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Shortcut Trick

Consecutive numbers ⇒ average = (first + last) ÷ 2

⇒ = (1 + 60) ÷ 2 = 61 ÷ 2

⇒ = 30.5

∴ Answer = 30.5

Alternate Method

Given:

Natural numbers from 1 to 60

Formula:

Average = (Sum of numbers) ÷ Total numbers

Sum of first n natural numbers = n(n + 1) ÷ 2

Calculation:

⇒ Sum = 60 × 61 ÷ 2 = 1830

⇒ Total numbers = 60

⇒ Average = 1830 ÷ 60 = 30.5

∴ Final Answer = 30.5

Additional Information

Consecutive Numbers: Average = (First + Last) ÷ 2

Sum Formula: n(n+1) ÷ 2

Key Insight: Middle value represents average

Tip: Use shortcut instead of full summation

Shortcut Trick

⇒ Amount doubles in 8 years ⇒ Growth factor = 2

⇒ 8 times = 2³

⇒ Time for 8 times = 3 × 8 = 24 years

∴ Answer = 24 years

Alternate Method

Given:

⇒ Amount doubles in 8 years

Concept:

⇒ A = P(1 + r)^t

Step 1:

⇒ 2P = P(1 + r)⁸

⇒ (1 + r)⁸ = 2

Step 2:

⇒ For 8 times:

⇒ 8P = P(1 + r)^t

⇒ (1 + r)^t = 8 = 2³

Step 3:

⇒ (1 + r)^t = [(1 + r)⁸]³

⇒ ⇒ t = 8 × 3 = 24 years

∴ Final Answer: 24 years

Additional Information

1. Compound Interest: A = P(1 + r)^t

2. Doubling Concept: If amount doubles → growth factor = 2

3. Exponential Growth: Multiples follow powers (2, 4, 8 = 2³)

4. Shortcut: Time multiplies with power

Shortcut Trick

Volume of Sphere = (4/3) × π × r³ = 38808 cm³.

Using 11 divisibility rule for Surface Area (4πr²): Only 5544 is divisible by 11.

Calculation: (5+4) − (5+4) = 0. Divisible by 11.

∴ The correct answer is 5544 cm².

Alternate Method

Given: Volume (V) = 38808 cm³

Formula Used: Volume = (4/3)πr³ and Surface Area = 4πr²

⇒ 38808 = (4/3) × (22/7) × r³

⇒ r³ = (38808 × 3 × 7) ÷ (4 × 22)

⇒ r³ = 441 × 21 = 9261

⇒ r = √³(9261) = 21 cm

⇒ Surface Area = 4 × (22/7) × 21 × 21

⇒ Surface Area = 88 × 3 × 21 = 5544 cm²

∴ The correct answer is 5544 cm².

Additional Information

Properties of Sphere

A Sphere is a perfectly round geometrical object in three-dimensional space. Unlike other solids, it has no edges or vertices.

Hemisphere Formulas

For a Hemisphere of radius r, the curved surface area is 2πr² and the total surface area is 3πr².

Volume Ratios

If the radius is doubled, the Surface Area increases by 4 times (2²) and the Volume increases by 8 times (2³).

Shortcut Trick

Ratio of Perimeter (P) to Breadth (B) is 5 : 1 ⇒ P = 5B

We know P = 2(L + B) ⇒ 2(L + B) = 5B ⇒ 2L = 3B ⇒ L : B = 3 : 2

Let Length = 3x and Breadth = 2x.

Area = 3x × 2x = 6x² = 216 ⇒ x² = 36 ⇒ x = 6

Length = 3x = 3 × 6 = 18 cm.

∴ The correct answer is 18 cm.

Alternate Method

Given:

Ratio of perimeter to breadth = 5 : 1

Area of the rectangle = 216 sq. cm

Formulas Used:

Perimeter of rectangle = 2(L + B)

Area of rectangle = L × B

Length (L) Breadth (B) Area = 216 sq. cm

Calculations:

⇒ Perimeter / Breadth = 5 / 1

⇒ 2(L + B) / B = 5

⇒ 2L + 2B = 5B

⇒ 2L = 3B

⇒ L / B = 3 / 2

⇒ Let Length (L) = 3x and Breadth (B) = 2x

⇒ Area = L × B = 216

⇒ 3x × 2x = 216

⇒ 6x² = 216

⇒ x² = 216 ÷ 6 = 36

⇒ x = √36 = 6

⇒ Length = 3x = 3 × 6 = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Perimeter of a Rectangle

The total distance around the outside of a rectangle. It is given by P = 2(L + B), where L is the length and B is the breadth.

