Quantitative Aptitude PYQs

Shortcut Trick

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding altitudes.

Ratio of areas = (Ratio of altitudes)²

⇒ Area Ratio = (2 ÷ 5)² = 4 : 25

∴ The correct answer is 4:25.

Alternate Method

Given:

Ratio of corresponding altitudes of two similar triangles (h₁ : h₂) = 2 : 5

Formula Used:

Area of Δ₁ ÷ Area of Δ₂ = (h₁ ÷ h₂)²

h1 = 2 Area = A1

h2 = 5 Area = A2

A1 : A2 = h1² : h2²

Calculations:

⇒ Area of Δ₁ ÷ Area of Δ₂ = (2 ÷ 5)²

⇒ Area of Δ₁ ÷ Area of Δ₂ = 4 ÷ 25

⇒ Ratio of their areas = 4 : 25

∴ The correct answer is 4:25.

Additional Information

Area of Similar Triangles

The ratio of the areas of two similar triangles is precisely equal to the square of the ratio of their corresponding sides, altitudes, or medians.

Perimeter of Similar Triangles

The ratio of the perimeters of two similar triangles is directly equal to the ratio of their corresponding sides, altitudes, or medians.

Inradius & Circumradius Ratio

For two similar triangles, the ratio of their corresponding inradii or circumradii follows the exact same ratio as their corresponding sides.

Shortcut Trick

Directly apply the algebraic identity: (x + y)² + (x − y)² = 2(x² + y²)

Here, replacing x with a and y with 1:

⇒ (a + 1)² + (a − 1)² = 2(a² + 1²)

Substitute a = √5 ⇒ a² = 5.

⇒ 2(5 + 1) = 2 × 6 = 12

∴ The correct answer is 12.

Alternate Method

Given:

a = √5

Formula Used:

(x + y)² = x² + 2xy + y²

(x − y)² = x² − 2xy + y²

Calculation:

⇒ (a + 1)² = a² + 2a + 1

⇒ (a − 1)² = a² − 2a + 1

Adding both expanded expressions together:

⇒ (a² + 2a + 1) + (a² − 2a + 1)

⇒ 2a² + 2

Substitute the given value a = √5:

⇒ 2(√5)² + 2

⇒ 2 × 5 + 2

⇒ 10 + 2 = 12

∴ The correct answer is 12.

Additional Information

Important Algebraic Identities

Another useful identity for subtraction is: (a + b)² − (a − b)² = 4ab.

Difference of Squares

The product of sum and difference is: a² − b² = (a − b)(a + b).

Square Root Properties

The square of a square root returns the original non-negative value: (√x)² = x. Also, roots can be multiplied directly: √x × √y = √(xy).

Shortcut Trick

Directly extract perfect squares from the square roots:

⇒ √50 + √18 − √8

⇒ √(25 × 2) + √(9 × 2) − √(4 × 2)

⇒ 5√2 + 3√2 − 2√2 = 6√2

∴ The correct answer is 6√2.

Alternate Method

Given:

Expression: √50 + √18 − √8

Method Used:

Prime factorization of the numbers inside the roots to extract terms outside the square root.

Calculations:

⇒ √50 = √(2 × 5 × 5) = 5√2

⇒ √18 = √(2 × 3 × 3) = 3√2

⇒ √8 = √(2 × 2 × 2) = 2√2

Substitute these values back into the expression:

⇒ 5√2 + 3√2 − 2√2

Take √2 as common:

⇒ (5 + 3 − 2) × √2

⇒ 6√2

∴ The correct answer is 6√2.

Additional Information

Similar Surds

Surds which have the same irrational factor are called similar surds. Only similar surds can be added or subtracted. Example: 3√2 and 5√2 are similar surds.

Multiplication and Division of Surds

Unlike addition and subtraction, surds with the same order can be multiplied or divided regardless of their base. √a × √b = √(ab) and √a ÷ √b = √(a/b).

Simplifying Surds

To simplify a square root, find the largest perfect square factor of the number inside the root, and bring its square root outside. Example: √(x²y) = x√y.

Shortcut Trick

The longest chord of a circle is its diameter.

Given the diameter of the circle is 25 cm.

Maximum length of the chord = Diameter = 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Diameter of the circle = 25 cm

Concept Used:

A chord is a line segment whose endpoints lie on the circumference of the circle.

