Shortcut Trick

Total weight of all 3 persons = 46 × 3 = 138 kg.

Total weight of Vipul and Mohan = 36 × 2 = 72 kg.

Total weight of Mohan and Mohammad = 43 × 2 = 86 kg.

Weight of the overlapping person (Mohan) = (Sum of first two + Sum of last two) − Total weight of all three.

Weight of Mohan = (72 + 86) − 138 = 158 − 138 = 20 kg.

∴ The correct answer is 20 kg.

Alternate Method

Given:

Average weight of Vipul, Mohan, and Mohammad = 46 kg

Average weight of Vipul and Mohan = 36 kg

Average weight of Mohan and Mohammad = 43 kg

Formula Used:

Total Value = Average × Number of Entities

Total Weight = 46 × 3 = 138 kg

Vipul

Mohan

Mohammad

V + M = 72 kg

M + Md = 86 kg

Mohan's Weight = (72 + 86) − 138 = 20 kg

Calculations:

⇒ Let the weights of Vipul, Mohan, and Mohammad be V, M, and Md respectively.

⇒ Total weight of all three (V + M + Md) = 46 × 3 = 138 kg

⇒ Total weight of Vipul and Mohan (V + M) = 36 × 2 = 72 kg

⇒ Total weight of Mohan and Mohammad (M + Md) = 43 × 2 = 86 kg

⇒ Adding both equations: (V + M) + (M + Md) = 72 + 86 = 158 kg

⇒ This simplifies to: V + 2M + Md = 158 kg

⇒ Subtracting the total weight from this equation yields Mohan's weight:

⇒ M = (V + 2M + Md) − (V + M + Md)

⇒ M = 158 − 138 = 20 kg

∴ The correct answer is 20 kg.

Additional Information

Combined Average

When two sets with n₁ and n₂ elements have averages A₁ and A₂ respectively, their combined average is computed as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Weighted Average

Calculated when different sets of values contribute differently to the overall measure. The formula is (w₁x₁ + w₂x₂ + ... ) ÷ (w₁ + w₂ + ...).

Net Deviation Method

The overall average can quickly be found by assigning an assumed average and calculating the net deviation of all values. True Average = Assumed Average + (Net Deviation ÷ Total Observations).

Shortcut Trick

By the concept of weighted average, if the overall literacy rate of a population is x% and the literacy rate of one component (males) is also exactly x%, then the literacy rate of the remaining component (females) must also be exactly x%.

Since the overall literacy is 63% and male literacy is also 63%, the female literacy rate must equal the overall average.

Therefore, no calculations are required to find the percentage of literate females.

∴ The correct answer is 63%.

Alternate Method

Given:

Total population = 364000

Total number of males = 178000

Total literate population = 63%

Literate males = 63%

Total Population = 364,000
Literate = 63%

Males = 178,000
Literate = 63%

Females = 186,000
Literate = ?

Calculations:

⇒ Total females = Total population − Total males

⇒ Total females = 364000 − 178000 = 186000

⇒ Total literate persons = (63 ÷ 100) × 364000 = 229320

⇒ Total literate males = (63 ÷ 100) × 178000 = 112140

⇒ Total literate females = Total literates − Male literates

⇒ Total literate females = 229320 − 112140 = 117180

⇒ Percentage of literate females = (Total literate females ÷ Total females) × 100

⇒ Percentage of literate females = (117180 ÷ 186000) × 100 = 63%

∴ The correct answer is 63%.

Additional Information

Weighted Average Property

If two distinct components A and B combine to form a whole, and both component A and the whole share the exact same average or percentage, then component B must also independently hold that same average or percentage.

Mixture and Alligation Principle

When two groups are merged, the ratio of their quantities is inversely proportional to the differences between their individual averages and the overall average. If a group's average equals the overall average, the difference is zero, forcing the other group's average to also equal the overall average.

Percentage Calculation Formula

The percentage of a specific subset relative to its base is always determined by the formula: (Subset Value ÷ Total Base Value) × 100.

Shortcut Trick

To find the length directly in kilometres, convert the given time into hours.

