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Shortcut Trick

Effective rate for 2 years at 20% p.a. = 20 + 20 + (20 × 20) ÷ 100 = 44%.

Compound Interest = 44% of ₹3,000.

CI = 3,000 × 0.44 = ₹1,320.

∴ The correct answer is ₹1,320.

Alternate Method

Given: Principal (P) = ₹3,000, Rate (R) = 20%, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + R ÷ 100)ⁿ and CI = A − P

⇒ A = 3,000 × (1 + 20 ÷ 100)²

⇒ A = 3,000 × (120 ÷ 100) × (120 ÷ 100)

⇒ A = 3,000 × 1.2 × 1.2

⇒ A = 3,000 × 1.44

⇒ A = ₹4,320

⇒ CI = 4,320 − 3,000

⇒ CI = ₹1,320

∴ The correct answer is ₹1,320.

Additional Information

CI and SI Difference (for 2 years)

The difference between Compound Interest and Simple Interest for 2 years is given by: D = P × (R ÷ 100)².

Effective Percentage Rate

For successive growth of x% and y%, the total growth is x + y + (xy ÷ 100)%. For 2 years at rate R, it is 2R + (R² ÷ 100)%.

Compounding Frequencies

If interest is compounded half-yearly, the rate becomes R ÷ 2 and the time period becomes 2n.

Shortcut Trick

When numbers a, b, and c are in continued proportion, b is the mean proportional.

Logic: b = √(a × c)

Applying the trick: x = √(2 × 18) = √36 = 6

∴ The correct answer is 6.

Alternate Method

Given:

The numbers 2, x, and 18 are in continued proportion.

Formula Used:

Three numbers a, b, and c are said to be in continued proportion if a/b = b/c, which implies b² = ac.

Calculations:

⇒ 2 ÷ x = x ÷ 18

⇒ x² = 2 × 18

⇒ x² = 36

⇒ x = √36

⇒ x = 6 (as we need the positive value)

∴ The correct answer is 6.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

If a, b, c are in continued proportion, then c is called the third proportional to a and b, where c = b²/a.

Fourth Proportional

If a:b = c:d, then d is called the fourth proportional to a, b, and c, calculated as d = (b × c) / a.

Shortcut Trick

Total volume of spherical tank = (4/3) × π × r³.

Given r = 2 m, Total Volume = (4/3) × 3.14159 × 8 ≈ 33.51 m³.

Water volume = 80% of 33.51 = 0.8 × 33.51 ≈ 26.81 m³.

Rounding to the nearest cubic meter, we get 27 m³.

∴ The correct answer is 27 m³.

Alternate Method

Given:

Radius of the sphere (r) = 2 m

Filled percentage = 80%

Formula Used:

Volume of a sphere = (4/3) × π × r³

Calculations:

⇒ Total volume of the tank = (4/3) × (22/7) × (2)³

⇒ Total volume = (4/3) × (22/7) × 8 = 704 / 21 m³

⇒ Total volume ≈ 33.5238 m³

⇒ Volume of water = 80% of 33.5238

⇒ Volume of water = (80 / 100) × 33.5238

⇒ Volume of water = 0.8 × 33.5238 = 26.819 m³

⇒ Rounded to nearest cubic meter = 27 m³

∴ The correct answer is 27 m³.

Additional Information

Surface Area of a Sphere

The total surface area is given by the formula A = 4 × π × r².

Volume of a Hemisphere

A hemisphere is half of a sphere; its volume is V = (2/3) × π × r³.

Volume to Liters Conversion

In practical applications, 1 cubic meter (m³) of volume is equivalent to 1,000 liters of water.

Shortcut Trick

Ratio of goods: 50% (Part 1) : 20% (40% of 50%, Part 2) : 30% (Rest, Part 3).

Simplified ratio = 5 : 2 : 3.

Apply weighted average for profit/loss: x = [(5 × 12) + (2 × 16) + (3 × −19)] ÷ (5 + 2 + 3).

x = [60 + 32 − 57] ÷ 10 = 35 ÷ 10 = 3.5.

∴ The correct answer is 3.5.