As a chord moves closer to the center of the circle, its length continuously increases.

The chord that passes exactly through the center is the longest possible chord, which is known as the diameter.

Shorter Chord

Longest Chord (Diameter) = 25 cm Center O

Calculation:

⇒ Maximum length of chord = Length of diameter

⇒ Maximum length of chord = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Perpendicular from Center to Chord

A perpendicular drawn from the center of a circle to a chord perfectly bisects the chord into two equal parts.

Equal Chords and Subtended Angles

Equal chords of a circle subtend equal angles at the center of the circle.

Distance of Chords from Center

Equal chords of a circle are always equidistant from the center. Conversely, chords that are equidistant from the center are exactly equal in length.

Shortcut Trick

⇒ sin A + cos A = √2 sin A

⇒ cos A = sin A(√2 − 1) ⇒ cot A = √2 − 1

⇒ tan A = 1 ÷ cot A = 1 ÷ (√2 − 1)

⇒ Rationalizing the denominator gives tan A = √2 + 1

∴ The correct answer is √2 + 1.

Alternate Method

Given:

sin A + cos A = √2 sin A

Formula Used:

tan A = sin A ÷ cos A

(a − b)(a + b) = a² − b²

Calculations:

⇒ Divide the entire given equation by cos A:

⇒ (sin A ÷ cos A) + (cos A ÷ cos A) = √2(sin A ÷ cos A)

⇒ tan A + 1 = √2 tan A

⇒ 1 = √2 tan A − tan A

⇒ 1 = tan A(√2 − 1)

⇒ tan A = 1 ÷ (√2 − 1)

⇒ Multiply numerator and denominator by (√2 + 1):

⇒ tan A = (√2 + 1) ÷ [(√2 − 1)(√2 + 1)]

⇒ tan A = (√2 + 1) ÷ [(√2)² − (1)²]

⇒ tan A = (√2 + 1) ÷ (2 − 1)

⇒ tan A = √2 + 1

∴ The correct answer is √2 + 1.

Additional Information

Basic Trigonometric Ratios

tan A = sin A ÷ cos A and cot A = cos A ÷ sin A.

Pythagorean Identities

sin² A + cos² A = 1 | 1 + tan² A = sec² A | 1 + cot² A = cosec² A.

Algebraic Rationalization

To rationalize a denominator of the form (a − b), multiply numerator and denominator by its conjugate (a + b) using the identity (a − b)(a + b) = a² − b².

Shortcut Trick

To find the ratio of the first and last terms (A : D), simply multiply all the given ratios together.

A/D = (A/B) × (B/C) × (C/D)

A/D = (7/5) × (2/3) × (4/5)

A/D = (7 × 2 × 4) / (5 × 3 × 5)

A/D = 56 / 75

∴ The correct answer is 56 : 75.

Alternate Method

Given:

A : B = 7 : 5

B : C = 2 : 3

C : D = 4 : 5

Method Used:

We can use the shift method to find the combined ratio A : B : C : D by copying the adjacent numbers into the empty spaces of each row.

ABCD75552233444556406075

Calculations:

⇒ Multiply down the columns to get the final combined ratio components:

⇒ A = 7 × 2 × 4 = 56

⇒ B = 5 × 2 × 4 = 40

⇒ C = 5 × 3 × 4 = 60

⇒ D = 5 × 3 × 5 = 75

⇒ The combined ratio is A : B : C : D = 56 : 40 : 60 : 75

⇒ Therefore, the required ratio is A : D = 56 : 75

∴ The correct answer is 56 : 75.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is called a compound ratio. For a:b and c:d, the compound ratio is (a × c) : (b × d).

Inverse Ratio

For any ratio a : b, its inverse ratio is b : a. The inverse of a compound ratio a : b : c is evaluated as 1/a : 1/b : 1/c.

Duplicate & Triplicate Ratio

The duplicate ratio of a : b is given by a² : b². The triplicate ratio of a : b is given by a³ : b³.

Shortcut Trick

Convert division to multiplication by taking the reciprocal of the second fraction in each bracket.

⇒ (7/5 × 15/14) × (10/7 × 3/5)

⇒ (3/2) × (6/7)

⇒ 18/14 = 9/7

∴ The correct answer is 9/7.