Time = 1.4 minutes = 1.4 ÷ 60 hours

Distance = Speed × Time

Distance = 75 × (1.4 ÷ 60) = (75 × 1.4) ÷ 60

Distance = 105 ÷ 60 = 1.75 km

∴ The correct answer is 1.75 km.

Alternate Method

Given:

Speed of the car = 75 km/hr

Time taken to cross the bridge = 1.4 minutes

Formula Used:

Distance = Speed × Time

1 km/hr = 5/18 m/s

Time taken = 1.4 minutes

Speed = 75 km/hr

Length of bridge = ?

Calculations:

⇒ Speed of the car = 75 × (5/18) = 125/6 m/s

⇒ Time taken = 1.4 × 60 = 84 seconds

⇒ Length of the bridge = Distance covered by the car

⇒ Distance = Speed × Time

⇒ Distance = (125/6) × 84 = 125 × 14

⇒ Distance = 1750 m

⇒ Distance in km = 1750 ÷ 1000 = 1.75 km

∴ The correct answer is 1.75 km.

Additional Information

Distance, Speed and Time Formula

Distance = Speed × Time. This is the fundamental relation used to calculate the length of the bridge covered by any vehicle.

Unit Conversion for Speed

To convert km/hr to m/s, multiply the speed value by 5/18. To convert back from m/s to km/hr, multiply by 18/5.

Average Speed Concept

When a journey is covered at varying speeds, Average Speed = Total Distance ÷ Total Time. For two equal distances covered at speeds A and B, Average Speed = (2 × A × B) ÷ (A + B).

Shortcut Trick

From the second condition, Mother's age M = 6D + 9.

Substitute directly into the first condition: 2(6D + 9) = 9D + 27

12D + 18 = 9D + 27 ⇒ 3D = 9 ⇒ Daughter's age D = 3 years.

Mother's age M = 6 × 3 + 9 = 27 years.

Difference in their ages = 27 − 3 = 24 years.

∴ The correct answer is 24 years.

Alternate Method

Given:

2 × Mother's age = 9 × Daughter's age + 27

6 × Daughter's age = Mother's age − 9

Mother

27 years

Daughter

3

Difference = 24 years

Calculations:

Let Mother's present age be M and Daughter's present age be D.

⇒ 2M = 9D + 27     --- (Equation 1)

⇒ 6D = M − 9

⇒ M = 6D + 9     --- (Equation 2)

Substitute Equation 2 into Equation 1:

⇒ 2 × (6D + 9) = 9D + 27

⇒ 12D + 18 = 9D + 27

⇒ 12D − 9D = 27 − 18

⇒ 3D = 9

⇒ D = 3

Substitute D = 3 into Equation 2:

⇒ M = 6 × 3 + 9

⇒ M = 18 + 9 = 27

⇒ Difference between ages = M − D

⇒ Difference = 27 − 3 = 24 years

∴ The correct answer is 24 years.

Additional Information

Problems on Ages

Generally, age problems are solved by assuming the present age of the unknowns as variables like x and y. All past and future ages must be formulated relative to these present ages.

Linear Equations Substitution Method

A system of equations can be effectively solved by isolating one variable from the first equation and directly substituting its expression into the second equation to eliminate unknowns.

Shortcut Trick

Year 2024 is a leap year, so total days from 23 Feb to 24 Apr = 6 (Feb) + 31 (Mar) + 24 (Apr) = 61 days.

Time (T) in years = 61 ÷ 366 = 1/6 year.

Direct formula: SI = P × R × T ÷ 100

SI = 4000 × 6.75 × (1/6) ÷ 100 = 40 × 1.125 = 45.

∴ The correct answer is ₹45.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate (R) = 6.75% per annum

Date of deposit = 23 February 2024

Date of withdrawal = 24 April 2024

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

23 Feb 2024
24 Apr 2024

Time (T) = 61 Days

P = ₹4000

R = 6.75%

SI = ₹45

Calculations:

⇒ Year 2024 is divisible by 4, hence it is a leap year and February has 29 days.

