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Shortcut Trick

Rate for 8 months = (30% ÷ 12) × 8 = 20%.

Total periods in 2 years (24 months) = 24 ÷ 8 = 3 periods.

Using Ratio Method (20% = 1/5): Principal : Amount = 5³ : 6³ = 125 : 216.

125 units = Rs. 24,000 ⇒ 1 unit = 192.

Amount (216 units) = 216 × 192 = Rs. 41472.

∴ The correct answer is Rs. 41472.

Alternate Method

Given: Principal (P) = Rs. 24,000, Annual Rate (R) = 30%, Time (T) = 2 years.

Formula Used: Amount (A) = P(1 + r/100)ⁿ

⇒ Effective Rate for 8 months = (30 × 8) ÷ 12 = 20%

⇒ Number of periods (n) = (2 × 12) ÷ 8 = 3

⇒ A = 24000 × (1 + 20/100)³

⇒ A = 24000 × (1.2)³

⇒ A = 24000 × 1.728

⇒ A = 24 × 1728 = 41472

∴ The correct answer is Rs. 41472.

Additional Information

Effective Rate for Compounding

When interest is compounded quarterly (4 times a year), the rate becomes R/4 and time becomes 4T. For 8-month compounding, the factor is 8/12 of the annual rate.

Difference between SI and CI

The difference between CI and SI for 2 years is P(R/100)² and for 3 years is P(R/100)² × (3 + R/100).

Successive Percentage Formula

For two successive cycles of r%, the effective rate is [r + r + (r×r)/100]%. For three cycles of 20%, it is 72.8%.

Shortcut Trick

Simplify ratios first: A ∶ B = 3 ∶ 6 = 1 ∶ 2 and B ∶ C = 7 ∶ 28 = 1 ∶ 4.

Make B common: Multiply (1 ∶ 2) by 1 and (1 ∶ 4) by 2.

A ∶ B = 1 ∶ 2 and B ∶ C = 2 ∶ 8.

Combined ratio A ∶ B ∶ C = 1 ∶ 2 ∶ 8.

∴ The correct answer is 1 ∶ 2 ∶ 8.

Alternate Method

Given:

Ratio A ∶ B = 3 ∶ 6

Ratio B ∶ C = 7 ∶ 28

Calculations:

⇒ A/B = 3/6 = 1/2

⇒ B/C = 7/28 = 1/4

⇒ To combine, we need the value of B to be the same in both ratios.

⇒ LCM of B values (2 and 1) is 2.

⇒ A ∶ B = 1 ∶ 2

⇒ B ∶ C = (1 × 2) ∶ (4 × 2) = 2 ∶ 8

⇒ Therefore, A ∶ B ∶ C = 1 ∶ 2 ∶ 8

∴ The correct answer is 1 ∶ 2 ∶ 8.

Additional Information

Compounding of Ratios

To combine two ratios (a ∶ b) and (b ∶ c), multiply the terms to make the common variable 'b' equal in both sets.

Simplification Rule

A ratio remains unchanged if both its terms are multiplied or divided by the same non-zero number.

Compound Ratio

The compound ratio of (a ∶ b) and (c ∶ d) is (ac ∶ bd).

Shortcut Trick

Net percentage = (Quantity₁ × Percentage₁ + Quantity₂ × Percentage₂) ÷ Total Quantity

Net water % = (2 × 12 + 3 × 5) ÷ (2 + 3)

Net water % = (24 + 15) ÷ 5 = 39 ÷ 5 = 7.8%

7.8% can be written as 7(4/5)%.

∴ The correct answer is 7(4/5) %.

Alternate Method

Given:

Quantity 1 = 2 litres, Water % = 12%

Quantity 2 = 3 litres, Water % = 5%

⇒ Water in first mixture = 12% of 2 = 0.12 × 2 = 0.24 litres

⇒ Water in second mixture = 5% of 3 = 0.05 × 3 = 0.15 litres

⇒ Total water in resulting mixture = 0.24 + 0.15 = 0.39 litres

⇒ Total quantity of resulting mixture = 2 + 3 = 5 litres

⇒ Resulting percentage = (0.39 ÷ 5) × 100

⇒ Resulting percentage = 39 ÷ 5 = 7.8%

⇒ 7.8 = 7 + 0.8 = 7 + 8/10 = 7(4/5)

∴ The correct answer is 7(4/5) %.

