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A number is divisible by 6 if it is both even (divisible by 2) and the sum of its digits is divisible by 3.

Checking 7,53,216: It is even (ends in 6). Sum of digits = 7 + 5 + 3 + 2 + 1 + 6 = 24 (divisible by 3).

Checking other options: 5,47,821 is odd; 6,42,538 (sum=28) and 8,91,452 (sum=29) are not divisible by 3.

∴ The correct answer is 7,53,216.

Alternate Method

Given Numbers: 7,53,216, 6,42,538, 8,91,452, 5,47,821

A number is divisible by 6 if it satisfies two divisibility rules simultaneously:

Step-by-step verification for 7,53,216:

⇒ Condition 1 (Divisibility by 2): The last digit is 6, which is even. ⇒ Passed.

⇒ Condition 2 (Divisibility by 3): Sum of digits = 7 + 5 + 3 + 2 + 1 + 6 = 24.

⇒ 24 ÷ 3 = 8 (No remainder). ⇒ Passed.

⇒ Since both conditions are satisfied, the number is divisible by 6.

∴ The correct answer is 7,53,216.

Additional Information

Divisibility by 4 and 8

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

Divisibility by 9

Similar to the rule for 3, a number is divisible by 9 if the sum of all its digits is divisible by 9.

Divisibility by 11

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

Shortcut Trick

Assume initial Cost Price = 100 units

Initial SP = 115 units and New CP = 90 units (10% less)

New SP = 90 × 120% = 108 units (20% gain)

Difference in SP = 115 − 108 = 7 units

7 units = ₹126 ⇒ 1 unit = ₹18 ⇒ 100 units = ₹1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given:

Initial Profit = 15%, Reduction in CP = 10%, Reduction in SP = ₹126, New Profit = 20%

Let the Cost Price (CP) of the product be ₹ x

⇒ Initial Selling Price (SP₁) = x + 15% of x = 1.15x

⇒ New Cost Price (CP₂) = x − 10% of x = 0.90x

⇒ New Selling Price (SP₂) = 0.90x + 20% of 0.90x

⇒ SP₂ = 0.90x × (120/100) = 1.08x

⇒ According to the question: SP₁ − SP₂ = 126

⇒ 1.15x − 1.08x = 126

⇒ 0.07x = 126

⇒ x = 126 ÷ 0.07

⇒ x = 12600 ÷ 7 = 1800

∴ The correct answer is ₹1,800.

Additional Information

Profit and Loss Formulas

Profit % = (Profit ÷ CP) × 100 and SP = CP × (100 + Profit%) ÷ 100.

Net Percentage Change

When a value changes by a% and then b%, the net change is given by (a + b + ab/100)%.

Relationship between CP and MP

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This is used for problems involving markup and discount.

Shortcut Trick

Compare the effective discount percentage for each scheme on 10 toys:

(A) 16 + 29 − (16 × 29)/100 = 45 − 4.64 = 40.36%

(B) Effective of (4, 34, 37) is approx. 60.1% (high discount)

(C) Average discount = [(4 × 12%) + (6 × 27%)] ÷ 10 = (48 + 162) ÷ 10 = 21%

(D) Discount = [Free ÷ Total] × 100 = (4 ÷ 14) × 100 ≈ 28.57%

The least beneficial scheme is the one with the lowest discount percentage.

∴ The correct answer is C.

Alternate Method

Given: Total toys to buy = 10. Let Marked Price (MP) of 1 toy = ₹100. Total MP = ₹1000.

Formula Used: Effective Discount = (Total Discount ÷ Total MP) × 100

Calculations:

Scheme (A): Successive 16% and 29%

⇒ Net Discount = 16 + 29 − (16 × 29)/100 = 45 − 4.64 = 40.36%

Scheme (B): Successive 4%, 34% and 37%

⇒ First two: 4 + 34 − (4 × 34)/100 = 36.64%

⇒ Final: 36.64 + 37 − (36.64 × 37)/100 = 73.64 − 13.55 = 60.09%

Scheme (C): 12% on 4 toys and 27% on 6 toys

⇒ Discount on 4 toys = 4 × 12 = ₹48

⇒ Discount on 6 toys = 6 × 27 = ₹162

⇒ Total Discount = 48 + 162 = ₹210

⇒ Effective Discount % = (210 ÷ 1000) × 100 = 21%

Scheme (D): 4 toys free on buying 10 toys

⇒ Effective Discount % = [Free ÷ (Paid + Free)] × 100

⇒ Discount % = [4 ÷ (10 + 4)] × 100 = 400 ÷ 14 ≈ 28.57%

Comparing all, Scheme C offers the minimum discount (21%), making it the least beneficial for the customer.

