Quantitative Aptitude PYQs

Shortcut Trick

Net multiplier method: Final Price = Initial CP × (100−L₁)% × (100+P₂)% × (100+P₃)%.

Convert percentages to fractions: 12.5% = 1/8 (loss ⇒ 7/8), 20% = 1/5 (profit ⇒ 6/5), 6(2/3)% = 1/15 (profit ⇒ 16/15).

Price for Priya = 12,000 × (7/8) × (6/5) × (16/15).

Price for Priya = 12,000 × (672/600) = 12,000 × 1.12 = 13,440.

∴ The correct answer is ₹13,440.

Alternate Method

Given → CP for Manu = ₹12,000; Loss₁ = 12.5%; Profit₂ = 20%; Profit₃ = 6(2/3)%.
Formulas: SP = CP × (100 ± P/L)% | Profit% = (Profit/CP) × 100.

⇒ Price for Anuj = 12,000 − (12.5% of 12,000) = 12,000 − 1,500 = ₹10,500.
⇒ Price for Rani = 10,500 + (20% of 10,500) = 10,500 + 2,100 = ₹12,600.
⇒ Price for Priya = 12,600 + (6 2/3% of 12,600).
⇒ 6 2/3% = 20/3% = 20/(3 × 100) = 1/15.
⇒ Price for Priya = 12,600 + (12,600 ÷ 15) = 12,600 + 840 = ₹13,440.

∴ The correct answer is ₹13,440.

Additional Information

Successive Percentage Change

When a value changes multiple times, the final value is found by multiplying the initial value by (100 ± r₁)% × (100 ± r₂)%...

Fractional Conversion

Using fractions like 1/8 for 12.5% or 1/15 for 6.67% simplifies calculations significantly in multi-step profit and loss problems.

Shortcut Trick

Second Number = (HCF × LCM) ÷ First Number

Second Number = (32 × 192) ÷ 64

Second Number = 192 ÷ 2 = 96

∴ The correct answer is 96.

Alternate Method

Given:

LCM of two numbers = 192

HCF of two numbers = 32

First number (n₁) = 64

Formula Used:

The product of two numbers is equal to the product of their HCF and LCM.

n₁ × n₂ = HCF × LCM

Calculations:

⇒ 64 × n₂ = 32 × 192

⇒ n₂ = (32 × 192) ÷ 64

⇒ n₂ = 1 × 192 ÷ 2

⇒ n₂ = 96

∴ The correct answer is 96.

Additional Information

HCF and LCM Relation

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

HCF of Fractions

HCF of fractions = HCF of Numerators ÷ LCM of Denominators.

LCM of Fractions

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

If each number in a data set is multiplied by a constant k, the average of the set is also multiplied by k.

Given Old Average = 7 and Multiplication Factor (k) = 2.

New Average = Old Average × k

⇒ 7 × 2 = 14.

∴ The correct answer is 14.

Alternate Method

Given: Number of observations (n) = 13, Initial Average = 7.

Formula Used: Sum = Average × Number of observations

⇒ Original Sum of 13 numbers = 13 × 7 = 91

⇒ Since each number is multiplied by 2, the New Sum = 91 × 2 = 182

⇒ New Average = New Sum ÷ Total Numbers

⇒ New Average = 182 ÷ 13

⇒ New Average = 14

∴ The correct answer is 14.

Additional Information

Linear Transformation of Average

If every observation x_i is changed to a × x_i + b, then the new average X' becomes a × X + b, where X is the original average.

Addition and Subtraction Rule

If a constant k is added to (or subtracted from) each observation, the new average is Old Average + k (or Old Average − k).

Division Rule

If each observation is divided by a non-zero constant k, the new average is Old Average ÷ k.

Shortcut Trick

Income Raman = 100%, Savings = 16%, Expenditure = 84%.

Change in Income (22% of 100) = +22 units.

Change in Expenditure (25% of 84) = +21 units.

Change in Savings = 22 − 21 = +1 unit.

Savings increase = 1% of ₹83,000 = ₹830.

∴ The correct answer is increase by ₹830.

Alternate Method

Given: Initial Income = ₹83,000 | Savings = 16% | Expenditure = 84%.

⇒ Initial Savings = 16% of 83,000 = 0.16 × 83,000 = ₹13,280.

⇒ Initial Expenditure = 84% of 83,000 = 0.84 × 83,000 = ₹69,720.

