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Shortcut Trick

Step 1: Calculate the LCM of 10, 11, and 8 ⇒ 440.

Step 2: Find the largest 4-digit multiple of 440 ⇒ 440 × 22 = 9680.

Step 3: Add the required remainder (7) to this multiple ⇒ 9680 + 7 = 9687.

∴ The correct answer is 9687.

Alternate Method

Given: Divisors are 10, 11, and 8; Remainder in each case is 7.

Formula Used: Required Number = (LCM of divisors × k) + Remainder

⇒ LCM(10, 11, 8) = 2³ × 5 × 11 = 8 × 5 × 11 = 440

⇒ The largest four-digit number is 9999.

⇒ Dividing 9999 by 440: 9999 ÷ 440 = 22 with a remainder of 319.

⇒ The largest multiple of 440 below 10000 = 9999 − 319 = 9680.

⇒ The required number = 9680 + 7 = 9687.

∴ The correct answer is 9687.

Additional Information

LCM (Least Common Multiple)

It is the smallest positive integer that is evenly divisible by all numbers in a given set.

The Remainder Property

If a number N leaves a constant remainder R when divided by a, b, and c, then N = LCM(a, b, c) × k + R.

Divisibility Rule of 11

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

Shortcut Trick

Total deviation from old average 38 = (50 − 38) + (62 − 38) = 12 + 24 = 36.

Total numbers now = 12 + 2 = 14.

Average increase = Total deviation ÷ Total count = 36 ÷ 14 ≈ 2.57.

New Average = 38 + 2.57 = 40.57.

∴ The correct answer is 40.57.

Alternate Method

Given: Average of 12 numbers = 38, Included numbers = 50 and 62.

Formula: New Average = (Old Sum + New Values) ÷ New Count

⇒ Sum of 12 numbers = 12 × 38 = 456

⇒ Sum of 14 numbers = 456 + 50 + 62 = 568

⇒ Total count = 12 + 2 = 14

⇒ New Average = 568 ÷ 14 = 40.5714...

⇒ Rounding to two decimal places, New Average = 40.57

∴ The correct answer is 40.57.

Additional Information

Standard Average Formula

Average = (Sum of all observations) ÷ (Total number of observations).

Weighted Average

When two groups are combined, New Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Inclusion/Exclusion Rule

If the newly added value is greater than the current average, the new average will increase.

Shortcut Trick

Ratio of Profit = (Capital × Time)_X : (Capital × Time)_Y : (Capital × Time)_Z

Ratio = (60 × 12) : (80 × 6) : (120 × 4) = 720 : 480 : 480

Simplified Ratio = 3 : 2 : 2

Total units = 3 + 2 + 2 = 7 units → ₹77,000

Z's share (2 units) = (77,000 ÷ 7) × 2 = ₹22,000

∴ The correct answer is ₹22,000.

Alternate Method

Given:

Investment of X = ₹60,000, Time = 12 months

Investment of Y = ₹80,000, Time = 6 months

Investment of Z = ₹1,20,000, Time = 4 months

Total Profit = ₹77,000

Formula Used:

Profit Share = Capital × Time Period

Calculations:

⇒ Profit share of X = 60,000 × 12 = 7,20,000

⇒ Profit share of Y = 80,000 × 6 = 4,80,000

⇒ Profit share of Z = 1,20,000 × 4 = 4,80,000

⇒ Ratio of profits X : Y : Z = 7,20,000 : 4,80,000 : 4,80,000

⇒ Ratio = 72 : 48 : 48 = 3 : 2 : 2

⇒ Sum of the ratio parts = 3 + 2 + 2 = 7

⇒ Share of Z = (2 ÷ 7) × 77,000

⇒ Share of Z = 2 × 11,000 = ₹22,000

∴ The correct answer is ₹22,000.

Additional Information

Compound Partnership

A partnership where the partners invest their capitals for different periods of time. The profit is divided based on the product of capital and time.