Additional Information

Alligation Rule

It relates concentrations and quantities: (Cheaper quantity) ÷ (Dearer quantity) = (Dearer rate − Mean rate) ÷ (Mean rate − Cheaper rate).

Weighted Average

Used when different groups have different weights: Average = (n₁w₁ + n₂w₂) ÷ (n₁ + n₂).

Replacement Method

Final Concentration = Initial Concentration × [1 − (Removed/Total)]ⁿ, where n is the number of operations.

Shortcut Trick

When a father says he was as old as the son is now at the time of the son's birth, the father's current age is always double the son's current age.

Son's current age = Father's age ÷ 2

Son's current age = 40 ÷ 2 = 20 years

Age 5 years ago = 20 − 5 = 15 years

∴ The correct answer is 15 years.

Alternate Method

Given:

Mallesh's current age = 40 years

Let the son's current age be x years.

According to the question, Mallesh's age at the time of his son's birth was equal to the son's current age (x).

⇒ Father's current age − Son's current age = Father's age at son's birth

⇒ 40 − x = x

⇒ 2x = 40

⇒ x = 20 years

⇒ Son's age 5 years ago = x − 5

⇒ 20 − 5 = 15 years

∴ The correct answer is 15 years.

Additional Information

Age Difference Property

The difference between the ages of two people remains constant regardless of the number of years that pass.

Ratio of Ages

If the ratio of ages is given at two different times, use the constant difference to balance the ratio units.

General Age Equation

If current age is A, then age 'n' years ago is (A − n) and age 'n' years hence is (A + n).

$\begin{array}{l} \frac{{\left( {0.0112 - 0.0012} \right);of;0.14; + ;0.25; \times ;0.2}}{{0.02; \times ;0.01}}\ = ;\frac{{\left( {0.01} \right)of;0.14; + ;0.05}}{{0.0002}}\ = ;\frac{{0.01; \times ;0.14; + ;0.05}}{{0.0002}}\ ; = ;\frac{{0.0014; + ;0.05}}{{0.0002}}\ = ;\frac{{0.0514}}{{0.0002}}\ = ;257 \end{array}$

Shortcut Trick

⇒ CP = 100

⇒ MP = 100 + 180% = 280

⇒ Required SP for 12% profit = 112

⇒ Discount% = (280 − 112) ÷ 280 × 100

⇒ = 168 ÷ 280 × 100 = 60%

∴ Answer = 60%

Alternate Method

Given:

⇒ Marked price is 180% above CP

⇒ Profit = 12%

Assume:

⇒ CP = 100

Calculation:

⇒ MP = 100 + 180 = 280

⇒ SP for 12% profit = 100 + 12 = 112

⇒ Discount = MP − SP = 280 − 112 = 168

⇒ Discount% = (168 ÷ 280) × 100

⇒ = 60%

∴ Final Answer: 60%

Additional Information

1. Marked Price: MP = CP + Markup%

2. Profit Formula: SP = CP × (1 + Profit%)

3. Discount Formula: Discount% = (MP − SP) ÷ MP × 100

4. Shortcut: Always assume CP = 100 for quick calculation

Shortcut Trick

Break the expression into two parts: A = 34% of 50 and B = (4/6) × ³√512 × √36.

Step 1: A = 34% of 50 = 0.34 × 50 = 17.

Step 2: B = (4/6) × 8 × 6 = 4 × 8 = 32.

Step 3: y = A − B = 17 − 32 = −15.

∴ Answer = −15

Alternate Method 

Given:

Expression: y = 34% of 50 − (4/6) × ³√512 × √36

Formula used:

VBODMAS Rule: Vinculum, Bracket, Of, Division, Multiplication, Addition, Subtraction.

34% of 50 17 (4/6) × 8 × 6 32 − −15 Result (y)

Calculations:

Step 1: Simplify 34% of 50

⇒ (34 ÷ 100) × 50 = 17

Step 2: Simplify the radical terms ³√512 and √36

⇒ ³√512 = 8 (since 8³ = 512)

⇒ √36 = 6 (since 6² = 36)

Step 3: Solve the multiplication part

⇒ (4 ÷ 6) × 8 × 6 = 4 × 8 = 32

Step 4: Perform the subtraction

⇒ y = 17 − 32

⇒ y = −15

∴ The correct answer is (3) -15.

