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Shortcut Trick

Directly calculate the perfect squares: √196 = 14 and √81 = 9.

Apply the square of a square root property: (√16)² = 16.

Perform the operations: 14 − 9 + 16 = 5 + 16 = 21.

∴ The correct answer is 21.

Alternate Method

Given: √196 − √81 + (√16)²

Evaluate the expression using the BODMAS rule and properties of square roots:

⇒ 14 − 9 + (4)²

⇒ 14 − 9 + 16

⇒ 5 + 16

⇒ 21

∴ The correct answer is 21.

Additional Information

Square Root Properties

The square root of a number is a value that gives the original number when squared. Specifically, (√x)² = x for any non-negative number x.

BODMAS / PEMDAS

This rule dictates the order of operations: Brackets, Orders (Powers and Roots), Division, Multiplication, Addition, and Subtraction.

Common Perfect Squares

Memorizing perfect squares up to 25 (e.g., 13² = 169, 14² = 196, 15² = 225) is essential for fast simplification calculation.

Shortcut Trick

The effective interest rate difference is calculated by multiplying the rate difference with the time period.

Difference in Rate per year = 7% − 5% = 2%

Total difference for 2 years = 2% × 2 = 4%

According to the question, 4% of Principal = ₹40

⇒ 1% of Principal = 40 ÷ 4 = ₹10

⇒ 100% (Principal) = 10 × 100 = ₹1,000

∴ The correct answer is ₹1,000.

Alternate Method

Given:

Time period (T) = 2 years

Rate 1 (R₁) = 5% per annum

Rate 2 (R₂) = 7% per annum

Difference in Simple Interest = ₹40

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Calculations:

Let the Principal be ₹P.

⇒ SI₁ = (P × 5 × 2) ÷ 100 = 10P/100

⇒ SI₂ = (P × 7 × 2) ÷ 100 = 14P/100

⇒ SI₂ − SI₁ = 40

⇒ (14P − 10P) ÷ 100 = 40

⇒ 4P ÷ 100 = 40

⇒ 4P = 40 × 100

⇒ P = 4000 ÷ 4

⇒ P = 1,000

Thus, the Principal of each sum is ₹1,000, which corresponds to Option (2).

∴ The correct answer is ₹1,000.

Additional Information

Simple Interest (SI) Properties

Simple interest is calculated only on the Principal amount and remains constant for every year if the rate and principal are unchanged.

Principal Formula

When the difference in interest (D) is given for two different rates (R₁ and R₂) over the same time (T), then P = (D × 100) ÷ [T × (R₂ − R₁)].

Total Amount

The total Amount after interest is calculated as A = Principal + Simple Interest = P × [1 + (R × T ÷ 100)].

Shortcut Trick

Assume Total Work = 24 units (LCM of 12 and 24).

Efficiency of A = 2 units/day and B = 1 unit/day.

Combined work in 4 days = 4 × (2 + 1) = 12 units.

Remaining work = 24 − 12 = 12 units.

Time taken by B = 12 ÷ 1 = 12 days.

∴ The correct answer is 12 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 24 days

Formula Used:

Total Work = Time × Efficiency

Calculations:

⇒ Total Work = LCM of 12 and 24 = 24 units

⇒ Efficiency of A = 24 ÷ 12 = 2 units/day

⇒ Efficiency of B = 24 ÷ 24 = 1 unit/day

⇒ Combined efficiency of A and B = 2 + 1 = 3 units/day

⇒ Work completed by both in 4 days = 4 × 3 = 12 units

⇒ Remaining work = 24 − 12 = 12 units

⇒ Days taken by B to finish remaining work = 12 ÷ 1 = 12 days

∴ The correct answer is 12 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken. If the ratio of time taken by two people is x:y, then the ratio of their efficiencies is y:x.

Work and Wages

Wages are distributed based on the amount of work done. If people work for the same duration, wages are divided in the ratio of their efficiencies.

