Quantitative Aptitude PYQs

Shortcut Trick

Sum of remaining 2 numbers = Total sum − (Sum of first 2 numbers + Third number)

Sum = 655 − (80 × 2 + 109) = 655 − 269 = 386

Average of remaining 2 numbers = 386 ÷ 2 = 193

∴ The correct answer is 193.

Alternate Method

Given:

Total sum of 5 numbers = 655

Average of the first 2 numbers = 80

The third number = 109

Formula Used:

Sum of numbers = Average × Number of items

Average = Sum of observations ÷ Number of observations

Total Sum of 5 numbers = 655

Sum = 160
Avg = 80 (2 nos)

109
(3^rd no)

Sum = 655 − 269 = 386
Avg = 386 ÷ 2 = 193

Calculations:

⇒ Sum of the first 2 numbers = 80 × 2 = 160

⇒ Sum of the first 3 numbers = 160 + 109 = 269

⇒ Sum of the remaining 2 numbers = Total sum − Sum of the first 3 numbers

⇒ Sum of the remaining 2 numbers = 655 − 269 = 386

⇒ Average of the remaining 2 numbers = 386 ÷ 2 = 193

∴ The correct answer is 193.

Additional Information

Weighted Average

When different groups have different averages, their combined average is calculated as (N₁A₁ + N₂A₂ + ...) ÷ (N₁ + N₂ + ...).

Change in Average

If a specific total is deducted from a group, the remaining sum distributed among the remaining items gives the new group average.

Shortcut Trick

Total parts in the ratio = 18 + 12 + 2 = 32

Difference between largest and smallest parts = 18 − 2 = 16

The difference is exactly half of the total parts (16 / 32 = 1/2).

Therefore, the difference in amount = 1/2 × Total Amount

Required difference = 658 ÷ 2 = 329

∴ The correct answer is ₹329.

Alternate Method

Given:

Total amount = ₹658

Ratio of shares = 18 : 12 : 2

Formula Used:

Share Difference = (Difference in ratio parts / Total ratio parts) × Total Amount

Total Amount = ₹658 (32 Parts)

18 Parts
Largest Share

12 Parts

2
Min

Largest - Smallest = 18 - 2 = 16 Parts
Difference = 16 / 32 × ₹658 = ₹329

Calculation:

⇒ Sum of all ratio parts = 18 + 12 + 2 = 32

⇒ Largest share part = 18

⇒ Smallest share part = 2

⇒ Difference in parts = 18 − 2 = 16

⇒ Required difference = (16 / 32) × 658

⇒ Required difference = (1 / 2) × 658

⇒ Required difference = 329

∴ The correct answer is ₹329.

Additional Information

Ratio Distribution

When a quantity N is divided in the ratio a : b : c, the share of the first part is calculated as (a / (a + b + c)) × N.

Difference in Shares

Instead of calculating individual shares and subtracting, directly calculate the difference in ratio parts and divide by the total ratio parts.

Proportion

An equality of two ratios is called a proportion. If a : b = c : d, then a × d = b × c (Product of extremes = Product of means).

Compound Ratio

For two ratios a : b and c : d, their compound ratio is obtained by multiplying the corresponding parts: (a × c) : (b × d).

Shortcut Trick

We can solve this quickly using the option elimination method based on basic divisibility rules.

→ Divisibility by 10: Last digit is 9, not 0, so it is not divisible by 10.

→ Divisibility by 12: The number is odd, so it is not divisible by 4 or 12.

→ Divisibility by 9: Sum of digits = 44, which is not a multiple of 9.

→ Since three options are completely eliminated, the number must be divisible by 19.

∴ The correct answer is 19.

Alternate Method

Given:

The given number is 444348269.

Method Used:

Direct long division method to check complete divisibility by 19.

Calculations:

⇒ 44 ÷ 19 = 2, Remainder = 6

⇒ Bring down 4: 64 ÷ 19 = 3, Remainder = 7

⇒ Bring down 3: 73 ÷ 19 = 3, Remainder = 16

⇒ Bring down 4: 164 ÷ 19 = 8, Remainder = 12

⇒ Bring down 8: 128 ÷ 19 = 6, Remainder = 14

⇒ Bring down 2: 142 ÷ 19 = 7, Remainder = 9

⇒ Bring down 6: 96 ÷ 19 = 5, Remainder = 1

⇒ Bring down 9: 19 ÷ 19 = 1, Remainder = 0

⇒ The quotient is 23386751 and the final remainder is 0.

