Shortcut Trick

For divisibility by 11, the difference of the sums of alternate digits must be 0 or a multiple of 11.

Sum of odd position digits = N + 9 + 1 = N + 10

Sum of even position digits = 2 + M + 1 = M + 3

Difference = (N + 10) − (M + 3) = N − M + 7

Equating to 0 for a valid single-digit relation: N − M + 7 = 0 ⇒ M − N = 7.

∴ The correct answer is M − N = 7.

Alternate Method

Given:

The 6-digit number is N29M11, and it is perfectly divisible by 11.

Concept Used:

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.

Calculations:

⇒ The given number = N29M11

⇒ Sum of digits at odd places (from left to right) = N + 9 + 1 = N + 10

⇒ Sum of digits at even places (from left to right) = 2 + M + 1 = M + 3

⇒ Difference = (Sum of odd places) − (Sum of even places)

⇒ Difference = (N + 10) − (M + 3)

⇒ Difference = N − M + 7

⇒ For the number to be divisible by 11, this difference must be equal to 0, 11, or a multiple of 11.

⇒ Case 1: Equating difference to 0

⇒ N − M + 7 = 0

⇒ M − N = 7

⇒ Case 2: Equating difference to 11

⇒ N − M + 7 = 11

⇒ N − M = 4

⇒ Since M and N are single digits, both relations are mathematically possible, but only M − N = 7 is present among the given choices.

∴ The correct answer is M − N = 7.

Additional Information

Divisibility Rule of 3 and 9

A number is perfectly divisible by 3 or 9 if the sum of all its digits is divisible by 3 or 9, respectively.

Divisibility Rule of 4 and 8

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if its last three digits form a number divisible by 8.

Combined Divisibility Rule of 7, 11, and 13

If the difference between the number formed by the last three digits and the number formed by the remaining leading digits is divisible by 7, 11, or 13, then the original number is divisible by 7, 11, or 13 respectively.

Shortcut Trick

Using the Ratio Method for successive changes (V ∝ r²h):

Radius ratio (20% decrease = −1/5) ⇒ 4/5

Height ratio (40% increase = +2/5) ⇒ 7/5

Volume ratio = (4/5)² × (7/5) = 16/25 × 7/5 = 112/125

Decrease = 125 − 112 = 13 parts ⇒ % Decrease = (13 ÷ 125) × 100 = 10.4%

∴ The correct answer is 10.4% decrease.

Alternate Method

r

h
Vol = 100%

→

0.8r

1.4h
Vol = 89.6%

Given:

Radius of cylinder is reduced by 20%.

Height of cylinder is increased by 40%.

Formula Used:

Volume of a cylinder = π × r² × h

Calculations:

⇒ Let original radius = 10 units and original height = 10 units.

⇒ Original Volume = π × (10)² × 10 = 1000π cubic units.

⇒ New radius = 10 − (20% of 10) = 8 units.

⇒ New height = 10 + (40% of 10) = 14 units.

⇒ New Volume = π × (8)² × 14

⇒ New Volume = π × 64 × 14 = 896π cubic units.

⇒ Decrease in Volume = Original Volume − New Volume

⇒ Decrease in Volume = 1000π − 896π = 104π cubic units.

⇒ Percentage Decrease = (104π ÷ 1000π) × 100

⇒ Percentage Decrease = 10.4%

∴ The correct answer is 10.4% decrease.

Additional Information

Volume Dependency in Cylinders

The volume of a cylinder changes in direct proportion to the square of its radius and linearly with its height (V ∝ r²h). Any change in the radius has a twofold compounded impact on the final volume.

Successive Percentage Formula

To calculate two continuous percentage changes x and y, use the formula: Net % change = x + y + (x × y) ÷ 100. This is useful for analyzing areas but must be applied twice for volumes involving r².

Multiplier/Ratio Logic

If values change to fraction 'a' and 'b' of their original, the new final value simply becomes (a² × b) of the initial state. This quickly bypasses complex intermediate decimal calculations.

Shortcut Trick

The average of the first n even numbers is directly given by the formula (n + 1).

Here, the number of even numbers (n) = 154.

Average = 154 + 1 = 155.