Types of Partners

An Active Partner manages the business and may receive an extra share or salary, while a Sleeping Partner only invests money without participating in daily operations.

Ratio of Capitals

If the profit and time periods are given, the ratio of capitals C₁ : C₂ is calculated as (P₁/T₁) : (P₂/T₂).

Shortcut Trick

Net percentage change formula: Net % = a + b + (ab/100)

Here, a = −30 (decrease) and b = +15 (increase).

Net change = −30 + 15 + [(−30 × 15) / 100] = −15 − 4.5 = −19.5%.

New Price = 800 − (19.5% of 800) = 800 − 156 = 644.

∴ The correct answer is ₹644.

Alternate Method

Given: Original Price of the shirt = ₹800, First change = 30% decrease, Second change = 15% increase.

Method: Step-by-step price calculation using basic percentage.

⇒ Price after 30% decrease = Original Price × [ (100 − 30) / 100 ]

⇒ Intermediate Price = 800 × (70 / 100) = ₹560

⇒ Final price after 15% increase = Intermediate Price × [ (100 + 15) / 100 ]

⇒ New Price = 560 × (115 / 100) = 5.6 × 115 = ₹644

Thus, the final price of the shirt is ₹644, which corresponds to Option (4).

∴ The correct answer is ₹644.

Additional Information

Successive Percentage Rule

When a value changes by x% and then y%, the effective change is [x + y + (xy/100)]%. Remember to use a negative sign for decrease/discount.

Multiplying Factor Method

For a decrease of d%, multiply the value by (1 − d/100). For an increase of i%, multiply the value by (1 + i/100).

Fractional Equivalents

Calculations are often faster using fractions: 30% = 3/10 (multiplier 7/10) and 15% = 3/20 (multiplier 23/20).

Shortcut Trick

Ratio of ages of Shyam and Rajat 4 years ago = 7 : 1 (Difference = 6 units)

Ratio of ages 8 years later = 3 : 1 (Multiply by 3 to balance difference) → 9 : 3 (Difference = 6 units)

Increase in ratio units = 9 − 7 = 2 units

Actual time interval = 4 + 8 = 12 years

2 units = 12 years ⇒ 1 unit = 6 years

Shyam's age 4 years ago = 7 × 6 = 42 years ⇒ Present age = 42 + 4 = 46 years

∴ The correct answer is 46 years.

Alternate Method

Given:

Age of Shyam was 7 times Rajat 4 years ago.

Age of Shyam will be 3 times Rajat 8 years later.

Calculations:

Let the age of Rajat 4 years ago be x years.

⇒ Age of Shyam 4 years ago = 7x years.

⇒ Present age of Rajat = x + 4

⇒ Present age of Shyam = 7x + 4

⇒ Age after 8 years: Rajat = (x + 4) + 8 = x + 12; Shyam = (7x + 4) + 8 = 7x + 12

⇒ According to the condition: 7x + 12 = 3 × (x + 12)

⇒ 7x + 12 = 3x + 36

⇒ 7x − 3x = 36 − 12

⇒ 4x = 24

⇒ x = 6

⇒ Present age of Shyam = 7(6) + 4 = 42 + 4 = 46 years.

⇒ Therefore, the age of Shyam is 46 years, which corresponds to Option (4).

∴ The correct answer is 46 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same throughout their lives, regardless of how many years pass.

Ratio Balancing Method

To solve age problems using ratios, ensure the difference between ratio parts is equal for both time periods by multiplying with the cross-difference.

Average Age Shift

If the average age of n people is A years, then after k years, their average age will be A + k years.

Shortcut Trick

Gain of 50% = 1/2 ⇒ Ratio of CP : SP for Article 1 = 2 : 3.

Since Selling Price is the same for both, SP of Article 2 = 3 units.

No profit, no loss condition implies: Profit on Article 1 = Loss on Article 2.

Profit on Article 1 = 3 − 2 = 1 unit. So, Loss on Article 2 = 1 unit.