⇒ Number of days in February (excluding deposit date) = 29 − 23 = 6 days

⇒ Number of days in March = 31 days

⇒ Number of days in April = 24 days

⇒ Total time (T) in days = 6 + 31 + 24 = 61 days

⇒ Since it is a leap year, total days in the year = 366 days

⇒ Time (T) in years = 61 ÷ 366 = 1/6 year

⇒ SI = (4000 × 6.75 × (1/6)) ÷ 100

⇒ SI = 40 × 6.75 × (1/6)

⇒ SI = 270 ÷ 6

⇒ SI = 45

∴ The correct answer is ₹45.

Additional Information

Simple Interest (SI)

It is calculated only on the original principal amount. The formula is SI = (P × R × T) ÷ 100.

Calculation of Days

Usually, the day of deposit is excluded and the day of withdrawal is included while calculating the total time period in days.

Leap Year Rule

A year is a leap year if it is exactly divisible by 4 (and by 400 if it is a century year). February has 29 days, making the total year 366 days.

Shortcut Trick

Multiplying factor for a 30% discount = (100 − 30)% = 0.70

Multiplying factor for a 12% tax = (100 + 12)% = 1.12

Final Price = Marked Price × 0.70 × 1.12

⇒ Final Price = 8000 × 0.70 × 1.12 = 5600 × 1.12 = &rupee;6272

∴ The correct answer is ₹6272.

Alternate Method

Given:

Marked Price (MP) of jacket = &rupee;8,000

Discount offered = 30%

Sales Tax added = 12%

Formula Used:

Discounted Price = MP × (100 − Discount%) ÷ 100

Final Price = Discounted Price × (100 + Tax%) ÷ 100

₹8000
Marked Price

₹5600
After Discount

₹6272
Final Price

− 30%
Discount

• 12% Tax Added

Final Price = 8000 × (70/100) × (112/100) = ₹6272

Calculations:

⇒ Discounted Price = 8000 × (100 − 30) ÷ 100

⇒ Discounted Price = 8000 × 70 ÷ 100 = &rupee;5600

⇒ Final Price = Discounted Price × (100 + 12) ÷ 100

⇒ Final Price = 5600 × 112 ÷ 100

⇒ Final Price = 56 × 112 = &rupee;6272

∴ The correct answer is ₹6272.

Additional Information

Successive Percentage Changes

When multiple percentage changes occur on a single item, the final value can be calculated directly by multiplying the initial value by the respective multiplying factors: Final = Initial × (1 ± R₁/100) × (1 ± R₂/100).

Discount vs Tax

A discount is always subtracted from the Marked Price, reducing the cost. Conversely, tax is always calculated on the discounted price and added to it, increasing the final amount paid by the customer.

Effective Percentage Formula

For two successive changes A% and B%, the net percentage change is given by: Net Change = A + B + (A × B)/100. For a 30% decrease and 12% increase, the net change = −30 + 12 + (−30 × 12)/100 = −18 − 3.6 = −21.6%.

Shortcut Trick

Use the value putting method by assuming A = 30° (avoid 45° as it makes the denominator zero).

Expression = (sin 30° − 2sin³30°) ÷ (2cos³30° − cos 30°) + √(cosec²30° − 1)

⇒ (1/2 − 2×(1/8)) ÷ (2×(3√3/8) − √3/2) + √(4 − 1)

⇒ (1/4) ÷ (√3/4) + √3 = 1/√3 + √3 = 4/√3

Checking Option 1: sec 30° × cosec 30° = (2/√3) × 2 = 4/√3 (Matches perfectly).

∴ The correct answer is sec A cosec A.

Alternate Method

Given:

(sin A − 2sin³A) / (2cos³A − cos A) + √(cosec²A − 1)

Formula Used:

cos 2A = 1 − 2sin²A = 2cos²A − 1

cosec²A − 1 = cot²A

Calculation:

⇒ [sin A × (1 − 2sin²A)] / [cos A × (2cos²A − 1)] + √(cot²A)

⇒ [sin A × cos 2A] / [cos A × cos 2A] + cot A

⇒ (sin A / cos A) + cot A

⇒ tan A + cot A

⇒ (sin A / cos A) + (cos A / sin A)

⇒ (sin²A + cos²A) / (sin A × cos A)

⇒ 1 / (sin A × cos A)

⇒ (1 / cos A) × (1 / sin A)

⇒ sec A × cosec A

∴ The correct answer is sec A cosec A.