Alternate Method

Given:

Total goods = 100 units (Assumed CP = ₹100)

Part 1: 50% of 100 = 50 units at 12% profit

Remaining: 100 − 50 = 50 units

Part 2: 40% of remaining = 0.40 × 50 = 20 units at 16% profit

Part 3: Still remaining = 50 − 20 = 30 units at 19% loss

Calculations:

⇒ Profit from Part 1 = 50 × 12 ÷ 100 = 6

⇒ Profit from Part 2 = 20 × 16 ÷ 100 = 3.2

⇒ Loss from Part 3 = 30 × 19 ÷ 100 = 5.7

⇒ Total Net Profit = 6 + 3.2 − 5.7 = 3.5

⇒ Overall Profit Percentage (x) = (Total Profit ÷ Total CP) × 100

⇒ x = (3.5 ÷ 100) × 100 = 3.5

∴ The correct answer is 3.5.

Additional Information

Weighted Average Profit/Loss

When goods are divided into parts, the net profit/loss is the sum of (weight × percentage) for each part divided by the total weight. Loss is always substituted as a negative value.

Successive Percentage Change

If a value is changed by a% and then by b%, the effective change is (a + b + ab ÷ 100)%. This is different from weighted averages used for parts of a whole.

Base Value Selection

Assuming a base value of 100 for calculations involving percentages simplifies arithmetic and helps avoid complex decimal fractions during competitive exams.

Shortcut Trick

46% of Maximum Marks = Girish score + 81 = 448 + 81 = 529

1% of Maximum Marks = 529 ÷ 46 = 11.5

Friend's Percentage = Friend's Score ÷ Value of 1%

Friend's Percentage = 805 ÷ 11.5 = 70%

∴ The correct answer is 70%.

Alternate Method

Given:

Girish's marks = 448

Difference to 46% = 81 marks

Friend's marks = 805

Formula Used:

Percentage of marks = (Obtained Marks ÷ Maximum Marks) × 100

Calculations:

⇒ Let Maximum Marks be 'M'.

⇒ 46% of M − 81 = 448

⇒ 0.46 × M = 448 + 81

⇒ 0.46 × M = 529

⇒ M = 529 ÷ 0.46 = 1150

⇒ Friend's Percentage = (805 ÷ 1150) × 100

⇒ Friend's Percentage = 80500 ÷ 1150 = 8050 ÷ 115

⇒ Friend's Percentage = 70%

∴ The correct answer is 70%.

Additional Information

Percentage Change

Percentage Increase/Decrease = (Difference ÷ Original Value) × 100.

Fraction to Percentage

To convert a fraction to a percentage, multiply by 100. Common values: 1/4 = 25%, 1/5 = 20%, 3/4 = 75%.

Base Value Concept

In problems involving percentages, the Maximum Marks or Total Value always acts as the base (100%).

Shortcut Trick

Total amount to be divided after deductions = 2288 − (39 + 47 + 40) = 2162.

Sum of ratio parts = 17 + 15 + 14 = 46 units.

Value of 46 units = 2162 ⇒ 1 unit = 2162 ÷ 46 = 47.

Meera’s share after deduction = 15 units = 15 × 47 = 705.

∴ The correct answer is ₹705.

Alternate Method

Given: Total sum = ₹2,288. Deductions for Pankaj, Meera, and Ashok are ₹39, ₹47, and ₹40 respectively. The ratio of their shares after deduction is 17 : 15 : 14.

Formula: Individual Share = (Individual Ratio ÷ Sum of Ratios) × Remaining Total

⇒ Total deduction = 39 + 47 + 40 = ₹126

⇒ Remaining sum = 2288 − 126 = ₹2,162

⇒ Ratio sum = 17 + 15 + 14 = 46

⇒ Let the shares after deduction be 17x, 15x, and 14x

⇒ 46x = 2162

⇒ x = 2162 ÷ 46 = 47

⇒ Meera’s share after deduction = 15x

⇒ Meera’s share after deduction = 15 × 47 = 705

∴ The correct answer is ₹705.

Additional Information

Ratio Distribution Rule

To divide a quantity A in ratio a:b:c, the shares are [a/(a+b+c)]×A, [b/(a+b+c)]×A, and [c/(a+b+c)]×A.

Original Share Calculation

The original share of a person is found by adding the deducted amount back to their share after deduction (Original = Share after deduction + Deducted amount).

Effective Ratio

When fixed amounts are added or subtracted from shares, the final ratio changes; calculations must always be performed on the adjusted total amount.

Shortcut Trick

Using the net deviation method relative to the average of 15:

Total deviation must be zero for the 4 numbers.

Deviation of known numbers: (6.5 − 15) + (10.8 − 15) + (20 − 15) = −8.5 − 4.2 + 5 = −7.7

The fourth number a must compensate this: a = 15 + 7.7 = 22.7

∴ The correct answer is 22.7.