∴ The correct answer is 155.

Alternate Method

Given:

Number of terms (n) = 154

Sequence of even numbers = 2, 4, 6, ...

Formula Used:

Average of an Arithmetic Progression (A.P.) = (First term + Last term) ÷ 2

2
1^st term

4
2^nd term

6

. . .

308
154^th term

Average = (2 + 308) / 2 = 155

Calculations:

⇒ First term (a) = 2

⇒ Common difference (d) = 2

⇒ Last term (154^th even number) = 154 × 2 = 308

⇒ Average = (2 + 308) ÷ 2

⇒ Average = 310 ÷ 2

⇒ Average = 155

∴ The correct answer is 155.

Additional Information

Average of first n natural numbers

The average of the first n natural numbers is given by the formula (n + 1) ÷ 2.

Average of first n odd numbers

The average of the first n odd numbers is exactly equal to n.

Average of squares of first n natural numbers

The formula to find the average of the squares of the first n natural numbers is (n + 1)(2n + 1) ÷ 6.

Sum of first n even numbers

The sum of the first n even numbers is calculated as n(n + 1), which directly leads to the average formula (n + 1).

Shortcut Trick

Equivalent discount % for successive discounts of 20% and 10% = 20 + 10 − (20 × 10) ÷ 100 = 28%.

Given that 28% of the Marked Price = ₹23,800.

⇒ MP = (23,800 ÷ 28) × 100 = 850 × 100 = ₹85,000.

∴ The correct answer is ₹85,000.

Alternate Method

Given:

Successive discounts = 20% and 10%

Total discount obtained = ₹23,800

Formula Used:

Effective Discount % = d₁ + d₂ − (d₁ × d₂) ÷ 100

Discount = Marked Price (MP) − Selling Price (SP)

MP (100%)

−20%

80%

−10%

SP (72%)

Total Discount = 100% − 72% = 28%
28% = ₹23,800

Calculations:

⇒ Let Marked Price (MP) = 100x

⇒ SP after 20% discount = 100x − 20% of 100x = 80x

⇒ SP after further 10% discount = 80x − 10% of 80x = 72x

⇒ Total Discount = MP − Final SP = 100x − 72x = 28x

⇒ We are given that the discount is ₹23,800.

⇒ 28x = 23,800

⇒ x = 23,800 ÷ 28 = 850

⇒ Marked Price (100x) = 850 × 100 = 85,000

∴ The correct answer is ₹85,000.

Additional Information

Successive Discount Formula

When two discounts d₁% and d₂% are given successively, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂) ÷ 100]%.

Marked Price & Selling Price Relation

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

Buy X Get Y Free

Discount % = [Free Items ÷ Total Items] × 100. This is another popular discount scheme in exams.

Shortcut Trick

Substitute the given powers directly into x^z = y⁶:

⇒ (103^0.2)^z = (103^0.43)⁶

⇒ 0.2 × z = 0.43 × 6

⇒ 0.2z = 2.58

⇒ z = 2.58 ÷ 0.2 = 12.9

∴ The correct answer is 12.9.

Alternate Method

Given:

x = 103^0.2

y = 103^0.43

x^z = y⁶

Formula Used:

(a^m)ⁿ = a^mn

If a^m = aⁿ, then m = n.

Calculations:

Substitute the values of x and y in the equation x^z = y⁶:

⇒ (103^0.2)^z = (103^0.43)⁶

⇒ 103^{(0.2 × z)} = 103^{(0.43 × 6)}

⇒ 103^0.2z = 103^2.58

Since the bases are identical, their exponents must be equal:

⇒ 0.2z = 2.58

⇒ z = 2.58 ÷ 0.2

⇒ z = 25.8 ÷ 2

⇒ z = 12.9

∴ The correct answer is 12.9.

Additional Information

Power of a Power Rule

When an exponent is raised to another exponent, the powers are multiplied: (a^m)ⁿ = a^mn.

Equality of Exponents

For any non-zero base except 1 and -1, if a^x = a^y, then x = y.

Product Rule for Exponents

When multiplying expressions with the same base, the exponents are added: a^m × aⁿ = a^m+n.