CP of Article 2 = SP + Loss = 3 + 1 = 4 units.

Loss % on Article 2 = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Selling Price (SP) of Article 1 = ₹2,973

Selling Price (SP) of Article 2 = ₹2,973

Gain % on Article 1 = 50%

Formula Used:

Cost Price (CP) = (SP × 100) ÷ (100 + Gain%)

Loss % = (Loss ÷ CP) × 100

Calculations:

⇒ CP of Article 1 = (2973 × 100) ÷ (100 + 50)

⇒ CP₁ = (2973 × 100) ÷ 150 = ₹1,982

⇒ Profit on Article 1 = 2973 − 1982 = ₹991

⇒ Since overall there is no profit, no loss:

⇒ Loss on Article 2 = Profit on Article 1 = ₹991

⇒ CP of Article 2 = SP of Article 2 + Loss

⇒ CP₂ = 2973 + 991 = ₹3,964

⇒ Loss % on Article 2 = (991 ÷ 3964) × 100

⇒ Loss % = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Additional Information

No Profit, No Loss Condition

This occurs when the total Cost Price of all articles equals the total Selling Price. Mathematically, Total CP = Total SP, or Profit from one item = Loss from another.

Cost Price with Gain/Loss

When gain is given, CP = SP × 100 / (100 + Gain%). When loss is given, CP = SP × 100 / (100 − Loss%).

Successive Changes

If an article's price is changed successively by x% and y%, the net change is given by the formula: [x + y + (xy ÷ 100)]%.

Shortcut Trick

For a regular polygon, the sum of an interior angle and an exterior angle at any vertex is always 180°.

⇒ Exterior angle = 180° − Interior angle = 180° − 162° = 18°.

The number of sides (n) of a regular polygon is given by 360° ÷ exterior angle.

⇒ n = 360° ÷ 18° = 20.

∴ The correct answer is 20 sides.

Alternate Method

Given:

Measure of each interior angle = 162°

Formula Used:

Each interior angle of a regular polygon with n sides = [(n − 2) × 180°] ÷ n

Calculations:

⇒ 162° = [(n − 2) × 180°] ÷ n

⇒ 162n = (n − 2) × 180

⇒ 162n = 180n − 360

⇒ 180n − 162n = 360

⇒ 18n = 360

⇒ n = 360 ÷ 18

⇒ n = 20

∴ The correct answer is 20 sides.

Additional Information

Sum of Interior Angles

The total sum of all interior angles of a polygon with n sides is always (n − 2) × 180°.

Sum of Exterior Angles

For any convex polygon, the sum of all exterior angles (one at each vertex) is exactly 360°.

Regular Polygon Properties

In a regular polygon, all sides are equal and all interior angles are equal. Each exterior angle = 360° ÷ n.

Shortcut Trick

The distance d between two objects in the same direction from a cliff of height h is given by: d = h × (cot θ₁ − cot θ₂).

Given: h = 80 m, θ₁ = 30°, and θ₂ = 60°.

d = 80 × (cot 30° − cot 60°) = 80 × (√3 − 1/√3) = 80 × (2/√3).

d = 160 / 1.732 ≈ 92.38 m.

∴ The correct answer is 92.38 m.

Alternate Method

Given: Height of the cliff (AB) = 80 m. Angles of depression of ships C and D are 60° and 30° respectively.

Formula Used: tan θ = Perpendicular / Base

In right-angled triangle ABC:

⇒ tan 60° = AB / BC

⇒ √3 = 80 / BC

⇒ BC = 80 / √3 m

In right-angled triangle ABD:

⇒ tan 30° = AB / BD

⇒ 1 / √3 = 80 / BD

⇒ BD = 80√3 m

Distance between ships (CD) = BD − BC:

⇒ CD = 80√3 − 80 / √3

⇒ CD = (80 × 3 − 80) / √3 = 160 / √3

⇒ CD = 160 / 1.73205 ≈ 92.376 m

Rounding off to two decimal places, we get 92.38 m.