Additional Information

Pythagorean Identities

sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Double Angle Formulas

cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Reciprocal Identities

tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Also, sec θ = 1 / cos θ and cosec θ = 1 / sin θ.

Shortcut Trick

Side of the square = Perimeter ÷ 4 = 248 ÷ 4 = 62 m.

Area = Side² = 62 × 62 = 3844 m².

∴ The correct answer is 3844 m².

Alternate Method

Given:

Perimeter of the square = 248 m

Formula Used:

Perimeter = 4 × a (where a is the side length)

Area = a²

a = 62 m
a = 62 m
P = 248 m
Area = ?

Calculations:

⇒ 4 × a = 248

⇒ a = 248 ÷ 4

⇒ a = 62 m

⇒ Area = a²

⇒ Area = 62 × 62

⇒ Area = 3844 m²

∴ The correct answer is 3844 m².

Additional Information

Rectangle

Perimeter = 2 × (l + b) | Area = l × b | Diagonal = √(l² + b²)

Diagonal of a Square

Diagonal (d) = a√2 | Area can also be calculated directly as (d²) ÷ 2.

Rhombus

Perimeter = 4 × a | Area = (d₁ × d₂) ÷ 2, where d₁ and d₂ are the length of the diagonals.

Circle

Circumference = 2πr | Area = πr²

Shortcut Trick

When investment time is the same for all partners, profit is directly proportional to the invested capital.

Total Investment = 1020 + 1440 + 1230 = 3690

Rahul's fractional share of the total capital = 1230 ÷ 3690 = 1/3

Rahul's Profit = 1/3 × Total Profit = 1/3 × 1260 = 420

∴ The correct answer is ₹420.

Alternate Method

Given:

Investment by Manoj = ₹1,020

Investment by Swati = ₹1,440

Investment by Rahul = ₹1,230

Total Profit at the end of the year = ₹1,260

Formula Used:

Profit Ratio = Capital₁ × Time₁ : Capital₂ × Time₂ : Capital₃ × Time₃

Total Investment = ₹3690

Manoj
₹1020

Swati
₹1440

Rahul
₹1230

Total Profit = ₹1260

Calculations:

⇒ Since the time period is 1 year for all, their profit ratio is equal to their investment ratio.

⇒ Profit Ratio (Manoj : Swati : Rahul) = 1020 : 1440 : 1230

⇒ Dividing by 30 ⇒ Simplified Ratio = 34 : 48 : 41

⇒ Sum of the ratio parts = 34 + 48 + 41 = 123 parts

⇒ Value of total 123 parts = ₹1,260

⇒ Rahul's Profit Share = (41 ÷ 123) × 1260

⇒ Rahul's Profit Share = (1 ÷ 3) × 1260 = 420

∴ The correct answer is ₹420.

Additional Information

Simple Partnership

When all partners invest their capitals for the exact same duration, the profits or losses are distributed precisely in the ratio of their original investments.

Compound Partnership

When partners invest for varying time periods, the profit sharing ratio is calculated as the product of their capital and time invested: Ratio = (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A working partner receives a fixed salary or percentage of the profit for managing the business. The remaining profit is then shared among all partners based on their respective investments.

Shortcut Trick

Let the assumed average be 50 kg.

Deviations of each weight from 50 kg: −10, −1, +6, +23, −11

Net sum of deviations = (−10 − 1 + 6 + 23 − 11) = +7

Average deviation = +7 ÷ 5 = +1.4 kg

Actual average = Assumed average + Average deviation = 50 + 1.4 = 51.4 kg

∴ The correct answer is 51.4 kg.

Alternate Method

Given:

Weights of the 5 family members are 40 kg, 49 kg, 56 kg, 73 kg, and 39 kg.

Formula Used:

Average = Sum of all observations ÷ Total number of observations

4049567339Average = 51.4Weights of 5 Family Members (kg)

Calculations:

⇒ Total weight of the family = 40 + 49 + 56 + 73 + 39

⇒ Total weight = 257 kg

⇒ Total number of members = 5

⇒ Average weight = 257 ÷ 5

⇒ Average weight = 51.4 kg

∴ The correct answer is 51.4 kg.