Alternate Method

Given:

Expression: (7/5 ÷ 14/15) × (10/7 ÷ 5/3)

Formula Used:

BODMAS Rule: Solve Brackets first, then Division (by converting to multiplication with reciprocal), then Multiplication.

Fraction Division: a/b ÷ c/d = a/b × d/c

Calculation:

Solving the first bracket:

⇒ 7/5 ÷ 14/15 = 7/5 × 15/14

⇒ (7 × 15) / (5 × 14) = 105 / 70 = 3/2

Solving the second bracket:

⇒ 10/7 ÷ 5/3 = 10/7 × 3/5

⇒ (10 × 3) / (7 × 5) = 30 / 35 = 6/7

Multiplying the results of both brackets:

⇒ 3/2 × 6/7

⇒ (3 × 6) / (2 × 7)

⇒ 18 / 14 = 9/7

∴ The correct answer is 9/7.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. Expressions must be solved in this exact sequential order.

Fraction Division

To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction: a/b ÷ c/d = a/b × d/c.

Simplifying Fractions

Always cancel out common factors from the numerator and denominator before multiplying to make calculations faster and avoid large numbers.

Shortcut Trick

Convert terms to simple fractions for faster and error-free calculation:

(3/2 + 3/8)³ − (9/5 × 5/4)

⇒ (15/8)³ − 9/4

⇒ 3375/512 − 1152/512 = 2223/512

⇒ 2223 ÷ 512 ≈ 4.3418

∴ The correct answer is 4.3418.

Alternate Method

Given:

(1.5 + 3/8)³ − (9/5 × 1.25)

Calculations:

⇒ Convert decimals to fractions to easily simplify the expression:

⇒ 1.5 = 15/10 = 3/2

⇒ 1.25 = 125/100 = 5/4

⇒ Substitute the fraction values back into the expression:

⇒ (3/2 + 3/8)³ − (9/5 × 5/4)

⇒ Make denominators equal inside the bracket:

⇒ (12/8 + 3/8)³ − 9/4

⇒ Add the terms together inside the bracket:

⇒ (15/8)³ − 9/4

⇒ Calculate the cube of the fraction (15³ = 3375, 8³ = 512):

⇒ 3375/512 − 9/4

⇒ Make denominators equal to perform subtraction (multiply 9/4 by 128/128):

⇒ 3375/512 − (9 × 128)/(4 × 128)

⇒ 3375/512 − 1152/512

⇒ Subtract the numerators:

⇒ (3375 − 1152) / 512

⇒ 2223 / 512

⇒ Convert the final fraction back to a decimal:

⇒ 2223 ÷ 512 = 4.341796875

⇒ Rounding to four decimal places, we get 4.3418.

∴ The correct answer is 4.3418.

Additional Information

BODMAS Rule

Always simplify complex mathematical expressions in the exact order of Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Memorizing common fraction values makes calculations significantly faster. For example, 1/8 = 0.125, 3/8 = 0.375, and 1/4 = 0.25.

Rounding Decimals

When rounding to 4 decimal places, check the 5^th decimal digit. If it is 5 or more, round up the 4^th digit by 1. Otherwise, leave it unchanged.

Shortcut Trick

Convert the given fractions directly to decimals for faster calculation:

⇒ 3/4 = 0.75 and 1/2 = 0.5

Substitute these values into the expression:

⇒ ((1.5 + 0.75) ÷ (2.25 − 0.5)) + 0.6

⇒ (2.25 ÷ 1.75) + 0.6 = (9 ÷ 7) + 0.6 ≈ 1.2857 + 0.6 = 1.8857

∴ The correct answer is 1.8857.

Alternate Method

Given:

Expression = ((1.5 + 3/4) ÷ (2.25 − 1/2)) + 0.6

Calculations:

To avoid decimal division, convert all values to fractions:

⇒ 1.5 = 3/2

⇒ 2.25 = 9/4

⇒ 0.6 = 3/5

Substitute the fractions back into the expression:

⇒ ((3/2 + 3/4) ÷ (9/4 − 1/2)) + 3/5

Make denominators equal to add and subtract the inner terms:

⇒ ((6/4 + 3/4) ÷ (9/4 − 2/4)) + 3/5

⇒ (9/4 ÷ 7/4) + 3/5

⇒ (9/4 × 4/7) + 3/5

⇒ 9/7 + 3/5

Take the LCM of 7 and 5, which is 35:

⇒ (45 + 21) / 35

⇒ 66 / 35 ≈ 1.885714...