⇒ New Income = 83,000 × (122/100) = ₹1,01,260.

⇒ New Expenditure = 69,720 × (125/100) = ₹87,150.

⇒ New Savings = New Income − New Expenditure = 1,01,260 − 87,150 = ₹14,110.

⇒ Change in Savings = 14,110 − 13,280 = ₹830.

∴ The correct answer is increase by ₹830.

Additional Information

Income Formula

Income = Expenditure + Savings. Any percentage change in components must balance this equation.

Alligation Method

You can use alligation for income-expenditure-savings problems where % change in income lies between % change of savings and expenditure.

Successive Percentage Change

If a value changes by a% then b%, net change = a + b + (a×b)/100.

Shortcut Trick

Given expression is in the form (a² − b²) ÷ (c × d).

Using the identity a² − b² = (a − b)(a + b):

Numerator = (8.4 − 2.8) × (8.4 + 2.8) = 5.6 × 11.2

Result = (5.6 × 11.2) ÷ (5.6 × 11.2) = 1

∴ The correct answer is 1.

Alternate Method

Given:

Expression: (8.4 × 8.4 − 2.8 × 2.8) ÷ (5.6 × 11.2)

Formula Used:

a² − b² = (a − b)(a + b)

Calculations:

⇒ Numerator = (8.4)² − (2.8)²

⇒ Applying the formula: (8.4 − 2.8)(8.4 + 2.8)

⇒ Numerator = 5.6 × 11.2

⇒ Denominator = 5.6 × 11.2

⇒ Expression = (5.6 × 11.2) ÷ (5.6 × 11.2)

⇒ Value = 1

∴ The correct answer is 1.

Additional Information

Algebraic Identities

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Cube Identities

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

BODMAS Rule

Simplify expressions following the exact order: Brackets, Orders (powers/roots), Division & Multiplication, Addition & Subtraction.

Shortcut Trick

Volume of cylinder depends on r² and h. Change in r = −17%, h = +138%.

Net change in r² = −17 − 17 + (−17 × −17)/100 = −34 + 2.89 = −31.11%.

Net change in Volume = −31.11 + 138 + (−31.11 × 138)/100.

⇒ 106.89 − 42.9318 = 63.9582% ≈ 64%.

∴ The correct answer is 64%.

Alternate Method

Given: Decrease in radius (r) = 17%, Increase in height (h) = 138%.

Formula: Volume of Cylinder (V) = π × r² × h

⇒ Let the initial radius be 100 units and initial height be 100 units.

⇒ Initial Volume (V₁) = π × (100)² × 100 = 1,000,000π

⇒ New radius (r') = 100 − 17% of 100 = 83 units.

⇒ New height (h') = 100 + 138% of 100 = 238 units.

⇒ New Volume (V₂) = π × (83)² × 238

⇒ V₂ = π × 6889 × 238 = 1,639,582π

⇒ Increase in Volume = 1,639,582π − 1,000,000π = 639,582π

⇒ Percentage Increase = (639,582π / 1,000,000π) × 100 = 63.9582%

⇒ Rounded to the closest integer = 64%.

∴ The correct answer is 64%.

Additional Information

Volume of Cylinder

The volume of a right circular cylinder is calculated as Area of base × Height = πr²h.

Successive Percentage Change

For two changes a% and b%, the net change is [a + b + (ab/100)]%. For volume, since r is squared, we apply the successive formula thrice (for r, r, and h).

Surface Area Properties

Curved Surface Area (CSA) = 2πrh and Total Surface Area (TSA) = 2πr(r + h). Changes in radius and height affect these differently than volume.

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

Required percentage = (1/20 ÷ 2/70) × 100

Required percentage = (1/20 × 70/2) × 100 = (70/40) × 100

Required percentage = 1.75 × 100 = 175%

∴ The correct answer is 175%.

Alternate Method

Given:

First number = 1/20

Second number = 2/70 = 1/35

Formula Used:

Percentage = (Part / Whole) × 100

Calculations:

⇒ Required % = [(1/20) / (1/35)] × 100

⇒ Required % = (1/20) × (35/1) × 100

⇒ Required % = (35 / 20) × 100

⇒ Required % = 35 × 5

⇒ Required % = 175%

∴ The correct answer is 175%.

Additional Information

Fraction to Percentage

To convert any fraction a/b into a percentage, multiply the fraction by 100 and add the % symbol.

Reciprocal Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Percentage Comparison

When calculating "x is what % of y", y always acts as the base (denominator).