Additional Information

Weighted Average

When different sets have different number of items, their combined average is calculated as: A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Change in Average

If a constant 'k' is added, subtracted, multiplied or divided to each observation, the new average also undergoes the exact same operation by 'k'.

Finding an Included Item

Value of the newly added item = Old Average + (New Average − Old Average) × New Total Number of Items.

Shortcut Trick

An increase of 100% means the original value is doubled (i.e., it becomes 200%).

⇒ Original Number × 2 = 2700

⇒ Original Number = 2700 ÷ 2

⇒ Original Number = 1350

∴ The correct answer is 1350.

Alternate Method

Given:

Total value after increase = 2700

Percentage increase = 100%

Original (100%)

• 100% Increase

Total (200%) = 2700

Value is doubled

Calculations:

Let the original number be N.

⇒ N + 100% of N = 2700

⇒ N + N = 2700

⇒ 2N = 2700

⇒ N = 2700 ÷ 2

⇒ N = 1350

∴ The correct answer is 1350.

Additional Information

Percentage Change

Percentage change is calculated as: ((Final Value − Initial Value) ÷ Initial Value) × 100.

Multiplying Factor Method

An increase of r% multiplies the base value by (1 + r/100). A decrease of r% multiplies it by (1 − r/100).

Successive Percentage

When a number undergoes successive percentage changes of a% and b%, the net percentage change is given by: (a + b + (a × b) ÷ 100)%.

Shortcut Trick

Given, 50 × SP = 5 × CP

⇒ SP / CP = 5 / 50 = 1 / 10

Here, CP = 10 parts and SP = 1 part.

⇒ Loss = 10 − 1 = 9 parts.

⇒ Loss % = (9 / 10) × 100% = 90%

∴ The correct answer is 90% loss.

Alternate Method

Given:

Selling price of 50 books = Cost price of 5 books

Formula Used:

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

Loss % = (Loss / CP) × 100

Cost Price (CP) = 10 units

SP = 1

Loss = 9 units (90%)

Calculations:

Let the CP of 1 book be ₹1.

⇒ Total CP of 50 books = ₹50

From the given condition, SP of 50 books = CP of 5 books

⇒ SP of 50 books = 5 × ₹1 = ₹5

Now, for the transaction of 50 books:

⇒ Total CP = ₹50

⇒ Total SP = ₹5

Since CP is greater than SP, there is a clear loss.

⇒ Loss = 50 − 5 = ₹45

⇒ Loss % = (45 / 50) × 100

⇒ Loss % = 45 × 2 = 90%

∴ The correct answer is 90% loss.

Additional Information

Article Equivalent Profit & Loss

When the CP of 'a' articles equals the SP of 'b' articles, Profit/Loss % = ((a − b) / b) × 100. A positive result indicates profit, while a negative result indicates a loss.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. The Discount % = (Discount / MP) × 100.

Dishonest Dealer

When a dealer pretends to sell goods at CP but uses false weight, Profit % = (Error / (True Weight − Error)) × 100.

Shortcut Trick

The given number is 159916901, which is an odd number.

An odd number can never be perfectly divided by even numbers, so options 18 and 28 are eliminated.

Its last two digits are 01, which means it is not divisible by 25 (as 25 requires the last two digits to be 00, 25, 50, or 75).

Thus, by the simple process of elimination, the number must be divisible by 19.

∴ The correct answer is 19.

Alternate Method

Given:

The number to be divided = 159916901

Checking the given options:

⇒ Option 1: 25

⇒ Since the last two digits of 159916901 are 01, it is not divisible by 25.

⇒ Option 3: 28

⇒ 28 is an even number, and 159916901 is an odd number. Hence, it is not divisible by 28.

⇒ Option 4: 18

⇒ 18 is an even number, so it cannot perfectly divide an odd number.

⇒ Option 2: 19

⇒ Dividing directly: 159916901 ÷ 19

⇒ 159916901 = 19 × 8416679

⇒ The remainder is 0, indicating perfect divisibility.

∴ The correct answer is 19.