∴ The correct answer is C.

Additional Information

Successive Discount Formula

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

Buy X Get Y Free

In such schemes, the effective discount percentage is calculated as [Y ÷ (X + Y)] × 100.

Customer's Benefit

A scheme is most beneficial when the discount is maximum and least beneficial when the discount is minimum.

Shortcut Trick

Initial Ratio = 3 : 5 (Difference = 2 units)

Final Ratio = 1 : 2 (Difference = 1 unit)

Equating the difference by multiplying the final ratio by 2: 1 × 2 : 2 × 2 = 2 : 4

Comparison: 3 → 2 and 5 → 4 (Decrease of 1 unit = 6)

Original Difference = 5 − 3 = 2 units ⇒ 2 × 6 = 12

∴ The correct answer is 12.

Alternate Method

Given:

Initial Ratio of numbers = 3 : 5

Value subtracted from each = 6

Resulting Ratio = 1 : 2

Formula Used:

If ratio is a : b, numbers are ax and bx. Difference = (b − a)x

Calculations:

Let the two numbers be 3x and 5x.

⇒ (3x − 6) / (5x − 6) = 1 / 2

⇒ 2 × (3x − 6) = 1 × (5x − 6)

⇒ 6x − 12 = 5x − 6

⇒ 6x − 5x = 12 − 6

⇒ x = 6

⇒ Original Difference = 5x − 3x = 2x

⇒ 2 × 6 = 12

∴ The correct answer is 12.

Additional Information

Ratio Difference Property

When the same quantity is added to or subtracted from both terms of a ratio, the absolute difference between the terms remains a constant multiple of the common variable x.

Cross-Multiplication Method

For any ratio problem involving changes, the Cross-product method (Product of extremes = Product of means) is the most reliable way to set up the linear equation.

Proportionality

A proportion is an equality of two ratios. If a/b = c/d, then a × d = b × c.

Given:

Ratios: 4 : 7, 3 : 5, 5 : 8, and 7 : 1

Formula used:

Ratio a : b = a ÷ b

Calculation:

To determine the greatest ratio, convert each into its decimal equivalent:

4 : 7 ⇒ 4 ÷ 7 ≈ 0.571

3 : 5 ⇒ 3 ÷ 5 = 0.6

5 : 8 ⇒ 5 ÷ 8 = 0.625

7 : 1 ⇒ 7 ÷ 1 = 7.0

Comparing these decimal values:

⇒ 7.0 > 0.625 > 0.6 > 0.571

The largest value is 7.0, which corresponds to the ratio 7 : 1.

∴ The correct answer is option (4).

Shortcut Trick

Surface Area of a sphere in terms of diameter (d) is π × d².

28962 ≈ (22/7) × d²

d² ≈ (28962 × 7) ÷ 22 ≈ 9215.18

Since 96² = 9216, the diameter d ≈ 96 cm.

∴ The correct answer is 96 cm.

Alternate Method

Given:

Surface Area (SA) of the sphere = 28962 cm²

Value of π = 22/7

Formula Used:

Surface Area of sphere = 4 × π × r²

Diameter (d) = 2 × r

Calculations:

⇒ Surface Area = 4 × (22/7) × r²

⇒ 28962 = (88/7) × r²

⇒ r² = (28962 × 7) ÷ 88

⇒ r² = 202734 ÷ 88

⇒ r² = 2303.795

⇒ r = √2303.795 ≈ 47.997

⇒ Diameter (d) = 2 × r

⇒ d = 2 × 47.997 = 95.994

⇒ Rounding to the closest integer, d ≈ 96 cm.

∴ The correct answer is 96 cm.

Additional Information

Volume of a Sphere

The volume (V) of a sphere is given by the formula V = (4/3) × π × r³.

Hemisphere Surface Area

For a solid hemisphere, Curved Surface Area (CSA) = 2πr² and Total Surface Area (TSA) = 3πr².

Circle vs Sphere

A circle is a 2D shape with area πr², while a sphere is its 3D counterpart with surface area 4πr².