⇒ Hence, the original number is completely divisible by 19.

∴ The correct answer is 19.

Additional Information

Divisibility by 19 (Osculator Method)

Add 2 × (last digit) to the remaining truncated number. Repeat the process until a 2-digit number is reached. If it is divisible by 19, the entire number is too.

Divisibility by 9

A number is divisible by 9 if the sum of all its digits is completely divisible by 9.

Divisibility by 12

A number is divisible by 12 if it is completely divisible by both its co-prime factors, i.e., 3 and 4.

Shortcut Trick

Convert the percentage changes into multiplying fractions: +25% = 5/4, +70% = 17/10, and −20% = 4/5.

Final Population = Initial Population × (5/4) × (17/10) × (4/5)

Notice that the fraction (5/4) and (4/5) perfectly cancel each other out.

Final Population = 14680 × (17/10) = 1468 × 17 = 24956.

∴ The correct answer is 24,956.

Alternate Method

Given:

Initial Population = 14,680

Population change in 1^st year = +25%

Population change in 2^nd year = +70%

Population change in 3^rd year = −20%

14,680
Initial

+25%

18,350

+70%

31,195

−20%

24,956
Final

Net Multiplier = 1.7

Formula Used:

Final Value = Initial Value × (1 ± R₁/100) × (1 ± R₂/100) × (1 ± R₃/100)

Calculations:

⇒ Final Population = 14680 × (1 + 25/100) × (1 + 70/100) × (1 − 20/100)

⇒ Final Population = 14680 × (125/100) × (170/100) × (80/100)

⇒ Final Population = 14680 × (5/4) × (17/10) × (4/5)

⇒ Final Population = 14680 × (17/10)

⇒ Final Population = 1468 × 17

⇒ Final Population = 24,956

∴ The correct answer is 24,956.

Additional Information

Successive Percentage Change

When a value is subjected to continuous percentage changes year on year, the final value is found directly by multiplying the initial value by all the respective consecutive multiplying factors. The order in which they are multiplied does not affect the final answer.

Effective Percentage Change

For two successive percentage changes of a% and b%, the net percentage change can also be determined using the specific formula: Net % = (a + b + ab/100)%.

Fractional Equivalents

Converting standard percentages to fractions significantly simplifies and accelerates calculation. For example, an increase of 25% corresponds to a multiplier of 5/4, and a decrease of 20% corresponds directly to 4/5.

Shortcut Trick

Assumed overall average = 42 (from the first 23 matches)

Deviation of the next 10 matches = 15 − 42 = −27

Total deviation = 10 × (−27) = −270

Net change in average = −270 ÷ 33 ≈ −8.18

Overall Average = 42 − 8.18 = 33.82

∴ The correct answer is 33.82 runs.

Alternate Method

Given:

Average runs of 23 matches = 42

Average runs of next 10 matches = 15

Formula Used:

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂)

23 Matches
Avg: 42

10 Matches
Avg: 15

Total 33 Matches
Overall Avg = ?

Calculations:

⇒ Total runs in first 23 matches = 23 × 42 = 966

⇒ Total runs in next 10 matches = 10 × 15 = 150

⇒ Total runs in all 33 matches = 966 + 150 = 1116

⇒ Combined Average = 1116 ÷ 33

⇒ Combined Average = 372 ÷ 11 = 33.8181...

⇒ Rounding off to two decimal places, the average is 33.82.

∴ The correct answer is 33.82 runs.

Additional Information

Weighted Average

Used when different groups have different sizes. The formula is A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method for Averages

An assumed average is taken and the net deviation of all values from this assumed average is calculated. The true average is Assumed Average + (Net Deviation ÷ Total Count).

Combined Average

It represents the single average value of two or more independent groups combined together, perfectly preserving the total sum of all values across the groups.