Additional Information

Divisibility Rule of 19

Multiply the last digit of the number by 2 and add it to the remaining truncated number. If the resulting sum is divisible by 19, then the original number is also divisible by 19.

Divisibility Rule of 25

A number is completely divisible by 25 only if its last two digits are exactly 00, 25, 50, or 75.

Divisibility Rule of Even Numbers

An even number will always have an even unit digit (0, 2, 4, 6, 8). Moreover, an even number can never perfectly divide any odd number as 2 will never be a prime factor of the odd number.

Shortcut Trick

Profit of M = (Total Profit + Difference) ÷ 2 = (4896 + 288) ÷ 2 = ₹2592

Profit of N = (Total Profit − Difference) ÷ 2 = (4896 − 288) ÷ 2 = ₹2304

Profit ratio (M : N) = 2592 : 2304 = 9 : 8

Ratio of Capitals = (Profit_M ÷ Time_M) : (Profit_N ÷ Time_N) = (9 ÷ 3) : (8 ÷ 5) = 3 : 1.6 = 15 : 8

Difference in capital ratio = 15 − 8 = 7 units ⇒ 7 units = ₹70,000 ⇒ 1 unit = ₹10,000

M's capital = 15 units × 10,000 = ₹1,50,000

∴ The correct answer is ₹1,50,000.

Alternate Method

Given:

Difference in Capital (M − N) = ₹70,000

Time for M (T_M) = 3 months

Time for N (T_N) = 5 months

Total Profit = ₹4896

Difference in Profit (M − N) = ₹288

Formula Used:

Profit Ratio = (Capital of M × Time of M) : (Capital of N × Time of N)

Capital Ratio
Time
Profit Ratio

M

15 units
×

3m

9 units (₹2592)

N

8 units
×

5m

8 units (₹2304)

Diff: 15 − 8 = 7 units
7 units = ₹70,000

Profit Diff: ₹288
Total: ₹4896

Calculations:

⇒ M's Profit = (Total Profit + Difference) ÷ 2

⇒ M's Profit = (4896 + 288) ÷ 2 = ₹2592

⇒ N's Profit = Total Profit − M's Profit = 4896 − 2592 = ₹2304

⇒ Profit Ratio (M : N) = 2592 : 2304 = 9 : 8

⇒ Capital Ratio (C_M : C_N) = (Profit_M ÷ T_M) : (Profit_N ÷ T_N)

⇒ C_M : C_N = (9 ÷ 3) : (8 ÷ 5) = 3 : (8 ÷ 5) = 15 : 8

⇒ Let M's Capital be 15x and N's Capital be 8x.

⇒ Difference in Capital = 15x − 8x = 7x

⇒ Given that 7x = ₹70,000 ⇒ x = ₹10,000

⇒ M's Capital = 15x = 15 × 10,000 = ₹1,50,000

∴ The correct answer is ₹1,50,000.

Additional Information

Profit Distribution in Partnership

Profit ratio is directly proportional to the product of Capital and Time. Formula: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Sum and Difference Rule

When the sum (S) and difference (D) of two quantities are given, the larger quantity = (S + D) ÷ 2, and the smaller quantity = (S − D) ÷ 2.

Sleeping vs Working Partner

A working partner receives a fixed salary or commission for managing the business before the remaining profit is divided among all partners according to their respective investment ratios.

Shortcut Trick

To find the LCM of numbers given in exponential form, simply take the product of the highest power of each distinct base present.

The distinct bases present in the given numbers are 3, 8, 14, and 19.

The highest powers for each are 3³, 8³, 14², and 19².

Multiply them together to obtain the LCM.

∴ The correct answer is 3³ × 8³ × 14² × 19².

Alternate Method

Given:

First number = 3³ × 8² × 14

Second number = 3² × 14² × 19

Third number = 8³ × 14² × 19²

Formula Used:

The Least Common Multiple (LCM) of numbers expressed in exponential form is the product of the highest power of all bases involved.

Calculations:

⇒ The distinct bases in all three expressions are 3, 8, 14, and 19.