Shortcut Trick

Let present ages of the father and son be F and S.

Past condition (4 years ago): (F − 4) = 2 × (S − 4) + 37 ⇒ F − 2S = 33.

Future condition (after x years): (F + x) = 2 × (S + x) ⇒ x = F − 2S.

By directly substituting the value from the past condition, we get x = 33.

∴ The correct answer is 33 years.

Alternate Method

Given:

4 years ago, the father's age was 37 years more than twice the son's age.

Formula Used:

Future Age = Present Age + Number of Years

Past Age = Present Age − Number of Years

Past (−4 yrs)
F − 4 =
2(S − 4) + 37

Present
F − 2S = 33

Future (+n yrs)
F + n = 2(S + n)
n = F − 2S

Calculation:

Let the present age of the father be F and the son be S.

⇒ Age of father 4 years ago = F − 4

⇒ Age of son 4 years ago = S − 4

According to the problem

Statement:

⇒ F − 4 = 2 × (S − 4) + 37

⇒ F − 4 = 2S − 8 + 37

⇒ F − 2S = 29 + 4

⇒ F − 2S = 33

Let the required number of years from now be n.

⇒ F + n = 2 × (S + n)

⇒ F + n = 2S + 2n

⇒ F − 2S = 2n − n

⇒ n = F − 2S

Substitute the value of F − 2S = 33 into the equation:

⇒ n = 33

∴ The correct answer is 33 years.

Additional Information

Constant Age Difference

The difference between the absolute ages of two individuals always remains constant regardless of the time period (past, present, or future).

Linear Equation Formation

In age-related problems, you can quickly solve for variables by defining past and future ages strictly in terms of present variables (e.g., Age − x for the past, Age + y for the future).

Cross-Multiplication Method

When the ratio of ages changes over a fixed number of years, cross-multiplication efficiently solves for the base ratio without writing full-length algebraic equations.

Shortcut Trick

The distance between the midpoints of the diagonals of a trapezium is always equal to half the difference of its parallel sides.

Given parallel sides: AB = 45 cm and DC = 53 cm.

Length of EF = 1/2 × (DC − AB)

Length of EF = 1/2 × (53 − 45) = 1/2 × 8 = 4 cm.

*(

Note:

The lengths of the non-parallel sides AD and BC are given only to distract and are not needed.)*

∴ The correct answer is 4 cm.

Alternate Method

A
B
C
D
E
F

AB = 45 cm
DC = 53 cm

EF = (DC − AB) ÷ 2

Given:

ABCD is a trapezium with AB || DC.

Length of the parallel side AB = 45 cm.

Length of the parallel side DC = 53 cm.

E and F are the midpoints of the diagonals AC and BD respectively.

Formula Used:

If a line segment connects the midpoints of the diagonals of a trapezium, it is parallel to the parallel sides, and its length is equal to half the difference of their lengths.

Calculation:

⇒ Length of EF = 1/2 × (DC − AB)

⇒ EF = 1/2 × (53 − 45)

⇒ EF = 1/2 × 8

⇒ EF = 4 cm.

∴ The correct answer is 4 cm.

Additional Information

Mid-segment of a Trapezium

The line segment joining the midpoints of the non-parallel sides of a trapezium is parallel to its bases. Its length is half the sum of the parallel sides. Length = (a + b) ÷ 2.

Area of a Trapezium

The area of a trapezium is calculated as half the product of the sum of its parallel sides and the perpendicular height between them. Area = 1/2 × (a + b) × h.

Isosceles Trapezium Properties

If the non-parallel sides of a trapezium are equal in length, it is called an isosceles trapezium. Its diagonals are of equal length, and the angles at the base are identical.

Shortcut Trick

Mohan's Profit = (Total + Difference) ÷ 2 = (4280 + 1070) ÷ 2 = ₹2675, Nishant's Profit = 4280 − 2675 = ₹1605.

Profit Ratio of Mohan to Nishant = 2675 : 1605 = 5 : 3.

Capital Ratio = (P₁ ÷ T₁) : (P₂ ÷ T₂) = (5 ÷ 3) : (3 ÷ 2) = 10 : 9.