Shortcut Trick

Total Work (LCM of 65, 26) = 130 units.

Efficiency: A = 2, B = 5.

Pipe A works for the entire 30 minutes: 2 × 30 = 60 units.

Remaining work for Pipe B: 130 − 60 = 70 units.

Time for Pipe B: 70 ÷ 5 = 14 minutes.

∴ The correct answer is 14 min.

Alternate Method

Given:

Time taken by Pipe A = 65 minutes

Time taken by Pipe B = 26 minutes

Total time to fill the cistern = 30 minutes

Calculations:

⇒ Let the total capacity of the cistern be the LCM of 65 and 26 = 130 units.

⇒ Efficiency of Pipe A = 130 ÷ 65 = 2 units/min.

⇒ Efficiency of Pipe B = 130 ÷ 26 = 5 units/min.

⇒ Since the cistern is filled in 30 minutes, Pipe A worked for the full 30 minutes.

⇒ Work done by Pipe A = 30 × 2 = 60 units.

⇒ Remaining work to be done by Pipe B = 130 − 60 = 70 units.

⇒ Time for which Pipe B was open = 70 ÷ 5 = 14 minutes.

∴ The correct answer is 14 min.

Additional Information

Pipes & Cisterns Efficiency

Efficiency is inversely proportional to the time taken to fill a tank. Total Work = Efficiency × Time.

Inlet and Outlet Pipes

Inlet pipes have positive efficiency (filling), while outlet pipes (leaks) have negative efficiency (emptying).

Combined Work Formula

If two pipes fill a tank in x and y hours, together they fill it in (x × y) ÷ (x + y) hours.

Shortcut Trick

Given: 8 × (2 × Base Area) = 5 × CSA

⇒ 8 × (2πr²) = 5 × (2πrh) ⇒ 16r = 10h ⇒ h = 1.6r

Since r = 21/2 = 10.5 cm, then h = 1.6 × 10.5 = 16.8 cm.

Volume = πr²h = (22/7) × 10.5 × 10.5 × 16.8 = 346.5 × 16.8 = 5821.2 cm³.

∴ The correct answer is 5821.2 cm³.

Alternate Method

Given:

Base diameter of cylinder (d) = 21 cm ⇒ Radius (r) = 10.5 cm

8 × (Sum of 2 circular faces) = 5 × (Curved Surface Area)

Formula Used:

Area of circular face = πr²

Curved Surface Area (CSA) = 2πrh

Volume of Cylinder = πr²h

Calculations:

⇒ 8 × (2 × πr²) = 5 × (2πrh)

⇒ 16πr² = 10πrh

⇒ 16r = 10h

⇒ h = (16 × 10.5) ÷ 10

⇒ h = 168 ÷ 10 = 16.8 cm

⇒ Volume = (22/7) × (10.5)² × 16.8

⇒ Volume = (22/7) × 110.25 × 16.8

⇒ Volume = 22 × 15.75 × 16.8

⇒ Volume = 346.5 × 16.8

⇒ Volume = 5821.2 cm³

∴ The correct answer is 5821.2 cm³.

Additional Information

Total Surface Area of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Hollow Cylinder Volume

If R is the external radius and r is the internal radius, the volume of the material is V = π(R² − r²)h.

Relation between Volume and Area

The volume can also be expressed as Base Area × Height. For a cylinder, it is (πr²) × h.

Shortcut Trick

Net selling price = Initial CP × (100 + Profit%)/100 × (100 − Loss%)/100

Selling Price = 5,000 × (120/100) × (90/100)

Selling Price = 5,000 × 1.2 × 0.9

Selling Price = 6,000 × 0.9 = 5,400

∴ The correct answer is ₹5,400.

Alternate Method

Given: CP of Ramesh = ₹5,000 | Profit% = 20% | Loss% = 10%
Formulas: SP = CP × (100 + P%)/100 | SP = CP × (100 − L%)/100

⇒ Selling price for Ramesh (which is Cost price for Suresh) = 5,000 × (120/100) = ₹6,000.

⇒ Selling price for Suresh = 6,000 × (90/100).

⇒ Suresh Selling Price = 60 × 90 = ₹5,400.