⇒ Rounding to 4 decimal places, we get 1.8857.

∴ The correct answer is 1.8857.

Additional Information

BODMAS Rule

Always follow the sequence of Brackets, Orders, Division/Multiplication, and Addition/Subtraction while simplifying expressions.

Fraction to Decimal Conversion

Memorising common fraction-to-decimal values like 1/2 = 0.5, 3/4 = 0.75, and 1/5 = 0.2 helps in quick calculation without the need to take an LCM.

Shortcut Trick

Equating the initial and new ratios for liquids A and B: (2x + 6) ÷ (3x + 12) = 3 ÷ 5 ⇒ 10x + 30 = 9x + 36 ⇒ x = 6.

New quantity of A = 2(6) + 6 = 18 liters (which represents 3 parts of the new ratio).

Since 3 parts = 18 liters, 1 part = 6 liters, and new C (8 parts) = 8 × 6 = 48 liters.

Initial quantity of C = 5x = 5(6) = 30 liters.

Quantity of C added = Final C − Initial C = 48 − 30 = 18 liters.

∴ The correct answer is 18 liters.

Alternate Method

Given:

Initial ratio of A, B, and C = 2 : 3 : 5

Quantity added to A = 6 liters

Quantity added to B = 12 liters

New ratio of A, B, and C = 3 : 5 : 8

Formula Used:

New Quantity = Initial Quantity + Quantity Added

Liquid A2x → 12 L+ 6 L18 L (3 parts)Liquid B3x → 18 L+ 12 L30 L (5 parts)Liquid C5x → 30 L+ y L48 L (8 parts)Initial Ratio = 2 : 3 : 5y = 48 − 30 = 18 Liters

Calculations:

⇒ Let the initial quantities of A, B, and C be 2x, 3x, and 5x respectively.

⇒ After adding 6 liters to A and 12 liters to B, their respective new quantities are:

⇒ New A = 2x + 6

⇒ New B = 3x + 12

⇒ According to the new given ratio of A to B (3 : 5), we can set up the equation:

⇒ (2x + 6) ÷ (3x + 12) = 3 ÷ 5

⇒ 5 × (2x + 6) = 3 × (3x + 12)

⇒ 10x + 30 = 9x + 36

⇒ 10x − 9x = 36 − 30

⇒ x = 6

⇒ Initial quantity of C = 5x = 5 × 6 = 30 liters

⇒ New quantity of A = 2(6) + 6 = 12 + 6 = 18 liters

⇒ According to the new ratio (3 : 5 : 8), 3 parts correspond to 18 liters of liquid A.

⇒ 1 part = 18 ÷ 3 = 6 liters

⇒ The new quantity of C corresponds to 8 parts of this ratio.

⇒ New C = 8 × 6 = 48 liters

⇒ Quantity of C added = New quantity of C − Initial quantity of C

⇒ Quantity of C added = 48 − 30 = 18 liters

∴ The correct answer is 18 liters.

Additional Information

Ratio Multiplier Concept

When components of a mixture are given in a ratio a : b : c, their absolute quantities can be modeled as ax, bx, and cx where x acts as a common variable multiplier.

Handling Independent Alterations

When distinct amounts are added to specific mixture components, forming an equation using only the known ratio components will yield the multiplier first, before trying to evaluate the unknown variables.

Cross-Multiplication Method

For simple rational equations structured as a ÷ b = c ÷ d, you can quickly simplify and solve by using direct cross-multiplication: a × d = b × c.

Shortcut Trick

In a partnership, the profit ratio is directly proportional to the product of Investment and Time.

Gautam's equivalent investment = 60000 × 24 months

Jatin's equivalent investment = 35000 × 16 months

Profit Ratio = (60000 × 24) : (35000 × 16) = (60 × 3) : (35 × 2) = 180 : 70 = 18 : 7

∴ The correct answer is 18:7.

Alternate Method

Given:

Gautam's investment (C₁) = ₹ 60,000

Jatin's investment (C₂) = ₹ 35,000

Total business duration = 2 years = 24 months

Jatin joins 8 months later.