Difference in Capital Ratio = 10 − 9 = 1 unit → ₹89,000.

Mohan's Capital = 10 units = 10 × 89,000 = ₹8,90,000.

∴ The correct answer is ₹8,90,000.

Alternate Method

Given:

Mohan's investment time (T_M) = 3 months

Nishant's investment time (T_N) = 2 months

Total Profit = ₹4,280

Difference in profit = ₹1,070

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Mohan

C + 89,000
×

3 months
→

Profit: 2675

Nishant

C
×

2 months
→

Profit: 1605

Capital Ratio = (P&sub1;/T&sub1;) : (P&sub2;/T&sub2;) = (5/3) : (3/2) = 10 : 9

Calculations:

⇒ Let Mohan's profit be P_M and Nishant's profit be P_N.

⇒ P_M + P_N = 4280

⇒ P_M − P_N = 1070

⇒ Adding both equations: 2P_M = 5350 ⇒ P_M = ₹2675

⇒ P_N = 4280 − 2675 = ₹1605

⇒ Profit ratio (P_M : P_N) = 2675 : 1605 = 5 : 3

⇒ Let Nishant's capital be C. Then Mohan's capital = C + 89,000.

⇒ [3 × (C + 89,000)] / (2 × C) = 5 / 3

⇒ 9 × (C + 89,000) = 10 × C

⇒ 9C + 8,01,000 = 10C

⇒ C = ₹8,01,000 (Nishant's Capital)

⇒ Mohan's Capital = 8,01,000 + 89,000 = ₹8,90,000

∴ The correct answer is ₹8,90,000.

Additional Information

Partnership Fundamentals

In any business partnership, the profit share is directly proportional to the product of capital invested and the time period. Formula: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Calculating Individual Values

When the sum (S) and difference (D) of two quantities are known: Larger Quantity = (S + D) ÷ 2, Smaller Quantity = (S − D) ÷ 2.

Capital Ratio Direct Relation

The ratio of capitals can be directly found if profit and time ratios are given by cross-dividing: C₁ : C₂ = (P₁ ÷ T₁) : (P₂ ÷ T₂).

Shortcut Trick

Increasing a number by 100% means the new value becomes 200% of the original.

200% of the number = 2800

⇒ 2 × Number = 2800

⇒ Number = 2800 ÷ 2 = 1400

∴ The correct answer is 1400.

Alternate Method

Given:

A number when increased by 100% gives 2800.

Original (100%)

• Increase (100%)

Total = 200% = 2800

Number to find = x

Formula Used:

Final Value = Original Value × (1 + Percentage Increase ÷ 100)

Calculation:

Let the original number be x.

⇒ x + 100% of x = 2800

⇒ x + (100 ÷ 100) × x = 2800

⇒ x + x = 2800

⇒ 2x = 2800

⇒ x = 2800 ÷ 2

⇒ x = 1400

∴ The correct answer is 1400.

Additional Information

Percentage Change

Percentage change = (Final Value − Initial Value) ÷ Initial Value × 100. It is used to calculate the relative difference between a final and initial quantity.

Successive Percentage Change

When a value is sequentially changed by a% and then by b%, the overall net percentage change can be calculated directly using the formula: (a + b + a×b ÷ 100)%.

Base of Percentage

A percentage is simply a ratio where the denominator is always 100. The original value is always considered as the base value (100%) before any modifications.

Shortcut Trick

Expand the brackets directly and combine like terms.

⇒ 7x² − 4x + 2x² − 4 + 11

⇒ 9x² − 4x + 7

∴ The correct answer is 9x² − 4x + 7.

Alternate Method

Given:

Expression: x(7x − 4) + 2(x² − 2) + 11

Calculations:

⇒ Distribute x into the first bracket and 2 into the second bracket:

⇒ x(7x) − x(4) + 2(x²) − 2(2) + 11

⇒ 7x² − 4x + 2x² − 4 + 11

⇒ Group the like terms together:

⇒ (7x² + 2x²) − 4x + (−4 + 11)

⇒ 9x² − 4x + 7

∴ The correct answer is 9x² − 4x + 7.

Additional Information

Distributive Property

It allows you to multiply a single term by two or more terms inside a set of parentheses: a(b − c) = ab − ac.