Alternate Method

Given: Observations = 6.5, 10.8, 20, a | Total observations = 4 | Average = 15

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum of observations = Average × Total number of observations

⇒ 6.5 + 10.8 + 20 + a = 15 × 4

⇒ 37.3 + a = 60

⇒ a = 60 − 37.3

⇒ a = 22.7

∴ The correct answer is 22.7.

Additional Information

Arithmetic Mean

The average or Arithmetic Mean is the central value of a set of numbers, calculated by dividing the total sum by the count of items.

Net Deviation Principle

The sum of the differences between each number in a set and the mean of the set is always zero.

Sum Formula

To find an unknown value in a sequence when the average is known: Sum = Average × Total Number of items.

Given:

Expression: ((12/24) / (12/6)) ÷ (2/9 × 18/6 + 4/8) + 6/9 ÷ 28/11 of 11/9

Formula used:

BODMAS Rule: Brackets, Of, Division, Multiplication, Addition, Subtraction

Calculations:

Applying the BODMAS rule to find the simplified value:

⇒ ((1/2) / 2) ÷ (2/9 × 3 + 1/2) + 6/9 ÷ (28/11 × 11/9)

⇒ 1/4 ÷ (2/3 + 1/2) + 2/3 ÷ 28/9

⇒ 1/4 ÷ 7/6 + (2/3 × 9/28)

⇒ (1/4 × 6/7) + (1/1 × 3/14)

⇒ 6/28 + 3/14

⇒ 3/14 + 3/14

⇒ 6/14

⇒ 3/7

∴ The correct answer is option (4).

Shortcut Trick

Present ratio of Father and Son = 3 : 1 (Difference = 2 units)

Ratio after 15 years = 2 : 1 (Difference = 1 unit)

Make the difference equal by multiplying future ratio by 2 ⇒ 4 : 2

Increase in units (4 − 3 or 2 − 1) = 1 unit → 15 years

Present age of Father = 3 units × 15 = 45 years

∴ The correct answer is 45 years.

Alternate Method

Given:

Let the present age of the Son be x years.

Present age of the Father = 3x years.

After 15 years, their ages will be (x + 15) and (3x + 15) respectively.

Calculations:

⇒ 3x + 15 = 2(x + 15)

⇒ 3x + 15 = 2x + 30

⇒ 3x − 2x = 30 − 15

⇒ x = 15

⇒ Father present age = 3x = 3 × 15 = 45 years

∴ The correct answer is 45 years.

Additional Information

Constant Age Gap

The mathematical difference between the ages of two individuals is a constant value and does not change over time.

Ratio Normalization

When solving age problems using ratios, if the time interval added is the same for both people, the difference between the ratio parts must be made equal.

Linear Equation Setup

For a person currently A years old, the age n years ago was (A − n) and the age n years later will be (A + n).

Shortcut Trick

Rate of interest = [(Difference in Amounts) ÷ (Amount after 2 years)] × 100

Rate (R) = [(8,651.50 − 7,865) ÷ 7,865] × 100 = (786.5 ÷ 7,865) × 100 = 10%

For R = 10%, Multiplying Factor for 2 years = (1.1)² = 1.21

Sum (P) = 7,865 ÷ 1.21 = ₹6,500

∴ The correct answer is ₹6,500.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹7,865

Amount after 3 years (A₃) = ₹8,651.50

Time gap = 1 year

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

The interest for the 3^rd year is the compound interest earned on the amount at the end of the 2^nd year.

⇒ Interest = A₃ − A₂

⇒ Interest = 8,651.50 − 7,865 = ₹786.50

⇒ Rate (R) = (786.50 ÷ 7,865) × 100 = 10%

Now, using the formula for 2 years:

⇒ 7,865 = P(1 + 10/100)²

⇒ 7,865 = P(1.1)²

⇒ 7,865 = P × 1.21

⇒ P = 7,865 ÷ 1.21

⇒ P = 6,500

∴ The correct answer is ₹6,500.

Additional Information

Successive Amount Property

In compound interest, the amount after n years becomes the Principal for the (n+1)^th year.

Constant Ratio in CI

The ratio of amounts for equal time intervals is constant: A₂/A₁ = A₃/A₂ = (1 + R/100).

SI vs CI for 2 Years

The difference between Simple Interest and Compound Interest for 2 years is given by D = P(R/100)².