Pipes and Cisterns Analogy

This concept is identical to Pipes and Cisterns, where filling a tank is positive work and a leak represents negative work (negative efficiency).

Shortcut Trick

When two equal halves of a distance are covered at speeds x and y, the Average Speed = (2 × x × y) ÷ (x + y).

Average Speed = (2 × 30 × 60) ÷ (30 + 60) = 3600 ÷ 90 = 40 km/h.

Total Time = Total Distance ÷ Average Speed = 120 ÷ 40 = 3 hr.

∴ The correct answer is 3 hr.

Alternate Method

Given:

Total Distance = 120 km

First half distance (D₁) = 120 ÷ 2 = 60 km

Second half distance (D₂) = 120 ÷ 2 = 60 km

Speed during first half (S₁) = 30 km/h

Speed during second half (S₂) = 60 km/h

Formula used: Time = Distance ÷ Speed

Calculations:

⇒ Time taken for the first half (T₁) = 60 ÷ 30 = 2 hr

⇒ Time taken for the second half (T₂) = 60 ÷ 60 = 1 hr

⇒ Total Time taken = T₁ + T₂ = 2 + 1 = 3 hr

Comparing the result with the options, it corresponds to Option 3.

∴ The correct answer is 3 hr.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. It is not the simple arithmetic mean of speeds unless the time intervals are equal.

Equal Distance Case

If a journey is divided into n equal distance parts with speeds v₁, v₂... v_n, the average speed is the Harmonic Mean of the speeds.

Speed-Time-Distance Relationship

Distance = Speed × Time. If distance is constant, speed is inversely proportional to time (S ∝ 1/T).

Shortcut Trick

A number is divisible by 68 if and only if it is divisible by both 4 and 17 (since 68 = 4 × 17).

Check divisibility by 4: The last two digits must be divisible by 4.

For ( i ) 17350 and ( iv ) 35350, the last two digits are 50. Since 50 is not divisible by 4 (50 ÷ 4 = 12.5), these numbers cannot be divisible by 68.

Numbers ( ii ) 23800 and ( iii ) 28560 end in 00 and 60, both of which are divisible by 4; checking with 17 confirms their divisibility by 68.

∴ The correct answer is ( i ) and ( iv ).

Alternate Method

Given: Numbers (i) 17350, (ii) 23800, (iii) 28560, (iv) 35350. Divisor = 68.

⇒ Divisibility of ( i ) 17350: 17350 ÷ 68 = 255.147 (Remainder ≠ 0)

⇒ Divisibility of ( ii ) 23800: 23800 ÷ 68 = 350 (Remainder = 0)

⇒ Divisibility of ( iii ) 28560: 28560 ÷ 68 = 420 (Remainder = 0)

⇒ Divisibility of ( iv ) 35350: 35350 ÷ 68 = 519.852 (Remainder ≠ 0)

∴ The correct answer is ( i ) and ( iv ).

Additional Information

Divisibility Rule for 4

A number is divisible by 4 if its last two digits are 00 or form a number that is exactly divisible by 4.

Divisibility by Composite Numbers

To check divisibility by a composite number L, the number must be divisible by all coprime factors of L. For 68, the coprime factors are 4 and 17.

Divisibility Rule for 17

Multiply the last digit of the number by 5 and subtract it from the rest of the number. If the result is divisible by 17, the whole number is divisible by 17.

Shortcut Trick

Compare the effective discount percentage of all four schemes for 4 toys:

(A) 6 + 39 − (6 × 39 / 100) = 45 − 2.34 = 42.66%

(B) Successive 48%, 16%, 11% ≈ 61.12%

(C) Average of (37 + 37 + 29 + 29) / 4 = 132 / 4 = 33%

(D) 2 free out of (4 + 2) = 2 / 6 × 100 = 33.33%

The lowest discount (least beneficial) is 33% from Scheme C.

∴ The correct answer is C.

Alternate Method

Assume the Marked Price (MP) of each toy is ₹100. A customer buys 4 toys, so total Marked Price = ₹400.