Like Terms

Terms whose variables and their exponents are exactly the same.

Only like terms can be added or subtracted to simplify an algebraic expression.

Shortcut Trick

Substitute α = 0° in the given expression.

⇒ cos 0° = 1 and sin 0° = 0

⇒ 1 / (1 − 0) − (1 + 0) / 1

⇒ 1 / 1 − 1 / 1 = 0

∴ The correct answer is 0.

Alternate Method

Given:

cos α / (1 − sin α) − (1 + sin α) / cos α

Formula Used:

⇒ a² − b² = (a − b)(a + b)

⇒ sin² α + cos² α = 1

Calculations:

⇒ Taking LCM of denominators: cos α × (1 − sin α)

⇒ [cos α × cos α − (1 + sin α) × (1 − sin α)] / [cos α × (1 − sin α)]

⇒ [cos² α − (1 − sin² α)] / [cos α × (1 − sin α)]

⇒ [cos² α − cos² α] / [cos α × (1 − sin α)]

⇒ 0 / [cos α × (1 − sin α)]

⇒ 0

∴ The correct answer is 0.

Additional Information

Pythagorean Identities

The three primary trigonometric identities are: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Value Substitution Trick

In simplification questions involving trigonometric expressions, substituting standard angles (like 0°, 30°, 45° or 60°) can directly yield the result. Always ensure the chosen angle does not make the denominator zero.

Algebraic Identities in Trigonometry

Identities such as a² − b² = (a − b)(a + b) are often used to simplify fractions involving (1 − sin θ) and (1 + sin θ), turning them into cos² θ.

Shortcut Trick

Calculate total days: Feb (17) + Mar (31) + Apr (13) = 61 days.

Since 2024 is a leap year, Time (T) = 61 ÷ 366 = 1/6 years.

Using the direct formula SI = (P × R × T) ÷ 100:

⇒ SI = 4000 × (7.5/100) × (1/6)

⇒ SI = 300 × (1/6) = 50

∴ The correct answer is ₹50.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate of interest (R) = 7.5% per annum

Date of deposit = 12 February 2024

Date of withdrawal = 13 April 2024

Formula Used:

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

12 Feb
29 Feb
31 Mar
13 Apr

17 d
31 d
13 d

Total Time = 61 days

P = ₹4000
R = 7.5%

SI = (4000 × 7.5 × 61) ÷ (100 × 366) = ₹50

Calculations:

⇒ Total days = Days in Feb (29 − 12) + Days in Mar (31) + Days in Apr (13)

⇒ Total days = 17 + 31 + 13 = 61 days

⇒ Since 2024 is a leap year, there are 366 days in the year.

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

⇒ SI = (4000 × 7.5 × 1) ÷ (100 × 6)

⇒ SI = 30000 ÷ 600

⇒ SI = 50

∴ The correct answer is ₹50.

Additional Information

Counting Days in Banking

When calculating interest for a specific period of days, generally the day the money is deposited is ignored, and the day of withdrawal is counted (or vice-versa).

Leap Year Rules

A leap year occurs every 4 years and has 366 days instead of 365, with 29 days in February. A century year must be divisible by 400 to be considered a leap year.

Simple Interest (SI) Formula

SI is directly proportional to the principal, rate, and time. SI = (P × R × T) ÷ 100. Time (T) must always be converted into years before substituting it into the standard formula.

Shortcut Trick

Use the (a − b)² algebraic identity to quickly evaluate squares near multiples of 100.

We can express 182 as (200 − 18).

Using (a − b)² = a² − 2ab + b²:

⇒ (200 − 18)² = 200² − 2 × 200 × 18 + 18²

⇒ 40000 − 7200 + 324 = 33124

∴ The correct answer is 33124.

Alternate Method

Given:

Expression: 182²

Formula Used:

(a + b)² = a² + 2ab + b²

Calculations:

⇒ 182² = (180 + 2)²

⇒ Here, a = 180 and b = 2

⇒ (180 + 2)² = 180² + 2 × 180 × 2 + 2²

⇒ 180² = 32400

⇒ 2 × 180 × 2 = 720

⇒ 2² = 4

⇒ Total sum = 32400 + 720 + 4

⇒ Total sum = 33124

∴ The correct answer is 33124.