∴ The correct answer is 92.38 m.

Additional Information

Angle of Depression

The angle formed between the horizontal line and the line of sight when an observer looks downward at an object.

Distance for Opposite Sides

If ships are on opposite sides of the cliff, the distance between them is d = h(cot α + cot β).

Trigonometric Identities

For calculations, remember tan 30° = 1/√3, tan 60° = √3, and √3 ≈ 1.732.

Given:

$\dfrac{3^{-5} \times 8^5 \times 125^3 \times 36^3}{100^4 \times 256 \times 216 \times 2^2}$

Formula used:

Exponent Laws:

(a^m)ⁿ = a^{m × n}

a^m × aⁿ = a^{m + n}

a^m ÷ aⁿ = a^{m − n}

a⁻ⁿ = 1 / aⁿ

Calculations:

Express each term in its prime factor form (2, 3, and 5):

⇒ $\dfrac{3^{-5} \times (2^3)^5 \times (5^3)^3 \times (2^2 \times 3^2)^3}{(2^2 \times 5^2)^4 \times 2^8 \times (2^3 \times 3^3) \times 2^2}$

⇒ $\dfrac{3^{-5} \times 2^{15} \times 5^9 \times 2^6 \times 3^6}{2^8 \times 5^8 \times 2^8 \times 2^3 \times 3^3 \times 2^2}$

Group the bases in the numerator and denominator:

⇒ $\dfrac{2^{15 + 6} \times 3^{6 - 5} \times 5^9}{2^{8 + 8 + 3 + 2} \times 3^3 \times 5^8}$

⇒ $\dfrac{2^{21} \times 3^1 \times 5^9}{2^{21} \times 3^3 \times 5^8}$

Divide by subtracting exponents of the same base:

⇒ 2^{21 − 21} × 3^{1 − 3} × 5^{9 − 8}

⇒ 2⁰ × 3⁻² × 5¹

⇒ 1 × (1 / 3²) × 5

⇒ 5 / 9

∴ The correct answer is option (2).

Shortcut Trick

Since the distance is the same for both ways (48 km), we use the Harmonic Mean formula for average speed.

Average Speed = (2 × s₁ × s₂) / (s₁ + s₂)

Average Speed = (2 × 32 × 24) / (32 + 24) = (2 × 32 × 24) / 56

Average Speed = (32 × 24) / 28 = (8 × 24) / 7 = 192 / 7 ≈ 27.43 km/hr.

∴ The correct answer is 27.43 km/hr.

Alternate Method

Given:

Distance from Warehouse A to Store B = 48 km

Speed during onward journey (s₁) = 32 km/hr

Speed during return journey (s₂) = 24 km/hr

Formula used:

Average Speed = Total Distance ÷ Total Time

⇒ Total Distance = 48 + 48 = 96 km

⇒ Time taken for onward journey (t₁) = Distance ÷ s₁ = 48 ÷ 32 = 1.5 hours

⇒ Time taken for return journey (t₂) = Distance ÷ s₂ = 48 ÷ 24 = 2 hours

⇒ Total Time = t₁ + t₂ = 1.5 + 2 = 3.5 hours

⇒ Average Speed = 96 ÷ 3.5

⇒ Average Speed = 960 ÷ 35 = 192 ÷ 7 ≈ 27.4285...

∴ The correct answer is 27.43 km/hr.

Additional Information

Average Speed (Fixed Distance)

When an object covers equal distances at speeds x and y, the average speed is 2xy ÷ (x + y).

Average Speed (Fixed Time)

When an object travels for the same duration at speeds x and y, the average speed is the arithmetic mean: (x + y) ÷ 2.

General Average Speed

For any journey, Average Speed = (d₁ + d₂ + ... + d_n) ÷ (t₁ + t₂ + ... + t_n).

Unit Conversion

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