Additional Information

VBODMAS Rule

It defines the order of operations: Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction. Here, 'Of' is treated as multiplication but solved before regular division/multiplication.

Percentage Interchange

x% of y is equal to y% of x. For example, 34% of 50 is the same as 50% of 34, which is 17.

Cube Roots and Square Roots

Memorizing basic roots helps speed up calculations: √36 = 6, √64 = 8, ³√216 = 6, ³√512 = 8, and ³√729 = 9.

Shortcut Trick

2/5 of Actual Selling Price (ASP) = 100% − 24% = 76% of CP.

⇒ Actual SP = 76% × 5 ÷ 2 = 190% of CP.

New Selling Price = 84% of Actual SP = 84% × 190% of CP.

⇒ New SP = 0.84 × 190% = 159.6% of CP.

Profit % = 159.6% − 100% = 59.6%.

∴ Answer = 59.6%

Alternate Method 

Given:

Loss = 24% when SP = 2/5 of Actual SP. Target SP = 84% of Actual SP.

Formula used:

Selling Price (SP) = CP × (100 − Loss%)/100 | Profit % = [(SP − CP) ÷ CP] × 100

CP = 100 New SP = 159.6 Actual SP = 190 Profit: 59.6% 2/5 ASP = 76

Calculations:

Let Cost Price (CP) = 100 units.

⇒ If Loss = 24%, then Selling Price (SP₁) = 100 − 24 = 76 units.

⇒ Given SP₁ = 2/5 × Actual SP.

⇒ 76 = 2/5 × Actual SP ⇒ Actual SP = (76 × 5) ÷ 2 = 190 units.

⇒ New Selling Price (SP₂) = 84% of Actual SP = 0.84 × 190.

⇒ New SP₂ = 159.6 units.

⇒ Profit = SP₂ − CP = 159.6 − 100 = 59.6 units.

⇒ Profit % = (59.6 ÷ 100) × 100 = 59.6%.

∴ The correct answer is option (1).

Additional Information

Successive Percentage

When a value is changed by a% and then by b%, net change = [a + b + (a × b)/100]%.

Marked Price and Discount

Selling Price = Marked Price × (100 − Discount%)/100. Also, CP × (100 + P%) = MP × (100 − D%).

Profit and Loss Basics

Profit % = (Profit/CP) × 100 and Loss % = (Loss/CP) × 100. Profit/Loss is always calculated on Cost Price unless stated otherwise.

Shortcut Trick

⇒ Net rate = 1/4 + 1/24 − 1/6

⇒ LCM = 24 ⇒ (6 + 1 − 4)/24 = 3/24 = 1/8

⇒ So tank fills in 8 hours

⇒ Half tank ⇒ 8/2 = 4 hours

∴ Answer = 4 hours

Alternate Method

Given:

Pipe A fills in 4 hr

Pipe B fills in 24 hr

Pipe C empties in 6 hr

Formula Used:

Net work rate = Sum of filling rates − emptying rate

Calculation:

⇒ Net rate = 1/4 + 1/24 − 1/6

⇒ = (6 + 1 − 4)/24 = 3/24 = 1/8

⇒ Time for full tank = 8 hr

⇒ Time for half tank = 4 hr

∴ The correct answer is 4 hours.

Additional Information

Net Rate: Add filling, subtract emptying

LCM Method: Use common denominator for speed

Half Work: Time halves if rate same

Exam Tip: Always find net rate first

Shortcut Trick

⇒ Final = 5P ⇒ Interest = 4P

⇒ SI = P × r × t / 100 ⇒ 4P = P × r × 12 / 100

⇒ r = (4 × 100) / 12 = 33.33%

∴ Answer = 33.33%

Alternate Method

Given:

Amount becomes 5 times in 12 years

Formula Used:

SI = (P × r × t) / 100

Amount = P + SI

Calculation:

⇒ Let principal = P

⇒ Amount = 5P

⇒ SI = 5P − P = 4P

⇒ Using SI formula:

⇒ 4P = (P × r × 12) / 100

⇒ Cancel P:

⇒ 4 = (r × 12) / 100

⇒ r = (4 × 100) / 12

⇒ r = 33.33%

∴ The correct answer is 33.33%.

Additional Information

Simple Interest: SI = (P × r × t)/100

Amount Relation: A = P + SI

Shortcut: Multiply factor − 1 gives interest multiple

Exam Tip: Cancel P early