Additional Information

Unit Digit Concept

The unit digit of a square is determined strictly by the unit digit of the base number. Since 182 ends in 2, its square must end in 4 (because 2² = 4).

Algebraic Identities

Key identities for calculation shortcuts: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Squaring Numbers Ending in 5

For any number ending in 5, say n5, the square is formed by multiplying n × (n + 1) and appending 25 at the end. Example: 85² → 8 × 9 = 72 → 7225.

Shortcut Trick

Ratio of Male to Female population = 178000 : (356000 − 178000) = 1 : 1

Since the population is equally divided into two parts, the overall literacy rate is the simple average of their individual literacy rates.

(Male Literacy % + Female Literacy %) ÷ 2 = Overall Literacy %

(59% + Female Literacy %) ÷ 2 = 39%

Female Literacy % = (39% × 2) − 59% = 78% − 59% = 19%

∴ The correct answer is 19%.

Alternate Method

Given:

Total population = 356000

Total males = 178000

Overall literacy rate = 39%

Male literacy rate = 59%

59%Malex%Female39%Overall39 − x201Ratio1

Calculations:

⇒ Number of females = 356000 − 178000 = 178000

⇒ Total literate population = 39% of 356000 = (39 ÷ 100) × 356000 = 138840

⇒ Literate males = 59% of 178000 = (59 ÷ 100) × 178000 = 105020

⇒ Literate females = Total literate population − Literate males

⇒ Literate females = 138840 − 105020 = 33820

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

⇒ Percentage of literate females = (33820 ÷ 178000) × 100 = 19%

∴ The correct answer is 19%.

Additional Information

Weighted Average Formula

When components have different base quantities, the overall average is determined by: A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Rule of Alligation

It allows us to quickly find the ratio of two ingredients/groups combined to achieve a mean value. Ratio = (Cheaper diff) : (Dearer diff).

Percentage Change

Calculated as (Final Value − Initial Value) ÷ Initial Value × 100%. Always identify the proper base value for accurate percentage determination.

Shortcut Trick

Total sum of the ratio parts = 5 + 17 + 7 = 29 parts.

Value of 29 parts = ₹638 ⇒ 1 part = 638 ÷ 29 = ₹22.

Difference in ratio between the largest (17) and smallest (5) share = 17 − 5 = 12 parts.

Required difference in amount = 12 × 22 = 264.

∴ The correct answer is ₹264.

Alternate Method

Given:

Total amount to be distributed = ₹638

Ratio of the distribution among three persons = 5 : 17 : 7

Total Amount = ₹638 (29x)

5x
17x
7x

Smallest
Largest

Diff = 17x − 5x = 12x

Calculations:

⇒ Let the respective shares of the three persons be 5x, 17x, and 7x.

⇒ Total sum of their shares = 5x + 17x + 7x = 29x

⇒ We are given that the total amount is ₹638.

⇒ 29x = 638

⇒ x = 638 ÷ 29 = 22

⇒ The largest share = 17x = 17 × 22 = ₹374

⇒ The smallest share = 5x = 5 × 22 = ₹110

⇒ Difference between the largest and the smallest shares = 374 − 110 = 264

∴ The correct answer is ₹264.

Additional Information

Distribution in a Ratio

To divide a total amount T in the ratio a : b : c, the individual shares are (a ÷ (a+b+c)) × T, (b ÷ (a+b+c)) × T, and (c ÷ (a+b+c)) × T respectively.

Compound Ratio

When two or more ratios are multiplied together, the resulting ratio is called a compound ratio. For a:b and c:d, the compound ratio is ac : bd.

Proportion Rule

When two ratios are equal (a:b = c:d), they are in proportion. In any proportion, the product of the extremes equals the product of the means: a × d = b × c.

Shortcut Trick

List the odd numbers in the given range: 393, 395, 397, 399, 401.

Eliminate 395 (divisible by 5) and 393, 399 (sum of digits divisible by 3).

The remaining candidates are 397 and 401.