Shortcut Trick

Net difference in interest rate per annum = 7.5% − 3.5% = 4%

Total difference for 4 years = 4% × 4 = 16%

Interest Difference = 16% of Principal (₹2,40,000)

Difference = 0.16 × 2,40,000 = ₹38,400

∴ The correct answer is ₹38,400.

Alternate Method

Given: Principal (P) = ₹2,40,000, Time (T) = 4 years, Rate A (R_A) = 3.5%, Rate B (R_B) = 7.5%

Formula Used: Simple Interest (SI) = (P × R × T) ÷ 100

⇒ SI from Bank A = (2,40,000 × 3.5 × 4) ÷ 100

⇒ SI_A = 2,400 × 14 = ₹33,600

⇒ SI from Bank B = (2,40,000 × 7.5 × 4) ÷ 100

⇒ SI_B = 2,400 × 30 = ₹72,000

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = ₹72,000 − ₹33,600 = ₹38,400

∴ The correct answer is ₹38,400.

Additional Information

Difference in Simple Interest

When the principal and time are the same, the difference in interest is given by: ΔSI = P × ΔR × T ÷ 100, where ΔR is the difference in annual interest rates.

Total Amount in SI

The total amount (A) to be paid at the end of the term is the sum of Principal and Simple Interest: A = P + SI = P(1 + RT/100).

Simple Interest Nature

Simple Interest remains constant for every year if the principal and rate of interest do not change.

Shortcut Trick

To find the mode, identify the observation with the maximum frequency (most repeated value).

Frequency of 5 → 3 | Frequency of 6 → 5 | Frequency of 7 → 3

Frequency of 8 → 2 | Frequency of 9 → 2 | Frequency of 10 → 2 | Frequency of 11 → 1

The value 6 appears the most number of times (5 times).

∴ The correct answer is 6.

Alternate Method

Given: Data set = {5, 7, 6, 5, 9, 8, 6, 7, 11, 10, 5, 7, 6, 8, 6, 9, 10, 6}

Formula Used: Mode = The value that occurs most frequently in a given set of data.

⇒ Frequency Count:

⇒ Number 5 occurs 3 times.

⇒ Number 6 occurs 5 times.

⇒ Number 7 occurs 3 times.

⇒ Number 8 occurs 2 times.

⇒ Number 9 occurs 2 times.

⇒ Number 10 occurs 2 times.

⇒ Number 11 occurs 1 time.

⇒ The maximum frequency is 5, which corresponds to the observation 6.

∴ The correct answer is 6.

Additional Information

Arithmetic Mean

It is the sum of all observations divided by the total number of observations. Formula: Mean = (Σx_i) ÷ n.

Median

The middle value of a data set when it is arranged in ascending or descending order. For even n, it is the average of the two middle terms.

Empirical Relationship

For moderately skewed distributions: Mode = 3 × Median − 2 × Mean.

Bimodal and Multimodal

A data set with two modes is called bimodal, and one with more than two modes is called multimodal.

Shortcut Trick

Find two numbers that multiply to 50 and add up to −15.

The numbers are −10 and −5.

The factors of the numerator are (x − 10) and (x − 5).

Divide by the denominator (x − 10) by cancelling the common term.

The remaining term is x − 5.

∴ The correct answer is x − 5.

Alternate Method

Given: Rational expression ((x² − 15x + 50)) ÷ (x − 10)

Method: Factorization of the quadratic numerator by splitting the middle term.

⇒ Numerator = x² − 15x + 50

⇒ Numerator = x² − 10x − 5x + 50

⇒ Numerator = x(x − 10) − 5(x − 10)

⇒ Numerator = (x − 10)(x − 5)

⇒ The roots of the quadratic equation x² − 15x + 50 = 0 are x = 10 and x = 5.

⇒ Total expression = [(x − 10)(x − 5)] ÷ (x − 10)

⇒ Total expression = x − 5

∴ The correct answer is x − 5.

Additional Information

Quadratic Factorization

For a quadratic ax² + bx + c, find two numbers whose sum is b and product is ac to split the middle term.

Factor Theorem

If a polynomial f(x) gives 0 when x = a, then (x − a) is a guaranteed factor of that polynomial.

Division of Polynomials

Simplifying a fraction involves removing common factors from both the numerator and the denominator through division.