∴ The correct answer is ₹5,400.

Additional Information

Chain Rule in Profit and Loss

When an article is sold through multiple hands, the Selling Price of the previous person becomes the Cost Price for the next buyer.

Effective Percentage Change

Net effect = a + b + (ab/100). Here, 20 − 10 − (200/100) = 8% net profit on initial CP. 5,000 + 8% of 5,000 = 5,400.

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

⇒ (1/20 ÷ 11/60) × 100

⇒ (1/20 × 60/11) × 100

⇒ (3/11) × 100 ≈ 27.27%

∴ The correct answer is 27.27%.

Alternate Method

Given: First fraction = 1/20, Second fraction = 11/60

Formula Used: Percentage = (Value ÷ Base) × 100

⇒ Let 1/20 be x% of 11/60

⇒ 1/20 = (x/100) × (11/60)

⇒ x = (1/20) × (60/11) × 100

⇒ x = (3/11) × 100

⇒ x = 300 ÷ 11

⇒ x = 27.2727...

∴ The correct answer is 27.27%.

Additional Information

Fraction to Percentage Conversion

To convert any fraction a/b to a percentage, multiply the fraction by 100 and add the % symbol.

Reciprocal Percentages

Standard values like 1/11 ≈ 9.09% are helpful; thus 3/11 = 3 × 9.09% = 27.27%.

Shortcut Trick

Speed of train = 63 × (5/18) = 17.5 m/s.

Time taken to cross Platform = 60 seconds.

Time taken to cross Man (Train length only) = 40 seconds.

Time difference to cover Platform length = 60 − 40 = 20 seconds.

Platform Length = Speed × Time Difference = 17.5 × 20 = 350 m.

∴ The correct answer is 350 metres.

Alternate Method

Given: Speed = 63 km/hr, Time (Platform) = 60 s, Time (Man) = 40 s.

Formula Used: Distance = Speed × Time; 1 km/hr = 5/18 m/s.

⇒ Speed in m/s = 63 × (5/18) = 17.5 m/s

⇒ Length of Train (L_t) = Speed × Time to cross man

⇒ L_t = 17.5 × 40 = 700 m

⇒ Total distance to cross Platform = L_t + L_p = Speed × 60

⇒ 700 + L_p = 17.5 × 60 = 1050 m

⇒ L_p = 1050 − 700 = 350 m

∴ The correct answer is 350 metres.

Additional Information

Relative Speed

When two objects move in the same direction, relative speed is S₁ − S₂. In opposite directions, it is S₁ + S₂.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Train Crossing Objects

For a stationary point object (man/pole), distance = train length. For a stationary long object (bridge/platform), distance = train length + object length.

Shortcut Trick

The Mode is the observation with the highest repetition (frequency).

Scanning the data: 35 appears 3 times, while 24 and 20 appear only 2 times.

All other numbers in the set appear only once.

Because 35 occurs the most, it is the Mode.

∴ The correct answer is 35.

Alternate Method

Given: The set of observations is 21, 29, 35, 24, 25, 33, 24, 23, 26, 28, 35, 27, 20, 35, 20.

Formula Used:Mode = Value that occurs most frequently in the distribution.

⇒ Arrange observations in ascending order: 20, 20, 21, 23, 24, 24, 25, 26, 27, 28, 29, 33, 35, 35, 35.

⇒ Count of 20 = 2

⇒ Count of 24 = 2

⇒ Count of 35 = 3

⇒ Count of others = 1

⇒ The maximum frequency is 3, which corresponds to the value 35.

∴ The correct answer is 35.

Additional Information

Mean (Arithmetic Average)

The sum of all observations divided by the total number of observations. Mean = (Σx_i) ÷ n.

Median

The middle value of a data set when it is arranged in ascending or descending order. For n = 15, Median is the ((15+1)÷2)^th i.e., 8^th term.

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Range

The difference between the maximum and minimum values in a data set. Range = Max − Min.

Shortcut Trick

If every value in a set is decreased by a constant 'k', the new average is simply the old average minus 'k'.

Given Old Average = 23 and Decrease (k) = 7.