Check divisibility by primes up to √401 ≈ 20 (primes: 7, 11, 13, 17, 19).

Neither 397 nor 401 is divisible by any of these primes, meaning both are prime numbers.

∴ The correct answer is 2.

Alternate Method

Given:

The numbers from 393 to 401, both included.

Method Used:

Primality test: A number N is prime if it is not divisible by any prime number ≤ √N.

Calculations:

⇒ Total numbers to check: 393, 394, 395, 396, 397, 398, 399, 400, 401

⇒ Even numbers greater than 2 are not prime. Discard 394, 396, 398, 400.

⇒ Numbers ending in 5 are not prime. Discard 395.

⇒ Numbers whose digit sum is a multiple of 3 are not prime. Discard 393 (3 + 9 + 3 = 15) and 399 (3 + 9 + 9 = 21).

⇒ Only 397 and 401 remain as potential prime numbers.

⇒ For 397: √397 ≈ 19.9. Prime numbers ≤ 19 are 2, 3, 5, 7, 11, 13, 17, 19. 397 is not divisible by any of them. ∴ 397 is a prime number.

⇒ For 401: √401 ≈ 20.02. Prime numbers ≤ 19 are 2, 3, 5, 7, 11, 13, 17, 19. 401 is not divisible by any of them. ∴ 401 is a prime number.

⇒ Total prime numbers between 393 and 401 = 2 (which are 397 and 401).

∴ The correct answer is 2.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It has exactly two distinct factors: 1 and itself.

Primality Test

To check if a number 'N' is prime, find its approximate square root (√N). If 'N' is not divisible by any prime number less than or equal to √N, then 'N' is a prime number.

Divisibility Rules

A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 5 if its last digit is 0 or 5. A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

Shortcut Trick

An odd number cannot be divisible by any even number.

Since the unit digit of 956169 is 9 (an odd number), it cannot be divisible by 2, 4, or 12.

Checking divisibility by 3: Sum of its digits = 9 + 5 + 6 + 1 + 6 + 9 = 36.

Since 36 is a multiple of 3, the given number is perfectly divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 956169

Concept Used:

Basic divisibility rules for the given options.

Calculations:

⇒ Checking Option 3 (Divisibility by 2):

⇒ The unit digit of 956169 is 9. Since 9 is not an even number, 956169 is not divisible by 2.

⇒ Checking Option 4 (Divisibility by 4):

⇒ The last two digits are 69. Since 69 is not divisible by 4, the entire number is not divisible by 4.

⇒ Checking Option 2 (Divisibility by 12):

⇒ A number is divisible by 12 only if it is divisible by both 3 and 4. Since it is not divisible by 4, it is not divisible by 12.

⇒ Checking Option 1 (Divisibility by 3):

⇒ Sum of the digits = 9 + 5 + 6 + 1 + 6 + 9 = 36.

⇒ Since 36 is perfectly divisible by 3 (36 ÷ 3 = 12), the number 956169 is divisible by 3.

∴ The correct answer is 3.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 (or 9) if the sum of all its individual digits is perfectly divisible by 3 (or 9).

Divisibility by 4 and 8

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. It is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

Composite Divisors

To check divisibility for composite numbers, check for its co-prime factors. For example, for 12, check 3 and 4; for 72, check 8 and 9.

Shortcut Trick

Let the average of the two consecutive even numbers be a. The numbers will be (a − 1) and (a + 1).

Sum of their squares = (a − 1)² + (a + 1)² = 2(a² + 1)

⇒ 2(a² + 1) = 1460 ⇒ a² + 1 = 730 ⇒ a² = 729 ⇒ a = 27

The numbers are 26 and 28. Their sum is 26 + 28 = 54.

∴ The correct answer is 54.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 1460.

Formula Used:

(a + b)² = a² + 2ab + b²

Calculation:

Let the two consecutive even natural numbers be x and (x + 2).