Shortcut Trick

Speed ratio (New : Normal) = 3 : 8

Since distance is constant, Time ratio is inverse of speed ratio → 8 : 3

Difference in time units = 8 − 3 = 5 units

Given 5 units = 10 minutes ⇒ 1 unit = 2 minutes

Normal time = 3 units = 3 × 2 = 6 minutes

∴ The correct answer is 6 minutes.

Alternate Method

Given: New Speed = 3/8 × Normal Speed; Time Delay = 10 minutes

Formula Used: Distance = Speed × Time

⇒ D = S × T_normal = (3/8)S × (T_normal + 10)

⇒ T_normal = (3/8) × (T_normal + 10)

⇒ 8 × T_normal = 3 × T_normal + 30

⇒ 5 × T_normal = 30

⇒ T_normal = 30 ÷ 5 = 6 minutes

∴ The correct answer is 6 minutes.

Additional Information

Inverse Variation of Speed and Time

When the distance is constant, speed is inversely proportional to time (S ∝ 1/T). If speed becomes a/b, time becomes b/a of the original.

Relationship Formula

Original Time = [Late time × Numerator] ÷ [Denominator − Numerator] when speed is reduced to (Numerator/Denominator).

Shortcut Trick

Use laws of exponents: (a^m ÷ aⁿ) = a^m−n

(3⁶ ÷ 3²)² = (3⁴)² = 3⁸

(3³ ÷ 3⁴) = 3⁻¹

Total = 3⁸ × 3⁻¹ = 3⁷

Alternate Method

Given:

(3⁶ ÷ 3²)² × (3³ ÷ 3⁴)

Concept Used:

BODMAS rule and laws of exponents

Calculation:

⇒ (3⁶ ÷ 3²)² × (3³ ÷ 3⁴)

⇒ (3⁴)² × (3³ ÷ 3⁴)

⇒ 3⁸ × (3³ ÷ 3⁴)

⇒ 3⁸ × 3⁻¹

⇒ 3⁷

∴ The correct answer is 3⁷.

Additional Information

BODMAS Rule

Brackets, Orders (powers), Division, Multiplication, Addition, Subtraction.

Exponent Laws

a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n

Negative Power

a⁻ⁿ = 1 ÷ aⁿ

Shortcut Trick

Ratio of Marked Price (MP) to Selling Price (SP) = 3,392 : 1,696.

Dividing both by 1,696, we get the ratio 2 :

1. Discount = 2 − 1 = 1 unit.

Discount % = (1 ÷ 2) × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: Marked Price (MP) = ₹3,392, Selling Price (SP) = ₹1,696.

Formulas: Discount = MP − SP | Discount % = (Discount ÷ MP) × 100

⇒ Discount = 3,392 − 1,696 = ₹1,696

⇒ Discount % = (1,696 ÷ 3,392) × 100

⇒ Discount % = (1 ÷ 2) × 100 = 50%

∴ The correct answer is 50%.

Additional Information

Marked Price & Discount

The Marked Price is the price printed on an item. Discount is always calculated on the Marked Price unless stated otherwise.

Selling Price Calculation

SP = MP × (100 − Discount %) ÷ 100. If the SP is exactly half of the MP, the discount is always 50%.

Shortcut Trick

Combine ratios M : N = 3 : 9 = 1 : 3 and N : T = 6 : 9 = 2 : 3.

Combined Ratio M : N : T = (1×2) : (3×2) : (3×3) = 2 : 6 : 9.

Total units = 2 + 6 + 9 = 17 units.

Share of N = (6 ÷ 17) × 1071 = 6 × 63 = 378.

∴ The correct answer is ₹378.

Alternate Method

Given: Total amount = ₹1,071, Ratio M : N = 3 : 9, Ratio N : T = 6 : 9.

Step 1: Simplify and Equate N:

⇒ M : N = 1 : 3

⇒ N : T = 2 : 3

⇒ Multiply first ratio by 2 and second by 3 to make N same.

⇒ M : N = 2 : 6

⇒ N : T = 6 : 9

⇒ M : N : T = 2 : 6 : 9

Step 2: Calculate Share:

⇒ Sum of ratio parts = 2 + 6 + 9 = 17

⇒ Value of 1 unit = 1071 ÷ 17 = 63

⇒ Share of N = 6 × 63 = 378

∴ The correct answer is ₹378.

Additional Information

Compound Ratio

To combine ratios a:b and b:c, multiply both to find a:b:c by making the common term 'b' equal in both sets.