Calculation for Scheme A:

⇒ Net Discount = x + y − (xy / 100)

⇒ 6 + 39 − (6 × 39 / 100) = 45 − 2.34 = 42.66%

Calculation for Scheme B:

⇒ Successive discount of 48% and 16% = 48 + 16 − (48 × 16 / 100) = 64 − 7.68 = 56.32%

⇒ Now combine 56.32% and 11% = 56.32 + 11 − (56.32 × 11 / 100) = 67.32 − 6.1952 = 61.1248%

Calculation for Scheme C:

⇒ Total MP for 4 toys = 4 × 100 = ₹400

⇒ Discount on first 2 toys = 2 × 37% of 100 = ₹74

⇒ Discount on next 2 toys = 2 × 29% of 100 = ₹58

⇒ Total Discount = 74 + 58 = ₹132

⇒ Discount % = (132 / 400) × 100 = 33%

Calculation for Scheme D:

⇒ Customer gets 2 toys free on buying 4 toys (Total = 6 toys)

⇒ Discount % = (Free Items / Total Items) × 100 = (2 / 6) × 100 = 33.33%

⇒ Comparing all: 42.66%, 61.12%, 33%, and 33.33%.

∴ The correct answer is C.

Additional Information

Successive Discount Formula

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

Buy X Get Y Free

In a 'Buy X Get Y Free' scheme, the effective discount percentage is given by [Y / (X + Y)] × 100.

Marked Price and Selling Price

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

Shortcut Trick

Ratio of employees in Branch A : B : C = 40 : 30 : 30 = 4 : 3 : 3.

Weighted Average = [(1,20,000 × 4) + (1,80,000 × 3) + (1,50,000 × 3)] ÷ (4 + 3 + 3)

Average = (4,80,000 + 5,40,000 + 4,50,000) ÷ 10

Average = 14,70,000 ÷ 10 = 1,47,000.

∴ The correct answer is ₹1,47,000.

Alternate Method

Given:

Branch A: 40 employees, Avg Sales = ₹1,20,000

Branch B: 30 employees, Avg Sales = ₹1,80,000

Branch C: 30 employees, Avg Sales = ₹1,50,000

Formula Used: Average = Total Sales ÷ Total Employees

⇒ Total Employees = 40 + 30 + 30 = 100

⇒ Total Sales = (40 × 1,20,000) + (30 × 1,80,000) + (30 × 1,50,000)

⇒ Total Sales = 48,00,000 + 54,00,000 + 45,00,000

⇒ Total Sales = ₹1,47,00,000

⇒ Average per employee = 1,47,00,000 ÷ 100 = ₹1,47,000

∴ The correct answer is ₹1,47,000.

Additional Information

Weighted Average Formula

It is used when data sets have different levels of importance or quantity. Formula: X_avg = (w₁x₁ + w₂x₂ + w₃x₃) ÷ (w₁ + w₂ + w₃).

Combined Mean

The combined mean of two or more groups is the sum of all observations of all groups divided by the total number of observations in all groups.

Average Property

If each value in a set is increased or decreased by a constant k, the average also increases or decreases by k.

Shortcut Trick

Profit Ratio P : Q : R = (20 × 8) : (15 × 9) : (12 × t) = 160 : 135 : 12t

Given: Q < R < P ⇒ 135 < 12t < 160

For complete months within a year (t ≤ 12), only t = 12 satisfies the condition.

R profit (144 units) = ₹72,000 ⇒ 1 unit = 72,000 ÷ 144 = ₹500

P profit (160 units) = 160 × 500 = ₹80,000

∴ The correct answer is ₹80,000.

Alternate Method

Given:

Investment of P = ₹2,00,000 for 8 months

Investment of Q = ₹1,50,000 for 9 months

Investment of R = ₹1,20,000 for t months

Profit of R = ₹72,000

Formula: Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : (Capital₃ × Time₃)

⇒ P's Profit Units = 2,00,000 × 8 = 16,00,000

⇒ Q's Profit Units = 1,50,000 × 9 = 13,50,000

⇒ R's Profit Units = 1,20,000 × t

⇒ Since Profit Q < Profit R < Profit P

⇒ 13,50,000 < 1,20,000 × t < 16,00,000

⇒ 11.25 < t < 13.33

⇒ For complete months in a business year, t = 12 months.