⇒ Comparing the powers of base 3: max(3, 2, 0) = 3

⇒ Comparing the powers of base 8: max(2, 0, 3) = 3

⇒ Comparing the powers of base 14: max(1, 2, 2) = 2

⇒ Comparing the powers of base 19: max(0, 1, 2) = 2

⇒ LCM = Product of the highest powers of all distinct bases

⇒ LCM = 3³ × 8³ × 14² × 19²

∴ The correct answer is 3³ × 8³ × 14² × 19².

Additional Information

HCF of Exponential Numbers

The Highest Common Factor (HCF) is found by taking the product of the lowest power of each common base present in all the given expressions.

LCM and HCF of Fractions

LCM of fractions = LCM of numerators ÷ HCF of denominators.

HCF of fractions = HCF of numerators ÷ LCM of denominators.

Product Rule

For any two numbers, the product of the numbers = LCM × HCF.

Shortcut Trick

Ratio of Expenditure to Savings = 225 : 100 = 9 : 4.

Net change in Income = (9 × −14% + 4 × 38%) ÷ (9 + 4)

Net change = (−126 + 152) ÷ 13 = 26 ÷ 13 = 2%.

∴ The correct answer is 2%.

Alternate Method

Given: Expenditure is 125% more than Savings.

⇒ Let Savings = 100

⇒ Expenditure = 100 + 125% of 100 = 225

⇒ Income = Expenditure + Savings = 225 + 100 = 325

⇒ New Expenditure = 225 × (100 − 14)/100 = 225 × 0.86 = 193.5

⇒ New Savings = 100 × (100 + 38)/100 = 138

⇒ New Income = 193.5 + 138 = 331.5

⇒ Increase in Income = 331.5 − 325 = 6.5

⇒ Percentage Increase = (6.5 ÷ 325) × 100 = 2%

∴ The correct answer is 2%.

Additional Information

Income-Expenditure Relationship

The fundamental accounting equation used in these problems is Income = Expenditure + Savings.

Weighted Average Application

Percentage change in Income is the weighted average of percentage changes in Expenditure and Savings based on their initial values.

Successive Percentage Change

For two successive changes a% and b%, the net effect is given by [a + b + (a×b)/100]%, though not directly applicable to additions like Income.

Shortcut Trick

Given: tanθ + cotθ = 4

Convert to sin and cos: (sinθ/cosθ) + (cosθ/sinθ) = 4 ⇒ (sin²θ + cos²θ) / (sinθ cosθ) = 4

Using identity sin²θ + cos²θ = 1: 1 / (sinθ cosθ) = 4

Squaring both sides: 1 / (sin²θ cos²θ) = 16

Since sec²θ = 1/cos²θ and csc²θ = 1/sin²θ, the expression is equal to 16.

∴ The correct answer is 16.

Alternate Method

Given: tanθ + cotθ = 4

Formula Used: tanθ + cotθ = secθ × cscθ

⇒ (sinθ/cosθ) + (cosθ/sinθ) = 4

⇒ (sin²θ + cos²θ) / (sinθ cosθ) = 4

⇒ 1 / (sinθ cosθ) = 4

⇒ secθ × cscθ = 4

⇒ (secθ × cscθ)² = 4²

⇒ sec²θ × csc²θ = 16

∴ The correct answer is 16.

Additional Information

Trigonometric Identity Relation

The expression tanθ + cotθ is always equal to secθ × cscθ or 1/(sinθcosθ).

Pythagorean Identities

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

Sum of Squares Identity

sec²θ + csc²θ = sec²θ × csc²θ. This unique property allows switching between sum and product.

Shortcut Trick

The Empirical Relationship between averages is: (Mode − Mean) = 3 × (Median − Mean).

From this, the ratio of differences is: (Mode − Median) : (Mode − Mean) = 2 : 3.

Given: (Mode − Mean) = 77.7, which corresponds to 3 units.

Value of 1 unit = 77.7 ÷ 3 = 25.9.

Required: (Mode − Median) = 2 units = 2 × 25.9 = 51.8.

∴ The correct answer is 51.8.