New Average = 23 − 7 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Number of observations (n) = 7

Initial Average = 23

Value decreased from each number = 7

Calculations:

⇒ Total Sum of 7 numbers = Average × n

⇒ Initial Sum = 23 × 7 = 161

⇒ Total decrease across all 7 numbers = 7 × 7 = 49

⇒ New Sum = 161 − 49 = 112

⇒ New Average = New Sum ÷ n

⇒ New Average = 112 ÷ 7 = 16

∴ The correct answer is 16.

Additional Information

Effect of Arithmetic Operations on Average

If every number is increased by 'k', the new average is Old Average + k. If every number is decreased by 'k', it is Old Average − k.

Multiplication and Division Rule

If every number is multiplied by 'k', the new average is Old Average × k. Similarly, if divided by 'k' (where k ≠ 0), it is Old Average ÷ k.

Sum of Observations

The Sum of Observations is always equal to the Average × Number of items. This is the fundamental basis for all average-related calculations.

Shortcut Trick

Let the Marked Price be MP.

Successive discounts of 35% and 20% mean the final Selling Price (SP) is 6,500.

Using the multiplier method: MP × (100 − 35) ÷ 100 × (100 − 20) ÷ 100 = SP

MP × 0.65 × 0.80 = 6,500

MP × 0.52 = 6,500 ⇒ MP = 6,500 ÷ 0.52 = 12,500

∴ The correct answer is ₹12,500.

Alternate Method

Given: Selling Price (SP) = ₹6,500, Discount₁ (d₁) = 35%, Discount₂ (d₂) = 20%.
Formulas: SP = MP × (100 − d₁)% × (100 − d₂)% | Net Discount = a + b − (a × b) ÷ 100

⇒ Equivalent single discount = 35 + 20 − (35 × 20) ÷ 100

⇒ Net Discount = 55 − 7 = 48%

⇒ If MP is 100%, then SP = 100% − 48% = 52%

⇒ 52% of MP = 6,500

⇒ MP = (6,500 × 100) ÷ 52 = 12,500

∴ The correct answer is ₹12,500.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The second discount is calculated on the price remaining after the first discount.

Marked Price (MP)

It is the price displayed on the product label. Discount is always calculated on the Marked Price unless stated otherwise.

Shortcut Trick

Income ratio = 5 : 9, Expenditure ratio = 1 : 3. Difference in income parts = 9 − 5 = 4. Difference in expenditure parts = 3 − 1 = 2.

To make difference equal, multiply expenditure ratio by 2 → 2 : 6.

Savings (Income − Expenditure) = 5 − 2 = 3 units or 9 − 6 = 3 units.

3 units = ₹45,000 ⇒ 1 unit = ₹15,000.

Mukund monthly income = 5 units = 5 × ₹15,000 = ₹75,000.

∴ The correct answer is ₹75,000.

Alternate Method

Given:

Ratio of monthly income of Mukund and Sachin = 5 : 9

Ratio of their monthly expenditure = 1 : 3

Savings of each = ₹45,000

Formula Used:

Income − Savings = Expenditure

Calculations:

Let income of Mukund be 5x and Sachin be 9x.

⇒ (5x − 45000) / (9x − 45000) = 1 / 3

⇒ 3 × (5x − 45000) = 1 × (9x − 45000)

⇒ 15x − 1,35,000 = 9x − 45,000

⇒ 15x − 9x = 1,35,000 − 45,000

⇒ 6x = 90,000

⇒ x = 15,000

⇒ Mukund monthly income = 5x = 5 × 15,000 = 75,000

∴ The correct answer is ₹75,000.

Additional Information

Income, Expenditure and Savings Relationship

The fundamental relation used in these problems is: Income = Expenditure + Savings.

Ratio Balancing Method

When the absolute change (savings) is same for both, we equalize the difference between the ratio terms of Income and Expenditure to find 1 unit value.

Cross Multiplication Method

Used for two ratios with constant or different additions: (a × d − b × c) units = (Constant × d − Constant × c).

Shortcut Trick

Test small values for x because the total ₹935 is close to a multiple of 180.

If x = 5, then cost of pens = 180 × 5 = ₹900.