⇒ R's actual units = 1,20,000 × 12 = 14,40,000

⇒ 14,40,000 units corresponds to ₹72,000 profit.

⇒ 1 unit value = 72,000 ÷ 14,40,000 = 1 ÷ 20

⇒ Profit of P = 16,00,000 × (1 ÷ 20) = ₹80,000

∴ The correct answer is ₹80,000.

Additional Information

Partnership Profit Sharing

Profit is distributed in the ratio of Effective Capital, where Effective Capital = Investment × Time period.

Compound Partnership

When partners invest their capitals for different time periods, it is known as a compound partnership.

Working vs. Sleeping Partner

A working partner manages the business and may receive an additional salary, while a sleeping partner only invests capital.

Shortcut Trick

Ratio of goods: 50 : (30% of 50) : (remaining) ⇒ 50 : 15 : 35 = 10 : 3 : 7.

Apply Weighted Average Method for overall profit percentage x.

x = [(10 × 21) + (3 × 43) + (7 × −17)] ÷ (10 + 3 + 7)

x = [210 + 129 − 119] ÷ 20 = 220 ÷ 20 = 11.

∴ The correct answer is 11.

Alternate Method

Given:

Let Total Goods = 100 units at ₹1/unit. Total CP = ₹100.

Part 1: 50% of the goods = 50 units sold at 21% profit.

Part 2: 30% of the remaining goods = 30% of 50 = 15 units sold at 43% profit.

Part 3: Still remaining goods = 100 − (50 + 15) = 35 units sold at 17% loss.

Formula Used:

Overall Profit/Loss % (x) = [Total Profit or Loss ÷ Total CP] × 100

Calculations:

⇒ Profit from Part 1 = 50 × (21 ÷ 100) = ₹10.5

⇒ Profit from Part 2 = 15 × (43 ÷ 100) = ₹6.45

⇒ Loss from Part 3 = 35 × (17 ÷ 100) = ₹5.95

⇒ Total Profit = 10.5 + 6.45 − 5.95

⇒ Total Profit = 16.95 − 5.95 = ₹11

⇒ Overall profit % (x) = (11 ÷ 100) × 100 = 11%

⇒ x = 11

∴ The correct answer is 11.

Additional Information

Weighted Average Formula

When a whole is divided into parts n₁, n₂... with individual changes p₁, p₂..., the net change is (Σ n_i × p_i) ÷ Σ n_i.

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Always calculate on Cost Price unless stated otherwise.

Successive Percentage Change

For two successive changes of a% and b%, the effective change is [a + b + (a × b) ÷ 100]%.

Shortcut Trick

A regular hexagon is composed of 6 equilateral triangles.

Area of 1 equilateral triangle = (√3 / 4) × (side)²

Total Area = 6 × (1.732 / 4) × 10²

Total Area = 1.5 × 1.732 × 100 = 259.8 m².

∴ The correct answer is 259.8 m².

Alternate Method

Given:

Side of the regular hexagon (s) = 10 m

Value of √3 ≈ 1.732

Formula Used:

Area of a regular hexagon = [3√3 / 2] × s²

Calculations:

⇒ Area = [3 × 1.732 / 2] × (10)²

⇒ Area = [5.196 / 2] × 100

⇒ Area = 2.598 × 100

⇒ Area = 259.8 m²

∴ The correct answer is 259.8 m².

Additional Information

Perimeter of a Regular Hexagon

The perimeter is the total length of the boundary, calculated as P = 6 × side.

Interior Angles

Each interior angle of a regular hexagon is 120°, and the sum of all interior angles is 720°.

Longest Diagonal

The longest diagonal connects two opposite vertices and its length is equal to twice the side (D = 2 × s).