Alternate Method

Given:

Difference between Mode and Mean (Mode − Mean) = 77.7

Formula Used:

Empirical Formula: Mode = 3 × Median − 2 × Mean

Calculations:

⇒ Mode = 3 × Median − 2 × Mean

⇒ Mode + 2 × Mean = 3 × Median

⇒ To find the difference between Mode and Median, subtract 3 × Mode from both sides:

⇒ Mode + 2 × Mean − 3 × Mode = 3 × Median − 3 × Mode

⇒ 2 × Mean − 2 × Mode = 3 × (Median − Mode)

⇒ −2 × (Mode − Mean) = −3 × (Mode − Median)

⇒ 2 × 77.7 = 3 × (Mode − Median)

⇒ Mode − Median = (2 × 77.7) ÷ 3

⇒ Mode − Median = 155.4 ÷ 3

⇒ Mode − Median = 51.8

∴ The correct answer is 51.8.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is Mode = 3 × Median − 2 × Mean.

Symmetric Distribution

In a perfectly symmetrical distribution, the Mean, Median, and Mode are all equal (Mean = Median = Mode).

Positively Skewed Distribution

In a positively skewed (right-skewed) distribution, the relation is usually Mean > Median > Mode.

Negatively Skewed Distribution

In a negatively skewed (left-skewed) distribution, the relation is usually Mode > Median > Mean.

Shortcut Trick

Convert percentages and fractions into a single multiplying factor.

Number = 303 ÷ (0.30 × 0.50 × 4/7)

Number = 303 × (100/30) × (100/50) × (7/4)

Number = 303 × (10/3) × 2 × (7/4) = 101 × 10 × 3.5 = 3535.

∴ The correct answer is 3535.

Alternate Method

Given: 30% of 50% of 4/7 of x = 303

Formula Used: Percentage = Value/100

⇒ (30/100) × (50/100) × (4/7) × x = 303

⇒ (3/10) × (1/2) × (4/7) × x = 303

⇒ (3 × 1 × 4) / (10 × 2 × 7) × x = 303

⇒ (12 / 140) × x = 303

⇒ (3 / 35) × x = 303

⇒ x = (303 × 35) / 3

⇒ x = 101 × 35 = 3535

∴ The correct answer is 3535.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert common percentages: 50% = 1/2, 25% = 1/4, 20% = 1/5, and 10% = 1/10.

Successive Percentage Rule

A% of B% of C is equivalent to (A × B / 100)% of C. This is useful for multi-step discount or growth problems.

Reciprocal Property

When solving for a number, multiply the result by the reciprocal of the total fraction: x = Result × (Denominator / Numerator).

Shortcut Trick

Calculate the semi-perimeter (s): (105 + 137 + 88) ÷ 2 = 165 cm.

Note that s = 165 (multiple of 11) and (s − c) = 165 − 88 = 77 (multiple of 7 and 11).

By Heron's Formula, the Area must be a multiple of 11 and 7.

Checking options for divisibility by 11 and 7: Only 4620 satisfies both conditions.

∴ The correct answer is 4620 cm².

Alternate Method

Given: Sides of the triangle are a = 105 cm, b = 137 cm, and c = 88 cm.

Formula Used: Area of a triangle = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.

⇒ s = (a + b + c) ÷ 2

⇒ s = (105 + 137 + 88) ÷ 2 = 330 ÷ 2 = 165 cm

⇒ Area = √[165 × (165 − 105) × (165 − 137) × (165 − 88)]

⇒ Area = √[165 × 60 × 28 × 77]

⇒ Area = √[(11 × 15) × (4 × 15) × (4 × 7) × (11 × 7)]

⇒ Area = √[11² × 15² × 4² × 7²]

⇒ Area = 11 × 15 × 4 × 7 = 165 × 28

⇒ Area = 4620 cm²

∴ The correct answer is 4620 cm².

Additional Information

Heron's Formula

Used for finding the area of any triangle when all three sides are known: Area = √[s(s − a)(s − b)(s − c)].

Equilateral Triangle Area

If all sides are equal to 'a', the area is given by (√3 ÷ 4) × a².

Right-Angled Triangle Area

The area of a triangle with base 'b' and height 'h' is 1/2 × b × h.

Inradius of a Triangle

The radius of the incircle (r) is related to the area (A) and semi-perimeter (s) by the formula: r = A ÷ s.