Search Previous Year Questions

Shortcut Trick

Apply the Successive Percentage Change formula: Net Profit% = a + b + (ab ÷ 100)

Where a (Markup) = +49 and b (Discount) = −18

Profit% (x) = 49 − 18 − (49 × 18 ÷ 100)

x = 31 − 8.82 = 22.18%

∴ The correct answer is 22.18%.

Alternate Method

Given: Markup Percentage = 49% and Discount Percentage = 18%

Formula used: Selling Price (SP) = Cost Price (CP) × (100 + Markup%)/100 × (100 − Discount%)/100

Let the Cost Price (CP) of the goods be 100 units.

⇒ Marked Price (MP) = 100 + 49% of 100 = 149 units

⇒ Discount Amount = 18% of MP = (18 ÷ 100) × 149

⇒ Discount = 0.18 × 149 = 26.82 units

⇒ Selling Price (SP) = MP − Discount = 149 − 26.82 = 122.18 units

⇒ Profit (x) = SP − CP = 122.18 − 100 = 22.18 units

⇒ Profit Percentage (x%) = (22.18 ÷ 100) × 100 = 22.18%

∴ The correct answer is 22.18%.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is given by: CP / MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

When two successive discounts d₁% and d₂% are given, the Equivalent Single Discount = [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%.

Markup Percentage

The percentage by which MP is higher than CP: Markup% = [(MP − CP) ÷ CP] × 100.

Shortcut Trick

Given Price₁ = ₹80, Price₂ = ₹95, and Ratio = 3 : 4.

Average price = (Price₁ × Ratio₁ + Price₂ × Ratio₂) ÷ (Ratio₁ + Ratio₂)

Average price = (80 × 3 + 95 × 4) ÷ 7 = (240 + 380) ÷ 7 = 620 ÷ 7

Average price = ₹88.5714... ≈ ₹88.57

∴ The correct answer is ₹88.57.

Alternate Method

Given:

Cost price of Variety A = ₹80 per kg

Cost price of Variety B = ₹95 per kg

Mixing Ratio = 3 : 4

Let the price of the resulting mixture be ₹x per kg.

By the Rule of Alligation:

⇒ (Price of dearer − Mean price) ÷ (Mean price − Price of cheaper) = Ratio₁ ÷ Ratio₂

⇒ (95 − x) ÷ (x − 80) = 3 ÷ 4

⇒ 4 × (95 − x) = 3 × (x − 80)

⇒ 380 − 4x = 3x − 240

⇒ 380 + 240 = 3x + 4x

⇒ 620 = 7x

⇒ x = 620 ÷ 7

⇒ x ≈ 88.5714

Rounding off to two decimal places, we get ₹88.57.

∴ The correct answer is ₹88.57.

Additional Information

Rule of Alligation

It is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a specified price. Ratio = (Price of dearer − Mean) : (Mean − Price of cheaper).

Weighted Average Formula

The mean value of a mixture is the weighted average of its components: Mean Value = (n₁ × C₁ + n₂ × C₂) ÷ (n₁ + n₂).

Cost Price of Mixture

When a mixture is sold at a certain profit percentage, always calculate the Cost Price (CP) first: CP = (Selling Price × 100) ÷ (100 + Profit%).

Shortcut Trick

For 1¹/₂ years, use interest rate of 16% for the 1^st year and 8% (half of 16%) for the remaining half year.

Effective interest rate = 16 + 8 + (16 × 8) ÷ 100 = 24 + 1.28 = 25.28%

Total Amount = 18000 × (100 + 25.28)% = 18000 × 1.2528

Total Amount = ₹22,550.40

∴ The correct answer is ₹22,550.40.

Alternate Method

Given:

Principal (P) = ₹18,000

Rate of Interest (R) = 16% per annum

Time (n) = 1¹/₂ years

Formula Used:

For a fractional time like T = n^a/_b years, Amount (A) = P × (1 + R/100)ⁿ × (1 + (R × a/b)/100)

Calculations:

⇒ Amount = 18000 × (1 + 16/100)¹ × (1 + (16 × 1/2)/100)

⇒ Amount = 18000 × (1 + 0.16) × (1 + 8/100)

⇒ Amount = 18000 × 1.16 × 1.08

⇒ Amount = 20880 × 1.08

⇒ Amount = ₹22,550.40

∴ The correct answer is ₹22,550.40.

Additional Information

Compound Interest for Fractional Time

When time is in fractions like t = 2³/₄ years, Amount = P(1 + R/100)² × [1 + (R × 3/4)/100].

Compounded Half-Yearly

If interest is compounded half-yearly, the rate becomes R/2 and the time becomes 2n. Formula: A = P(1 + R/200)²ⁿ.