Unitary Method in Ratios

Individual Share = (Individual Ratio Part ÷ Sum of all Ratio Parts) × Total Amount.

Simplification Rule

Always simplify individual ratios (e.g., 3:9 to 1:3) before combining to keep calculations easier.

Shortcut Trick

In a 30°−60°−90° triangle, the ratio of sides opposite to these angles is 1 : √3 : 2.

The side adjacent to 60° (base) is 12 m.

The side opposite to 60° (height) = Base × √3.

Height = 12 × √3 ≈ 12 × 2 = 24 m.

∴ The correct answer is 24 m.

Alternate Method

Given:

Distance from the flagpole (Base AB) = 12 m

Angle of elevation (θ) = 60°

Height of the boy = 0 m

Formula Used:

tan θ = Perpendicular / Base

Calculations:

⇒ tan 60° = Height / 12

⇒ √3 = Height / 12

⇒ Height = 12 × √3

⇒ Height ≈ 12 × 2 (Since √3 ≈ 2 is given)

⇒ Height = 24 m

∴ The correct answer is 24 m.

Additional Information

Angle of Elevation

The angle formed between the horizontal line and the line of sight when looking up at an object.

Trigonometric Ratios

For a right-angled triangle, tan θ = Height/Base, sin θ = Height/Hypotenuse, and cos θ = Base/Hypotenuse.

Standard Values

Common values include tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Shortcut Trick

Total work = LCM of days (20, 12, 2) = 60 units.

Efficiencies: A = 60 ÷ 20 = 3, B = 60 ÷ 12 = 5, A+B+C = 60 ÷ 2 = 30.

Efficiency of C = 30 − (3 + 5) = 22.

Time for C = Total Work ÷ C's efficiency = 60 ÷ 22 = 30/11 = 2 ⁸⁄₁₁ days.

∴ The correct answer is 2 ⁸⁄₁₁ days.

Alternate Method

Given:

Time taken by A = 20 days

Time taken by B = 12 days

Time taken by A, B and C together = 2 days

Calculations:

⇒ Efficiency of C = Efficiency of (A + B + C) − Efficiency of A − Efficiency of B

⇒ Efficiency of C = 30 − 3 − 5 = 22 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time taken by C alone = 60 ÷ 22 = 30/11

⇒ Time taken by C alone = 2 ⁸⁄₁₁ days

∴ The correct answer is 2 ⁸⁄₁₁ days.

Additional Information

Total Work Concept

Total work is assumed to be the LCM of the given time periods. Efficiency is inversely proportional to time.

Combined Efficiency

When multiple people work together, their combined efficiency is the sum of their individual efficiencies: E(A+B) = E(A) + E(B).

Fractional Work Formula

If someone can complete a task in N days, they can complete 1/N of the task in exactly 1 day.

Shortcut Trick

Directly apply known square roots: √324 = 18 and √841 = 29.

For decimal roots, simply halve the number of decimal places.

⇒ √324 + √0.0324 − √8.41

⇒ 18 + 0.18 − 2.9

⇒ 18.18 − 2.90 = 15.28

∴ The correct answer is 15.28.

Alternate Method

Given:

√324 + √0.0324 − √8.41

Calculations:

Convert the decimals to fractions to simplify the roots clearly:

⇒ √324 + √(324 ÷ 10000) − √(841 ÷ 100)

⇒ 18 + (18 ÷ 100) − (29 ÷ 10)

⇒ 18 + 0.18 − 2.9

Align the decimals to perform addition and subtraction:

⇒ 18.18 − 2.90

⇒ 15.28

∴ The correct answer is 15.28.

Additional Information

Square Root of Decimals

When finding the square root of a decimal number, the number of decimal places in the answer will be exactly half the number of decimal places in the original number (e.g., √0.0004 = 0.02).

Important Squares to Remember

Memorizing perfect squares up to 30 significantly improves calculation speed. For example, 18² = 324 and 29² = 841.

Simplification Rules

Always follow the basic order of operations. Evaluate the roots first, followed by addition and then subtraction, ensuring decimal points are properly aligned during the calculation.

Shortcut Trick

The offer states that the first product is sold at full price and the second gets a 30% discount.

Total amount paid = Price of 1^st product + 70% of Price of 2^nd product

Total amount paid = 350 + (350 × 70 ÷ 100)

Total amount paid = 350 + 245 = ₹595

∴ The correct answer is ₹595.