Area of a Rectangle

The total space enclosed within the boundary of a rectangle. It is calculated as Area = L × B.

Diagonal of a Rectangle

A straight line connecting the opposite corners of a rectangle. It is calculated using the Pythagoras theorem: Diagonal = √(L² + B²).

Shortcut Trick

Let Total Work be LCM(3, 4, 12) = 12 units.

Efficiency of 1 Man = 4, 1 Woman = 3, 1 Boy = 1.

Target time = 1 day, so total efficiency needed = 12 units/day.

Efficiency of (1 Man + 1 Woman) = 4 + 3 = 7 units.

Efficiency required from boys = 12 − 7 = 5 units.

Number of boys required = 5 ÷ 1 = 5 boys.

∴ The correct answer is 5 boys.

Alternate Method

Given: Time taken by 1 Man = 3 days, 1 Woman = 4 days, 1 Boy = 12 days.

Formula Used: Total Work = Efficiency × Time

⇒ Efficiency of 1 Man + 1 Woman + x Boys = 12 (to complete in 1 day)

⇒ (4 × 1) + (3 × 1) + (1 × x) = 12

⇒ 7 + x = 12

⇒ x = 12 − 7

⇒ x = 5

∴ The correct answer is 5 boys.

Additional Information

Work Equivalence Formula

M₁D₁H₁ = M₂D₂H₂ where M is number of workers, D is days, and H is hours.

Efficiency Concept

Efficiency is inversely proportional to time. If a person completes work in 'n' days, their 1-day work (efficiency) is 1/n.

Combined Work

When multiple people work together, their individual efficiencies are added to find the total work rate.

Shortcut Trick

Find the common difference (k): (36 − 16) = (72 − 52) = (80 − 60) = (88 − 68) = 20.

The least number is LCM(36, 72, 80, 88) − k.

LCM(36, 72, 80, 88) = 7920.

Required Number = 7920 − 20 = 7900.

Sum of digits = 7 + 9 + 0 + 0 = 16.

∴ The correct answer is 16.

Alternate Method

Given divisors and their respective remainders:

To find the Least Common Multiple (LCM) of 36, 72, 80, and 88:

⇒ Prime factorisation:

⇒ 36 = 2² × 3²

⇒ 72 = 2³ × 3²

⇒ 80 = 2⁴ × 5

⇒ 88 = 2³ × 11

⇒ LCM = 2⁴ × 3² × 5 × 11

⇒ LCM = 16 × 9 × 5 × 11 = 7920

⇒ The least number (N) = LCM − Common Difference

⇒ N = 7920 − 20 = 7900

⇒ Sum of digits of N = 7 + 9 + 0 + 0 = 16

∴ The correct answer is 16.

Additional Information

LCM with Constant Difference

When a number leaves remainders r₁, r₂, r₃ when divided by a, b, c such that (a−r₁) = (b−r₂) = (c−r₃) = k, the number is LCM(a, b, c) − k.

LCM with Same Remainder

If the number leaves the same remainder 'r' in each case when divided by a, b, c, then the least such number is LCM(a, b, c) + r.

Divisibility of Sum of Digits

If the sum of digits of a number is divisible by 3 or 9, then the number itself is divisible by 3 or 9 respectively.

Shortcut Trick

Let Savings = 1 unit.

Expenditure = 1 + 300% of 1 = 4 units.

Income = Expenditure + Savings = 4 + 1 = 5 units.

Using Net Deviation: Increase in (Exp × %E) + (Sav × %S) = Inc × %I

(4 × %E) + (1 × 20) = 5 × 30 ⇒ 4 × %E = 150 − 20 = 130 ⇒ %E = 32.5%.

∴ The correct answer is 32.5%.

Alternate Method

Given: Expenditure is 300% more than savings.

⇒ Let Initial Savings = ₹100

⇒ Initial Expenditure = 100 + (300% of 100) = ₹400

⇒ Initial Income = 400 + 100 = ₹500

⇒ New Income (30% increase) = 500 × 1.30 = ₹650

⇒ New Savings (20% increase) = 100 × 1.20 = ₹120

⇒ New Expenditure = New Income − New Savings = 650 − 120 = ₹530

⇒ Increase in Expenditure = 530 − 400 = ₹130

⇒ Percentage Increase = (130 ÷ 400) × 100 = 32.5%

∴ The correct answer is 32.5%.

Additional Information

Income-Expenditure Relation

Income is the sum of Expenditure and Savings. Income = Expenditure + Savings.