Difference between CI and SI

For 2 years, the difference (D) between CI and SI is given by D = P(R/100)².

Shortcut Trick

Ratio of payment of X : Y = 200% : 100% = 2 : 1

Total units = 2 + 1 = 3 units

3 units → ₹440

1 unit (Payment to Y) = 440 ÷ 3 = ₹146.67

∴ The correct answer is ₹146.67.

Alternate Method

Given:

Total payment of X and Y = ₹440 per day.

Payment of X = 200% of payment of Y.

Calculations:

⇒ Let the daily payment to Y be ₹a.

⇒ Payment to X = 200% of a = 2a.

⇒ Total payment = X + Y = 440

⇒ 2a + a = 440

⇒ 3a = 440

⇒ a = 440 ÷ 3

⇒ a = 146.666...

⇒ a ≈ 146.67

∴ The correct answer is ₹146.67.

Additional Information

Percentage and Ratios

A statement like 'A is 200% of B' means the ratio A:B is 200:100 or 2:1. If 'A is 200% more than B', then A = B + 200% of B = 3B (Ratio 3:1).

Distribution of Amount

To distribute a total amount T in the ratio m:n, the shares are calculated as T × [m/(m+n)] and T × [n/(m+n)].

Rounding Rules

When dividing by 3, the decimal part is usually .333... or .666... . For monetary values, these are rounded to two decimal places as .33 or .67.

Shortcut Trick

Total numbers from 1 to 60 = 60.

In every set of 10 numbers (e.g., 1-10, 11-20), there are 2 numbers with unit digit 1 or 8.

There are 6 such sets (60 ÷ 10 = 6).

Total favorable numbers = 2 × 6 = 12.

Required percentage = (12 ÷ 60) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Range of numbers = 1 to 60

Unit digits required = 1 or 8

Formula Used:

Percentage = (Number of favorable outcomes ÷ Total number of outcomes) × 100

List of numbers with unit digit 1: 1, 11, 21, 31, 41, 51 (Total 6)

List of numbers with unit digit 8: 8, 18, 28, 38, 48, 58 (Total 6)

⇒ Total favorable numbers = 6 + 6 = 12

⇒ Total numbers = 60

⇒ Required percentage = (12 ÷ 60) × 100

⇒ Required percentage = (1 ÷ 5) × 100

⇒ Required percentage = 20%

∴ The correct answer is 20%.

Additional Information

Unit Digit Frequency

In any consecutive range of 10 integers, every digit from 0 to 9 appears exactly once in the unit's place.

Basic Percentage Formula

Percentage = (Value ÷ Total Value) × 100. It represents a fraction with a denominator of 100.

Probability Connection

The probability of picking such a number is 12/60 = 1/5. Converting probability to percentage: 1/5 × 100 = 20%.

Shortcut Trick

Solve the expression using the BODMAS priority rule.

Solve the Bracket first: (3 − 1) = 2.

Calculate Division and Multiplication: 12 ÷ 2 = 6 and 5 × 2 = 10.

Perform Addition: 10 + 6 = 16.

∴ The correct answer is 16.

Alternate Method

Given Expression: 5 × 2 + 12 ÷ (3 − 1)

To simplify, follow the VBODMAS rule (Vinculum, Bracket, Of, Division, Multiplication, Addition, Subtraction):

⇒ 5 × 2 + 12 ÷ 2 (Solving the parentheses)

⇒ 5 × 2 + 6 (Performing the division)

⇒ 10 + 6 (Performing the multiplication)

⇒ 16 (Final addition)

∴ The correct answer is 16.

Additional Information

BODMAS Rule

The acronym BODMAS stands for Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction, defining the correct sequence for operations.

Priority of Operations

Operations of Division and Multiplication hold equal priority and should be calculated from left to right as they appear in the expression.

Types of Brackets

Always solve in the following order: Round Brackets (), then Curly Brackets {}, and finally Square Brackets [].

Shortcut Trick

Apply the successive percentage change formula: Net change = a + b + (a × b)/100.

Here, increase (a) = +20% and decrease (b) = −20%.

Net Change = 20 − 20 + [20 × (−20)] / 100 = 0 − 400 / 100 = −4%.

A negative sign indicates a percentage decrease.

∴ The correct answer is 4% decrease.

Alternate Method

Given:

Initial length of rectangle = x cm

Initial breadth of rectangle = y cm

Formula Used:

Area of Rectangle = Length × Breadth

Percentage Change = [(New Value − Original Value) / Original Value] × 100

Calculations:

⇒ Original Area (A₁) = x × y = xy cm²

⇒ New Length (x') = x + 20% of x = 1.2x

⇒ New Breadth (y') = y − 20% of y = 0.8y

⇒ New Area (A₂) = 1.2x × 0.8y = 0.96xy cm²

⇒ Change in Area = A₂ − A₁ = 0.96xy − xy = −0.04xy

⇒ Percentage Change = (−0.04xy / xy) × 100

⇒ Percentage Change = −4%

∴ The correct answer is 4% decrease.