Remaining amount = 935 − 900 = ₹35.

Number of pencils y = 35 ÷ 3.5 = 10.

Check total items: x + y = 5 + 10 = 15 (Satisfied).

Calculation: x² + y² = 5² + 10² = 25 + 100 = 125.

∴ The correct answer is 125.

Alternate Method

Given:

Price of branded pen = ₹180

Price of pencil = ₹3.50

Total items (x + y) = 15

Total cost = ₹935

Formula Used:

Total Cost = (Price per Pen × Number of Pens) + (Price per Pencil × Number of Pencils)

Calculations:

⇒ 180x + 3.5y = 935 ...

(i) ⇒ x + y = 15 → y = 15 − x ...

(ii) ⇒ Substituting (ii) in (i):

⇒ 180x + 3.5(15 − x) = 935

⇒ 180x + 52.5 − 3.5x = 935

⇒ 176.5x = 935 − 52.5

⇒ 176.5x = 882.5

⇒ x = 882.5 ÷ 176.5 = 5

⇒ y = 15 − 5 = 10

⇒ x² + y² = 5² + 10²

⇒ x² + y² = 25 + 100 = 125

∴ The correct answer is 125.

Additional Information

Algebraic Identity

The sum of squares can be related to the sum of terms using: x² + y² = (x + y)² − 2xy.

Substitution Method

In a system of two linear equations, express one variable in terms of the other from the simpler equation and substitute it into the complex one.

Averages & Sums

If the average price is A, then Total Cost = A × (x + y). Here, Average ≈ 935 ÷ 15 ≈ 62.33.

Shortcut Trick

Use the Algebraic Identity: (a + b)² − (a − b)² = 4ab

Here, let a = 9x and b = 4y.

Numerator = 4 × (9x) × (4y) = 144xy.

Divide by 24x: 144xy ÷ 24x = 6y.

∴ The correct answer is 6y.

Alternate Method

Given: [(9x + 4y)² − (9x − 4y)²] ÷ 24x

Formula Used: Expansion of (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ Expansion 1: (9x + 4y)² = 81x² + 16y² + 72xy

⇒ Expansion 2: (9x − 4y)² = 81x² + 16y² − 72xy

⇒ Subtracting: (81x² + 16y² + 72xy) − (81x² + 16y² − 72xy)

⇒ Result of Numerator: 81x² − 81x² + 16y² − 16y² + 72xy + 72xy = 144xy

⇒ Final Division: 144xy ÷ 24x = (144/24) × (x/x) × y

⇒ Simplified Value: 6y

∴ The correct answer is 6y.

Additional Information

Standard Algebraic Identities

1. (a + b)² + (a − b)² = 2(a² + b²)
2. a² − b² = (a + b)(a − b)

Simplification Rules

Always apply BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) to solve multi-operational algebraic expressions efficiently.

Shortcut Trick

The angle between the bisectors of two adjacent angles of a quadrilateral is half the sum of the other two angles:

∠STR = (∠P + ∠Q) ÷ 2

⇒ 36° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 72°

Given the ratio of ∠P : ∠Q = 3 : 5, the total is 3 + 5 = 8 units.

⇒ 8 units = 72° ⇒ 1 unit = 9°

The difference between ∠Q and ∠P is 5 − 3 = 2 units ⇒ 2 × 9° = 18°.

∴ The correct answer is 18°.

Alternate Method

Given:

In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at T.

∠STR = 36°

Ratio of ∠P : ∠Q = 3 : 5

Formula Used:

Sum of angles in a triangle = 180°

Sum of angles in a quadrilateral = 360°

Calculations:

In triangle STR:

⇒ ∠STR + ∠TSR + ∠TRS = 180°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2 and ∠TRS = ∠R ÷ 2

⇒ 36° + (∠S ÷ 2) + (∠R ÷ 2) = 180°

⇒ (∠S + ∠R) ÷ 2 = 180° − 36°

⇒ (∠S + ∠R) ÷ 2 = 144°

⇒ ∠S + ∠R = 288°

In quadrilateral PQRS, the sum of all interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 288° = 360°

⇒ ∠P + ∠Q = 360° − 288° = 72°

Let the measures of ∠P and ∠Q be 3x and 5x respectively:

⇒ 3x + 5x = 72°

⇒ 8x = 72°

⇒ x = 9°

The difference between the measures of ∠Q and ∠P is:

⇒ ∠Q − ∠P = 5x − 3x = 2x

⇒ Difference = 2 × 9° = 18°

∴ The correct answer is 18°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles. For adjacent angles R and S, ∠STR = (∠P + ∠Q) ÷ 2.