According to the given condition:

⇒ x² + (x + 2)² = 1460

⇒ x² + x² + 4x + 4 = 1460

⇒ 2x² + 4x + 4 = 1460

⇒ 2x² + 4x − 1456 = 0

Dividing the entire equation by 2:

⇒ x² + 2x − 728 = 0

Factorising the quadratic equation:

⇒ x² + 28x − 26x − 728 = 0

⇒ x(x + 28) − 26(x + 28) = 0

⇒ (x − 26)(x + 28) = 0

⇒ x = 26 or x = −28

Since the numbers are natural numbers, x must be positive. Therefore, x = 26.

⇒ The second consecutive even number = x + 2 = 26 + 2 = 28.

⇒ Sum of the numbers = 26 + 28 = 54.

∴ The correct answer is 54.

Additional Information

Consecutive Even and Odd Numbers

Consecutive even integers are always represented as 2n, 2n+2, 2n+4, etc. Consecutive odd integers are represented as 2n+1, 2n+3, 2n+5, etc.

Quadratic Formula

For a quadratic equation ax² + bx + c = 0, the exact roots can be found using the formula: x = (−b ± √(b² − 4ac)) ÷ 2a.

Sum and Product of Roots

For the general quadratic equation ax² + bx + c = 0, the sum of the roots is equal to −b/a and the product of the roots is equal to c/a.

Shortcut Trick

Convert 40% directly to its simplified fraction value: 40% = 2/5

Equate the fractional parts of both numbers: (2/5) × I = (2/7) × II

Cancel out the common numerator 2 from both sides of the equation.

I / II = 5 / 7 ⇒ I : II = 5 : 7

∴ The correct answer is 5 : 7.

Alternate Method

Given:

40% of the first number is equal to 2/7 of the second number.

40% of 1^st

2/7 of 2^nd

2/5 × 1^st
2/7 × 2^nd

Calculations:

Let the first number be x and the second number be y.

⇒ 40% of x = 2/7 of y

⇒ (40 / 100) × x = (2 / 7) × y

⇒ (2 / 5) × x = (2 / 7) × y

⇒ x / y = (2 / 7) ÷ (2 / 5)

⇒ x / y = (2 / 7) × (5 / 2)

⇒ x / y = 5 / 7

⇒ x : y = 5 : 7

∴ The correct answer is 5 : 7.

Additional Information

Percentage to Fraction Conversion

Knowing standard conversion equivalents saves calculation time during exams: 20% = 1/5, 40% = 2/5, 60% = 3/5, 80% = 4/5.

Ratio and Proportion

A ratio simply indicates how many times one number contains another. If a/b = c/d, the ratio of a to b is the same as c to d, represented standardly as a : b :: c : d.

Cross-Multiplication Principle

When comparing or equating two fractions a/b = c/d, you can apply cross-multiplication to rapidly solve for unknown variables: a × d = b × c.

Shortcut Trick

Convert the given percentage discounts into fractional multiplying factors:

25% discount → Factor = 1 − 1/4 = 3/4

20% discount → Factor = 1 − 1/5 = 4/5

Final Price = MP × 3/4 × 4/5

Final Price = 4000 × 3/5 = 2400

∴ The correct answer is ₹2,400.

Alternate Method

Given:

Marked Price (MP) = ₹4,000

First Discount (D₁) = 25%

Second Discount (D₂) = 20%

MP = ₹4000

−25%

₹3000

−20%

SP = ₹2400

Total Equivalent Discount = 40%

Formula Used:

Selling Price (SP) = MP × (1 − D₁/100) × (1 − D₂/100)

Calculations:

⇒ SP = 4000 × (1 − 25/100) × (1 − 20/100)

⇒ SP = 4000 × (75/100) × (80/100)

⇒ SP = 4000 × (3/4) × (4/5)

⇒ SP = 4000 × (12/20)

⇒ SP = 4000 × (3/5)

⇒ SP = 2400

∴ The correct answer is ₹2,400.

Additional Information

Equivalent Successive Discount

When two discounts D₁ and D₂ are applied successively, the single equivalent discount percentage is computed as D₁ + D₂ − (D₁ × D₂)/100.

Marked Price (MP)

It is the list price or printed tag price of an article. Any discount or promotional offer given by a seller is always calculated on the Marked Price, not the Cost Price.

Buy X Get Y Free

In this common discount format, the equivalent discount percentage is given by the formula [Y ÷ (X + Y)] × 100.