Additional Information

Constant Percentage Rule

When one dimension increases by r% and another decreases by r%, the net area always decreases by (r/10)² or (r²/100)%.

Successive Percentage Formula

For two successive changes x% and y%, the effective change is [x + y + (xy/100)]%. Use positive for increase and negative for decrease.

Area and Perimeter Relation

For a fixed perimeter, a square has the maximum area compared to any other rectangle.

Shortcut Trick

Use the Effective Percentage Formula for Profit: Profit% = m − d − (m × d)/100

Where m = Mark-up percentage = 50% and d = Discount percentage = 12%.

x = 50 − 12 − (50 × 12)/100

x = 38 − 6 = 32

∴ The correct answer is 32.

Alternate Method

Given:

Mark-up Percentage = 50%

Discount Percentage = 12%

Formula Used:

1. Marked Price (MP) = CP × (100 + Mark-up%)/100
2. Selling Price (SP) = MP × (100 − Discount%)/100
3. Profit% (x) = [(SP − CP) / CP] × 100

Calculations:

⇒ Let the Cost Price (CP) of the goods be 100 units.

⇒ Marked Price (MP) = 100 + 50% of 100 = 100 + 50 = 150 units.

⇒ Selling Price (SP) = MP − 12% of MP

⇒ SP = 150 − (12/100 × 150) = 150 − 18 = 132 units.

⇒ Gain = SP − CP = 132 − 100 = 32 units.

⇒ Gain Percentage (x) = (32 / 100) × 100 = 32%.

∴ The value of x is 32.

∴ The correct answer is 32.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is given by the formula: CP / MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

If two successive discounts d₁ and d₂ are given, the single equivalent discount is [d₁ + d₂ − (d₁ × d₂)/100]%.

Dishonest Dealer Concept

Profit% = [Error / (True Value − Error)] × 100, when a dealer uses false weights while selling goods.

Shortcut Trick

Apply the Chain Rule formula: M₁D₁ = M₂D₂

Here, M₁ = 36, D₁ = 17, and D₂ = 9

36 × 17 = M₂ × 9

M₂ = (36 × 17) ÷ 9 = 4 × 17 = 68

∴ The correct answer is 68 workers.

Alternate Method

Given:

Initial workers (M₁) = 36

Initial days (D₁) = 17

Target days (D₂) = 9

Formula Used:

Total Work = Number of Workers × Number of Days

Calculations:

⇒ Total Work = 36 × 17 man-days

⇒ Let the required number of workers be 'x'

⇒ Total Work = x × 9 man-days

⇒ 36 × 17 = x × 9

⇒ x = (36 × 17) ÷ 9

⇒ x = 4 × 17

⇒ x = 68

∴ The correct answer is 68 workers.

Additional Information

Inverse Proportion in Work

The number of workers is inversely proportional to the time taken to complete a constant piece of work. If workers increase, days decrease.

General Chain Rule Formula

For different work amounts, use: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where H is hours and W is work units.

Man-Day Concept

One man-day represents the amount of work done by one person in one day. Total work is usually expressed in these units for Time and Work problems.

Shortcut Trick Product of Divisor and Quotient = Dividend − Remainder Product = 2369 − 17 = 2352 Since Divisor = 1/3 × Quotient, then Quotient = 3 × Divisor. 3 × (Divisor)2 = 2352 ⇒ (Divisor)2 = 784 ⇒ Divisor = 28 ∴ The correct answer is 28.

Alternate Method Given: Dividend = 2369 Remainder = 17 Divisor = 1/3 × Quotient Formula Used: Dividend = (Divisor × Quotient) + Remainder Calculations: Let the Divisor be x. ⇒ Quotient = 3x Substitute the values in the Division Algorithm formula: ⇒ 2369 = (x × 3x) + 17 ⇒ 2369 − 17 = 3x2 ⇒ 2352 = 3x2 ⇒ x2 = 2352 ÷ 3 ⇒ x2 = 784 ⇒ x = √784 ⇒ x = 28 ∴ The correct answer is 28.

Additional Information Division Algorithm For any two integers a and b (b ≠ 0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < |b|. Remainder Property The remainder must always be non-negative and strictly smaller than the divisor. Square Root Calculation To find √784 quickly: 202 = 400 and 302 = 900. Since 784 ends in 4, the root ends in 2 or 8. Only 28 fits the range.