Sum of Interior Angles of a Polygon

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

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral intersect, the angle between them is half the difference of the other two opposite angles.

Shortcut Trick

Substitute standard trigonometric values: sin 60° = √3/2, cos 30° = √3/2, and tan 45° = 1.

Expression = (√3/2 × √3/2) ÷ 1 = 3/4.

∴ The correct answer is 3/4.

Alternate Method

Given:

Expression: (sin 60° × cos 30°) ÷ tan 45°

Standard Trigonometric Values:

⇒ sin 60° = √3/2

⇒ cos 30° = √3/2

⇒ tan 45° = 1

Calculation:

⇒ Expression = [(√3/2) × (√3/2)] ÷ 1

⇒ Expression = [(√3 × √3) / (2 × 2)] ÷ 1

⇒ Expression = (3 / 4) / 1

⇒ Expression = 3/4

∴ The correct answer is 3/4.

Additional Information

Complementary Angles

For any angle θ, sin θ = cos(90° − θ). Therefore, sin 60° = cos 30° = √3/2.

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.

Tangent of 45 Degrees

At 45°, the opposite and adjacent sides of a right-angled triangle are equal, so tan 45° = 1.

Shortcut Trick

Total Work = LCM of (2, 4, 4) = 4 units.

Individual efficiencies: A = 2, B = 1, C = 1.

Combined efficiency = 2 + 1 + 1 = 4 units/day.

Thrice the work = 3 × 4 = 12 units.

Time taken = 12 ÷ 4 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given: Time taken by A = 2 days, B = 4 days, C = 4 days.

Formula Used: Efficiency = Total Work ÷ Time Taken; Time = Total Work ÷ Combined Efficiency.

⇒ Total Efficiency of (A + B + C) = 2 + 1 + 1 = 4 units/day.

⇒ Work to be completed = 3 × Total Work = 3 × 4 = 12 units.

⇒ Required Time = 12 ÷ 4 = 3 days.

∴ The correct answer is 3 days.

Additional Information

Total Work Concept

Total work is usually assumed as the LCM (Least Common Multiple) of the number of days taken by each person to simplify calculations.

Work and Efficiency

Efficiency is inversely proportional to the time taken. If a person completes work in 'n' days, their efficiency is 1/n per day.

Combined Work Rule

If multiple people work together, their individual efficiencies are added to find the total rate of work per day.

Shortcut Trick

Net discount for 6% and 5% = 6 + 5 − (6 × 5 ÷ 100) = 11 − 0.3 = 10.7%.

Net discount for 10.7% and 11% = 10.7 + 11 − (10.7 × 11 ÷ 100).

⇒ 21.7 − (117.7 ÷ 100) = 21.7 − 1.177 = 20.523%.

∴ The correct answer is 20.52%.

Alternate Method

Given: Three consecutive discounts d₁ = 6%, d₂ = 5%, and d₃ = 11%.

Formula: Net SP = MP × (100 − d₁)% × (100 − d₂)% × (100 − d₃)%

Let the Marked Price (MP) be 100.

⇒ SP = 100 × (94 ÷ 100) × (95 ÷ 100) × (89 ÷ 100)

⇒ SP = 100 × 0.94 × 0.95 × 0.89

⇒ SP = 89.3 × 0.89 = 79.477

⇒ Single Equivalent Discount = MP − SP = 100 − 79.477 = 20.523%

∴ The correct answer is 20.52%.

Additional Information

Successive Discount Formula

For two discounts a% and b%, the single equivalent discount is (a + b − ab ÷ 100)%.

Effective Price Calculation

When multiple discounts are applied, the order of application does not change the final Selling Price.