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Gautam (₹ 60,000)

24 months

Jatin (₹ 35,000)

16 months

Joined 8 months later

Profit Ratio = (60k × 24) : (35k × 16)

Calculations:

⇒ Gautam's time period (T₁) = 24 months

⇒ Jatin's time period (T₂) = 24 − 8 = 16 months

⇒ Profit Ratio = (60000 × 24) : (35000 × 16)

⇒ Simplifying the capitals by removing common zeros:

⇒ Profit Ratio = (60 × 24) : (35 × 16)

⇒ Profit Ratio = 1440 : 560

⇒ Dividing both sides by 80:

⇒ Profit Ratio = 18 : 7

∴ The correct answer is 18:7.

Additional Information

Compound Partnership

When the capitals of partners are invested for different time periods, it is called a compound partnership. The share of profit for each partner is calculated by multiplying their respective capital with the time period.

Joining Mid-way

If a partner joins the business t months after the business started, and the total duration to calculate profit is T months, their capital is actively invested for only (T − t) months.

Active vs Sleeping Partner

An active partner manages the operations and usually receives a fixed percentage or salary from the total profit before the remainder is distributed. A sleeping partner only provides capital and shares the remaining profit based entirely on the investment ratio.

Shortcut Trick

Weight of the left person = Old Average + (Remaining People × Decrease in Average)

Old Average = 65 kg, Remaining People = 11

Decrease in Average = 65 − 64.5 = 0.5 kg

Weight = 65 + (11 × 0.5)

Weight = 65 + 5.5 = 70.5 kg

∴ The correct answer is 70.5 kg.

Alternate Method

Given:

Initial average weight of 12 people = 65 kg

New average weight after 1 person leaves = 64.5 kg

12 People | Average: 65 kg (Total: 780 kg)

11 People | Avg: 64.5 kg

1 Left

Total: 709.5 kg ? kg

Calculations:

⇒ Total initial weight of 12 people = 12 × 65 = 780 kg

⇒ Total weight of remaining 11 people = 11 × 64.5 = 709.5 kg

⇒ Weight of the person who left = (Total initial weight) − (Total new weight)

⇒ Weight of the person who left = 780 − 709.5

⇒ Weight of the person who left = 70.5 kg

∴ The correct answer is 70.5 kg.

Additional Information

When a Person Joins

If a new person joins and the average increases, New Weight = Old Average + (New total people × Increase in average).

When a Person is Replaced

If one person is replaced by another and the average changes, Weight of New Person = Weight of Replaced Person ± (Total number of people × Change in average).

Weighted Average

When combining two groups, Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Let the assumed average be 65.

Deviations of Q's sales (65, 68, 62, 64) from 65 are 0, +3, −3, and −1.

Sum of deviations = 0 + 3 − 3 − 1 = −1

Actual Average = Assumed Average + (Sum of deviations ÷ 4)

Actual Average = 65 − (1 ÷ 4) = 64.75

∴ The correct answer is ₹64.75.

Alternate Method

Given:

Sales of salesperson Q in different months:

Jan = 65, Feb = 68, March = 62, April = 64

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Total Sales of Q = 65 + 68 + 62 + 64

⇒ Total Sales of Q = 259

⇒ Total number of months = 4

⇒ Average Sales = 259 ÷ 4

⇒ Average Sales = 64.75

∴ The correct answer is ₹64.75.

Additional Information

Average Formula

Average is calculated by dividing the sum of all values by the total number of values. Average = Sum of observations ÷ Total number of observations.

Net Deviation Method

By selecting an assumed average, calculate the positive or negative deviation of each term. Net average = Assumed average + (Sum of deviations ÷ Total number of items).