Percentage Increase Shortcut

Percentage Change = (Final Value − Initial Value) ÷ Initial Value × 100.

Weighted Average in Income

Income % change = (Exp Ratio × Exp % change + Sav Ratio × Sav % change) ÷ Total Ratio.

Shortcut Trick

In a partnership, Profit Ratio = Capital Ratio × Time Ratio

Let Naveen's investment time be T months.

Ratio of their profits = (Sunil's capital × Sunil's time) ÷ (Naveen's capital × Naveen's time)

2 / 3 = (11 × 8) / (12 × T)

24T = 264 ⇒ T = 11

∴ The correct answer is 11 months.

Alternate Method

Given:

Investment ratio of Sunil and Naveen = 11 : 12

Time period for Sunil = 8 months

Profit ratio of Sunil and Naveen = 2 : 3

Formula Used:

Profit = Investment × Time

Calculations:

⇒ Let the time period for Naveen be T months.

⇒ (Investment of Sunil × Time of Sunil) / (Investment of Naveen × Time of Naveen) = Profit of Sunil / Profit of Naveen

⇒ (11 × 8) / (12 × T) = 2 / 3

⇒ 88 / 12T = 2 / 3

⇒ 88 × 3 = 24 × T

⇒ 264 = 24T

⇒ T = 264 ÷ 24

⇒ T = 11 months

∴ The correct answer is 11 months.

Additional Information

Simple Partnership

When investments of all partners are for the exact same time period, the profit or loss is distributed among the partners purely in the ratio of their original investments. (Profit ∝ Capital)

Compound Partnership

When investments are for different time periods, equivalent capitals are calculated for a unit of time by multiplying capital with time. Profit ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Working vs Sleeping Partner

A working partner actively manages the business and receives a fixed salary or a specific percentage of the total profit first. The remaining profit is then distributed among all partners (including the working partner) in their capital ratio.

Given: 

$81^{x-3}$ ÷ $81^4$ = $\sqrt[6]{9^{24}}$

Formula used:

a^m ÷ aⁿ = a^m-n

(a^m)ⁿ = a^mn

)\(\sqrt[n]{x} = x^{\frac{1}{n}}

If a^x = a^y, then x = y

Calculation:

$81^{x-3}$ ÷ $81^4$ = $\sqrt[6]{9^{24}}$

⇒ $81^{x-3-4}$ = $\sqrt[6]{9^{24}}$

⇒ $81^{x-3-4}$ = $\sqrt[6]{9^{2(12)}}$

⇒ $81^{x-3-4}$ = $\sqrt[6]{81^{12}}$

⇒ $81^{x-3-4}$ = ${81^{\frac{12}{6}}}$

⇒ $81^{x-3-4}$ = $81^{2}$

Now, on comparing the power with same base, we get,

⇒ x - 3 - 4 = 2

⇒ x - 7 = 2

⇒ x = 9

∴ The required value of 'x' is 9.

Shortcut Trick

Profit Ratio P : Q : R = 1430 : 1870 : 2420 = 13 : 17 : 22.

Total Profit units = 13 + 17 + 22 = 52 units.

Since time is constant, Investment Ratio = Profit Ratio.

52 units = Rs. 52,000 ⇒ 1 unit = Rs. 1,000.

Q investment = 17 units × 1,000 = Rs. 17,000.

∴ The correct option is C.

Alternate Method

Given:

Total investment = Rs. 52,000

Profit of P = Rs. 1430, Profit of Q = Rs. 1870, Profit of R = Rs. 2420

Formula Used:

Investment Ratio = Profit Ratio (When time period of investment is same for all partners)

Profit Ratio Analysis

P: 13u

Q: 17u

R: 22u

Total Units = 52 = Rs. 52,000

Calculations:

Profit Ratio P : Q : R = 1430 : 1870 : 2420

⇒ P : Q : R = 143 : 187 : 242 (Dividing by 10)

⇒ P : Q : R = 13 : 17 : 22 (Dividing by 11)

Sum of ratio parts = 13 + 17 + 22 = 52

⇒ Q’s Investment = (17 / 52) × 52000

⇒ Q’s Investment = 17 × 1000

⇒ Q’s Investment = Rs. 17,000

∴ The correct option is C.

Additional Information

Simple Partnership

When investments are for the same duration, Profit₁ : Profit₂ = Capital₁ : Capital₂.

Compound Partnership

When investments are for different durations, Profit Ratio = (C₁ × T₁) : (C₂ × T₂), where C is capital and T is time.

Calculating Capital

Capital = Profit ÷ Time. If profit ratio and time ratio are known, the capital ratio can be derived directly.