = Rs. 30,000

Average salary of 4 senior employees = Rs. 60,000

Formula Used:

Combined Average = (N₁ × A₁ + N₂ × A₂) ÷ (N₁ + N₂)

 Rs. 30,000 Rs. 60,000 Combined Avg 5 Juniors 4 Seniors  Distance: 4 parts  Distance: 5 parts

Calculations:

⇒ Total salary of juniors = 5 × 30,000 = Rs. 150,000

⇒ Total salary of seniors = 4 × 60,000 = Rs. 240,000

⇒ Combined total salary = 150,000 + 240,000 = Rs. 390,000

⇒ Total number of employees = 5 + 4 = 9

⇒ Combined Average = 390,000 ÷ 9

⇒ Combined Average = 130,000 ÷ 3 = 43,333.33

∴ The correct answer is Rs. 43,333.33.

Additional Information

Combined Average

It represents the overall average of two or more groups combined. Formula: A = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Weighted Average Method (Net Deviation)

Assume a base average (usually the smaller value), then calculate the total excess deviation for the other groups and distribute it equally among the total count of all entities.

Alligation Method

Often used as an alternate ratio technique to mix groups with different averages. The ratio of the quantities is universally calculated as: Ratio = (A₂ − Combined Avg) : (Combined Avg − A₁).

Shortcut Trick

Initial quantity of acid = 100 × (4/5) = 80 L.

First step: 20 L mixture replaced by water. Acid left = 80 × (1 − 20/100) = 64 L.

Second step: 20 L mixture replaced by pure acid. Acid left = 64 × (1 − 20/100) + 20 = 71.2 L.

Since the total volume remains 100 L, the final percentage of acid is 71.2%.

∴ The correct answer is 71.2%.

Alternate Method

Initial Acid: 80 L Water: 20 L -20L Mix +20L Water Step 1 Acid: 64 L Water: 36 L -20L Mix +20L Acid Step 2 Acid: 71.2 L Water: 28.8 L Final Acid Percentage = 71.2%

Given:

Total mixture = 100 L

Initial ratio of acid to water = 4 : 1

Calculations:

⇒ Initial Acid = 100 × (4/5) = 80 L

⇒ Initial Water = 100 × (1/5) = 20 L

Step 1: 20 L of mixture is removed and replaced with water.

⇒ Fraction of mixture removed = 20 / 100 = 1/5

⇒ Acid removed = 80 × (1/5) = 16 L

⇒ Remaining Acid = 80 − 16 = 64 L

⇒ Since total volume is maintained at 100 L, New Water = 100 − 64 = 36 L

Step 2: 20 L of mixture is removed and replaced with pure acid.

⇒ Acid removed = 64 × (1/5) = 12.8 L

⇒ Remaining Acid = 64 − 12.8 = 51.2 L

⇒ Pure acid added = 20 L

⇒ Final Acid = 51.2 + 20 = 71.2 L

⇒ Final percentage of acid = (71.2 / 100) × 100 = 71.2%

∴ The correct answer is 71.2%.

Additional Information

Standard Replacement Formula

When a quantity y is removed and replaced n times, Final Quantity = Initial Quantity × (1 − y / Total Volume)ⁿ.

Fractional Change in Mixtures

When removing a fraction of a mixture, all components of the mixture are removed in exactly the same fraction. It's best to track the component that is not being externally added in that step.

Successive Replacement

If the replaced liquid is different in each step, calculate the remaining amount step-by-step. Just multiply the previous remaining amount by the fraction left over and add any pure substance poured back in.

Shortcut Trick

In a right-angled triangle, a standard Pythagorean triplet is (5, 12, 13).

Given secA = Hypotenuse ÷ Base = 13 / 12.

So, the Perpendicular side must be 5.

We know sinA = Perpendicular ÷ Hypotenuse.

sinA = 5 / 13.

∴ The correct answer is 5/13.

Alternate Method

Given:

secA = 13 / 12

Angle A is an acute angle.

Formula Used:

secθ = Hypotenuse ÷ Base

sinθ = Perpendicular ÷ Hypotenuse

Perpendicular² + Base² = Hypotenuse²

A P Q R Base = 12k Perp = 5k Hypotenuse = 13k

Calculations:

⇒ Let Hypotenuse = 13k and Base = 12k.

⇒ According to Pythagoras theorem: Perpendicular² + Base² = Hypotenuse²

⇒ Perpendicular² + (12k)² = (13k)²

⇒ Perpendicular² + 144k² = 169k²

⇒ Perpendicular² = 169k² − 144k² = 25k²

⇒ Perpendicular = √(25k²) = 5k

⇒ sinA = Perpendicular ÷ Hypotenuse

⇒ sinA = 5k ÷ 13k = 5 / 13

∴ The correct answer is 5/13.

Additional Information

Trigonometric Ratios

In a right-angled triangle, sinθ = Perpendicular ÷ Hypotenuse, cosθ = Base ÷ Hypotenuse, and tanθ = Perpendicular ÷ Base.

Pythagorean Triplets

Sets of numbers that satisfy the Pythagoras theorem (a² + b² = c²) include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Fundamental Trigonometric Identities

The three basic identities are: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.