Combined Average

When combining two distinct groups with quantities n₁ and n₂ and averages A₁ and A₂ respectively, the combined average is: (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick

Rate = 10% = 1/10 ⇒ Multiplying factor per year = 11/10

Ratio of Amount to Principal = 66550 ÷ 50000 = 1331 ÷ 1000

We know that (11/10)³ = 1331 / 1000

Since the cube corresponds to 3 periods, the time taken is 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹50,000

Amount (A) = ₹66,550

Rate of Interest (R) = 10% per annum

Formula Used:

A = P × (1 + R/100)ⁿ

Step-by-Step Compound Interest Growth

₹50,000

₹55,000

₹60,500

₹66,550

+10% 1 Yr

+10% 1 Yr

+10% 1 Yr

Total Time = 3 Years

Calculations:

⇒ 66,550 = 50,000 × (1 + 10/100)ⁿ

⇒ 66,550 ÷ 50,000 = (1 + 1/10)ⁿ

⇒ 6,655 ÷ 5,000 = (11/10)ⁿ

⇒ 1,331 ÷ 1,000 = (11/10)ⁿ

⇒ (11/10)³ = (11/10)ⁿ

⇒ n = 3

∴ The correct answer is 3 years.

Additional Information

Effective Rate of CI

For a 10% annual interest rate, the effective CI rate for 2 years is 21%, and for 3 years it is 33.1%. Here, 33.1% of ₹50,000 exactly equals the interest of ₹16,550.

Difference between CI and SI (2 Years)

The difference between compound interest and simple interest for a period of 2 years is calculated directly as: Difference = P × (R ÷ 100)².

Difference between CI and SI (3 Years)

For a period of 3 years, the formula extends to: Difference = P × (R ÷ 100)² × (3 + R ÷ 100).

Shortcut Trick

Given the ratio of Profit to Cost Price is 2 : 7.

Let Profit = 2 units and Cost Price = 7 units.

Selling Price = Cost Price + Profit = 7 + 2 = 9 units.

Required Ratio (Selling Price : Profit) = 9 : 2.

∴ The correct answer is 9:2.

Alternate Method

Given:

Ratio of Profit to Cost Price (CP) = 2 : 7

Cost Price (CP) = 7x Profit = 2x

Selling Price (SP) = 7x + 2x = 9x Profit & Loss Visualisation

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Calculation:

⇒ Let the Profit be 2x and the Cost Price be 7x.

⇒ SP = 7x + 2x

⇒ SP = 9x

⇒ The required ratio of Selling Price to Profit is SP : Profit

⇒ Ratio = 9x : 2x

⇒ Ratio = 9 : 2

∴ The correct answer is 9:2.

Additional Information

Profit and Loss Percentage

Both profit percentage and loss percentage are always calculated on the Cost Price (CP) unless specified otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Selling Price under Loss

When a product is sold at a loss, the Selling Price is less than the Cost Price. Formula: SP = CP − Loss.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP) of the article. Selling Price = Marked Price − Discount.

Shortcut Trick

Let the Marked Price (MP) of 1 box = 100.

Cost Price (CP) of 1 box = (MP of 8 boxes) ÷ 10 = 800 ÷ 10 = 80.

Selling Price (SP) of 1 box = MP − 2% discount = 100 − 2 = 98.

Profit % = [(SP − CP) ÷ CP] × 100 = [(98 − 80) ÷ 80] × 100.

Profit % = (18 ÷ 80) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Alternate Method

Given:

Total number of boxes bought by the shop owner = 10

Price paid for 10 boxes = Price of 8 boxes

Discount allowed on each box during sale = 2% on Marked Price

Formula Used:

Profit = Selling Price (SP) − Cost Price (CP)

Profit % = (Profit ÷ CP) × 100

Cost Price (CP) of 1 box = 80

Marked Price (MP) of 1 box = 100

Selling Price (SP) of 1 box = 98 (-2% discount)

Profit = 18

Profit % = (18 ÷ 80) × 100 = 22.5%

Calculations:

⇒ Let the Marked Price (MP) of 1 box be 100.

⇒ The shopkeeper buys 10 boxes for the price of 8 boxes.

⇒ Cost Price (CP) of 10 boxes = 8 × 100 = 800.

⇒ Cost Price (CP) of 1 box = 800 ÷ 10 = 80.

⇒ He sells each box at a 2% discount on its Marked Price.

⇒ Selling Price (SP) of 1 box = 100 − (2% of 100) = 98.

⇒ Profit earned on 1 box = SP − CP = 98 − 80 = 18.

⇒ Profit % = (18 ÷ 80) × 100

⇒ Profit % = (9 ÷ 40) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Additional Information

Marked Price and Discount

Discount is always calculated strictly on the Marked Price (MP) unless specified otherwise. The Selling Price is derived as SP = MP − Discount.