Given:

3/4 + (5/6 × 9/10) − 2/3

Formula used:

The BODMAS rule is used to solve the expression:

B = Bracket

O = Of

D = Division

M = Multiplication

A = Addition

S = Subtraction

Calculations:

First, solve the Multiplication inside the bracket:

⇒ 5/6 × 9/10

⇒ 45/60

⇒ 3/4

Now, the expression becomes:

⇒ 3/4 + 3/4 − 2/3

Next, perform the Addition:

⇒ (3 + 3) / 4 − 2/3

⇒ 6/4 − 2/3

⇒ 3/2 − 2/3

Finally, perform the Subtraction by finding the LCM of 2 and 3, which is 6:

⇒ (3 × 3 − 2 × 2) / 6

⇒ (9 − 4) / 6

⇒ 5/6

∴ The correct answer is option (2).

Shortcut Trick

Total distance = Train length + Bridge length = 180 + 350 = 530 m.

Speed in m/s = 72 × 5/18 = 20 m/s.

Required Time taken = 530 ÷ 20 = 26.5 sec.

∴ The correct answer is 26.5 sec.

Alternate Method

Given:

Train length = 180 m

Bridge length = 350 m

Speed = 72 km/h

Formula Used:

Time = Total Distance ÷ Speed

Calculations:

⇒ Total distance to cross = 180 + 350 = 530 m

⇒ Speed in m/s = 72 × (5 ÷ 18) = 20 m/s

⇒ Time taken = 530 ÷ 20

⇒ Time taken = 26.5 sec

∴ The correct answer is 26.5 sec.

Additional Information

Unit Conversion

To convert speed from km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Relative Speed

When two objects move in the same direction, relative speed is the difference. In opposite directions, it is the sum.

Distance Concept

When crossing stationary objects like a platform or tunnel, total distance is the sum of the train length and the object length.

Shortcut Trick

Average Speed for equal distances = (2 × s₁ × s₂) ÷ (s₁ + s₂)

⇒ Average Speed = (2 × 5 × 10) ÷ (5 + 10) = 100 ÷ 15 = 20/3 km/hr.

Total Distance covered (round trip) = Average Speed × Total Time = (20/3) × 3 = 20 km.

Distance between Point P and Point Q = Total Distance ÷ 2 = 20 ÷ 2 = 10 km.

∴ The correct answer is 10 km.

Alternate Method

Given:

Speed from P to Q (s₁) = 5 km/hr

Speed from Q to P (s₂) = 10 km/hr

Total Time (T) = 3 hr

Formula Used:

Time = Distance ÷ Speed

Calculations:

Let the Distance between P and Q be 'd' km.

⇒ Time taken from P to Q (t₁) = d ÷ 5

⇒ Time taken from Q to P (t₂) = d ÷ 10

⇒ Total Time = t₁ + t₂ = 3

⇒ d/5 + d/10 = 3

⇒ (2d + d) ÷ 10 = 3

⇒ 3d ÷ 10 = 3

⇒ 3d = 30

⇒ d = 10 km

∴ The correct answer is 10 km.

Additional Information

Average Speed (Equal Distances)

When a person covers two equal distances at speeds x and y, the average speed for the whole journey is given by 2xy ÷ (x + y).

Ratio of Speed and Time

If the distance is constant, the ratio of speeds is inversely proportional to the ratio of time taken (Speed₁ : Speed₂ = Time₂ : Time₁).

Average Speed (Equal Time Intervals)

When a person travels for two equal time intervals at speeds x and y, the average speed is the arithmetic mean (x + y) ÷ 2.

Shortcut Trick

The present sum of ages is 58 − (2 × 6) = 46 years.

If the son is s, then the man is 2s + 4.

Sum: s + 2s + 4 = 46 ⇒ 3s = 42 ⇒ s = 14.

∴ The correct answer is 14 years.

Alternate Method

Given:

Present age relationship: Man = 2 × Son + 4

Future sum (after 6 years): 58 years

Calculations:

⇒ Let the present age of the son be x years.

⇒ The present age of the man = 2x + 4 years.

⇒ After 6 years, son's age = x + 6.

⇒ After 6 years, man's age = (2x + 4) + 6 = 2x + 10.

⇒ Sum of their ages after 6 years = (x + 6) + (2x + 10) = 58.

⇒ 3x + 16 = 58.

⇒ 3x = 58 − 16.

⇒ 3x = 42.

⇒ x = 42 ÷ 3 = 14.

∴ The correct answer is 14 years.

Additional Information

Age Sum Calculation

When calculating the sum of ages after n years for k people, the sum increases by n × k. For two people, the sum increases by 2n.

Linear Equations in Ages

Age problems are typically solved by assigning a variable to the youngest person's present age to simplify the algebraic expression.

Age Ratio Rule

The difference between the ages of two individuals remains constant at any point in time, regardless of how many years pass.