Alternate Method

Given:

Marked price of each product = ₹350

Offer given = Buy one get one 30% off

First Product
Price: ₹350
Discount: 0%
Paid: ₹350

Second Product
Price: ₹350
30% OFF
Paid: ₹245

Total Paid = ₹350 + ₹245 = ₹595

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100

Calculation:

⇒ Amount paid for the first product = ₹350

⇒ Amount paid for the second product = 350 × (100 − 30) ÷ 100

⇒ Amount paid for the second product = 350 × 70 ÷ 100 = ₹245

⇒ Total amount paid = Price of 1^st product + Price of 2^nd product

⇒ Total amount paid = 350 + 245 = ₹595

∴ The correct answer is ₹595.

Additional Information

Buy X Get Y Free Discount

Equivalent Discount % = [Free Items ÷ (Paid Items + Free Items)] × 100

Successive Discounts

For two successive discounts of a% and b%, Single Equivalent Discount = a + b − (a × b ÷ 100)%

Selling Price Formula

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100

Shortcut Trick

If the ratio of two numbers is a : b and their HCF is H, then their LCM is calculated directly as:

LCM = a × b × H

Here, the ratio parts are a = 5, b = 6, and HCF = 7.

LCM = 5 × 6 × 7 = 210

∴ The correct answer is 210.

Alternate Method

Given:

Ratio of two numbers = 5 : 6

Highest Common Factor (HCF) = 7

HCF = 7

Num 1 = 5 × 7 = 35

Num 2 = 6 × 7 = 42

LCM = 5 × 6 × 7 = 210

Formula Used:

First Number = a × HCF

Second Number = b × HCF

Product of Numbers = LCM × HCF

Calculations:

⇒ Let the two numbers be 5x and 6x, where x is the HCF.

⇒ Given that the HCF is 7, we get x = 7.

⇒ First Number = 5 × 7 = 35

⇒ Second Number = 6 × 7 = 42

⇒ We know that: First Number × Second Number = LCM × HCF

⇒ 35 × 42 = LCM × 7

⇒ LCM = (35 × 42) ÷ 7

⇒ LCM = 35 × 6 = 210

∴ The correct answer is 210.

Additional Information

Product of Two Numbers Rule

For any two numbers, the product of the numbers is always equal to the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM). Formula: N₁ × N₂ = HCF × LCM.

Co-prime Numbers and LCM

When a ratio a : b is given, a and b are essentially co-prime parts of the numbers. If two numbers are co-prime (HCF = 1), their LCM is simply their product.

LCM & HCF of Fractions

To find the LCM of fractions: (LCM of Numerators) ÷ (HCF of Denominators). To find the HCF of fractions: (HCF of Numerators) ÷ (LCM of Denominators).

Shortcut Trick

Investment Ratio = Profit Ratio = 58 : 72 : 20

Total profit units = 58 + 72 + 20 = 150 units

Difference in profit units of Y and Z = 72 − 20 = 52 units

Required difference = (52 / 150) × 3750

Required difference = 52 × 25 = ₹1300

∴ The correct answer is ₹1,300.

Alternate Method

Given:

Investment ratio of X, Y and Z = 58 : 72 : 20

Total profit earned = ₹3750

Formula Used:

Profit Ratio = Investment Ratio (when the time period is constant for all partners)

X
58x

Y
72x

Z
20x

Diff = 52x

Total Profit = 150x = ₹3750

Calculations:

⇒ Let the profit shares of X, Y and Z be 58x, 72x, and 20x respectively.

⇒ Total profit = 58x + 72x + 20x = 150x

⇒ 150x = 3750

⇒ x = 3750 ÷ 150 = 25

⇒ Profit share of Y = 72x = 72 × 25 = ₹1800

⇒ Profit share of Z = 20x = 20 × 25 = ₹500

⇒ Difference between shares of Y and Z = 1800 − 500 = 1300

∴ The correct answer is ₹1,300.

Additional Information

Simple Partnership

When investments are made for the exact same time period by all partners, the profit is distributed directly in the ratio of their invested capitals.

Compound Partnership

When investments are made for different time periods, the profit sharing ratio is equivalent to the product of their capital and time period (Profit Ratio = C₁×T₁ : C₂×T₂).

Working Partner vs Sleeping Partner

A working partner receives a fixed percentage of the profit or a regular salary for managing the business. The remaining profit is then distributed among all the partners strictly in the ratio of their effective capital.