Buy X Get Y Free Offers

In promotional schemes, the discount percentage is based on the total number of items taken. Discount % = [Free Items ÷ Total Items (X + Y)] × 100.

Successive Discounts

When two consecutive discounts of a% and b% are applied on an item, the equivalent single discount percentage is given by the formula: (a + b − (a × b) ÷ 100)%.

Dishonest Dealer Profit

When a dealer claims to sell at cost price but uses a false weight, the actual profit percentage is calculated as: [(True Weight − False Weight) ÷ False Weight] × 100.

Shortcut Trick

Let the Cover Price be 100 units.

Discounted Price = 100 − 30% = 70 units.

Total Cost Price = 70 + (5% of 70) = 73.5 units.

Profit = 40% of Cost Price = 0.40 × 73.5 = 29.4 units.

Since 29.4 units = ₹1620, Cover Price (100 units) = (1620 ÷ 29.4) × 100 ≈ ₹5510.20

∴ The correct answer is ₹5,510.20.

Alternate Method

Given:

Discount on Cover Price = 30%

Shipping charge on discounted price = 5%

Profit percentage = 40%

Profit amount (sold for ₹1620 more) = ₹1620

Cover Price(x)-30% disc+5% shipCost Price(0.735x)+40% Profit= ₹1620Selling Price(CP + 1620)Equation: 40% of 0.735x = 1620

Formula Used:

Cost Price (CP) = Discounted Price × (1 + Shipping Rate)

Profit = Profit % × CP

Calculations:

⇒ Let the original Cover Price of the novel be x.

⇒ Discounted Price = x − 30% of x = 0.70x

⇒ Cost Price (CP) = 0.70x + 5% of 0.70x

⇒ CP = 0.70x + 0.035x = 0.735x

⇒ Profit = 40% of CP = 0.40 × 0.735x = 0.294x

⇒ We are given that the Profit earned is ₹1620.

⇒ 0.294x = 1620

⇒ x = 1620 ÷ 0.294

⇒ x = 1620000 ÷ 294

⇒ x ≈ 5510.20

∴ The correct answer is ₹5,510.20.

Additional Information

Marked Price (Cover Price)

It is the original price printed on an item. Successive discounts or flat discounts are always calculated on the Marked Price before deciding the Selling Price or Purchase Cost.

Effective Cost Price with Overheads

When extra charges like shipping, repairs, or transportation are incurred, they are added to the initial discounted purchase price to evaluate the total effective Cost Price.

Profit and Loss Application

All Profit and Loss calculations and percentage margins are applied purely on the final Cost Price, not on the original marked price. Formula: Profit % = (Profit ÷ CP) × 100.

Shortcut Trick

Fraction of syrup replaced each time = 10 ÷ 100 = 1/10.

Fraction of syrup left after 2 replacements = (1 − 1/10)² = (9/10)² = 81/100.

Final syrup = 81/100 × 100 L = 81 L.

Final water = Total volume − Final syrup = 100 L − 81 L = 19 L.

Ratio of syrup to water = 81 : 19.

∴ The correct answer is 81:19.

Alternate Method

Given:

Initial volume of pure syrup (V) = 100 L

Volume drawn and replaced by water (x) = 10 L

Number of operations (n) = 2

Formula Used:

Final amount of pure liquid = V × (1 − x/V)ⁿ

Initial 100L Pure Syrup −10L Mixture +10L Water (2 Times) Final 81L Syrup 19L Water Ratio 81:19

Calculations:

⇒ Final amount of syrup = 100 × (1 − 10/100)²

⇒ Final amount of syrup = 100 × (1 − 1/10)²

⇒ Final amount of syrup = 100 × (9/10)²

⇒ Final amount of syrup = 100 × 81/100 = 81 L

⇒ Volume of water in the final mixture = Total Volume − Final amount of syrup

⇒ Volume of water = 100 − 81 = 19 L

⇒ Ratio of syrup to water = 81 : 19

∴ The correct answer is 81:19.

Additional Information

Successive Replacement Formula

When a constant volume 'x' is replaced 'n' times from an initial volume 'V', Final amount = V × (1 − x/V)ⁿ.

Multiple Different Replacements

If replaced amounts are different (x, y, z) from initial volume V, Final amount = V × (1 − x/V) × (1 − y/V) × (1 − z/V).

Alligation Cross Method

Used to find the ratio in which two mixtures should be mixed. The ratio of quantities is inversely proportional to the differences in their unit concentrations.

Shortcut Trick

Volume is proportional to the cube of the radius, while surface area is proportional to the square of the radius.

Since 1 large sphere is melted into 8 identical smaller spheres, the volume ratio of 1 Large : 1 Small = 8 : 1.

Therefore, the radius ratio of Large : Small = 8^1/3 : 1^1/3 = 2 : 1.

The surface area ratio of 1 Large : 1 Small = 2² : 1² = 4 : 1.

Ratio of the surface area of 1 Large sphere to 8 Small spheres = 4 : (8 × 1) = 4 : 8 = 1 : 2.

∴ The correct answer is 1:2.

Alternate Method

Given:

Radius of original metal sphere (R) = 10 cm

Number of identical smaller spheres (n) = 8

Formula Used:

Volume of a sphere = (4/3) × π × r³

Surface Area of a sphere = 4 × π × r²

R = 10 Original Sphere

Melted into 8

r × 8 Smaller Spheres

Calculations:

⇒ Total volume remains constant when melting and recasting.

⇒ Volume of original sphere = 8 × Volume of one smaller sphere

⇒ (4/3) × π × R³ = 8 × (4/3) × π × r³

⇒ R³ = 8r³

⇒ 10³ = 8r³

⇒ 1000 = 8r³

⇒ r³ = 125 ⇒ r = 5 cm

⇒ Surface Area of Original Sphere (S₁) = 4 × π × 10²

⇒ S₁ = 400π cm²

⇒ Total Surface Area of 8 Smaller Spheres (S₂) = 8 × (4 × π × 5²)

⇒ S₂ = 8 × 100π = 800π cm²

⇒ Required Ratio = S₁ : S₂ = 400π : 800π

⇒ Ratio = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Melting and Recasting Law

When a solid object is melted and recast into new shapes, the total volume of the material is always conserved, though the total surface area will change.

Dimensional Scaling Factors

If the linear dimension (radius, side) of a 3D figure changes by a factor of k, then its surface area scales by k² and its volume scales by k³.

Sphere vs Hemisphere Properties

For a full sphere, Total Surface Area = 4πr². For a solid hemisphere, the Curved Surface Area is 2πr² while the Total Surface Area is 3πr² because of the flat circular base.

Shortcut Trick

When a large shape is melted into smaller identical shapes, the number of objects is the ratio of their volumes.

For identical shapes like hemispheres, constants cancel out, leaving only the ratio of the radii cubed.

Number of small hemispheres = (Radius of large ÷ Radius of small)³

⇒ Number = (12 ÷ 3)³ = 4³

⇒ Number = 64

∴ The correct answer is 64.

Alternate Method

12 cm Large Hemisphere

Melted

3 cm

...

n = ? n Small Hemispheres

Given:

Radius of large hemisphere (R) = 12 cm

Radius of small hemisphere (r) = 3 cm

Formula Used:

Volume of a hemisphere = (2 ÷ 3) × π × radius³

Calculation:

Let the total number of small hemispheres produced be n.

Since the large hemisphere is melted to form smaller ones, the total volume of the material remains constant.

⇒ Volume of large hemisphere = n × Volume of small hemisphere

⇒ (2 ÷ 3) × π × R³ = n × (2 ÷ 3) × π × r³

Canceling out (2 ÷ 3) × π from both sides:

⇒ R³ = n × r³

⇒ 12³ = n × 3³

⇒ n = 12³ ÷ 3³

⇒ n = (12 ÷ 3)³

⇒ n = 4³

⇒ n = 64

∴ The correct answer is 64.

Additional Information

Melting and Recasting

When a solid object is melted and recast into new shapes, the total volume remains constant regardless of the new shape. Number of shapes = Total initial volume ÷ Volume of one new shape.

Volume of a Hemisphere

The space occupied by a hemisphere is exactly half the volume of a full sphere. It is given by V = (2 ÷ 3) × π × r³, where r is the radius.

Surface Areas of a Hemisphere

The Curved Surface Area (CSA) of a hemisphere is 2 × π × r². The Total Surface Area (TSA) includes the flat circular top, making it 3 × π × r².