Quantitative Aptitude PYQs

Shortcut Trick

To find the maximum advantage, calculate the equivalent single discount for each option.

Successive discount formula: a + b − (a × b) ÷ 100.

Option 1: 44 + 18 − (44 × 18) ÷ 100 = 62 − 7.92 = 54.08%.

Option 2: 30 + 30 − (30 × 30) ÷ 100 = 60 − 9 = 51%.

Option 3: Single discount = 22%.

Option 4: 24 + 25 − (24 × 25) ÷ 100 = 49 − 6 = 43%.

∴ The correct answer is Two successive discounts of 44% and 18%.

Alternate Method

Given: Marked Price (MP) = ₹5,000. Goal: Find the highest Effective Discount Rate.
Formulas: Effective Discount = a + b − (ab ÷ 100) | SP = MP × (100 − D%) ÷ 100.

⇒ Option 1: Effective Discount = 44 + 18 − (44 × 18 ÷ 100) = 62 − 7.92 = 54.08%.
⇒ Option 2: Effective Discount = 30 + 30 − (30 × 30 ÷ 100) = 60 − 9 = 51%.
⇒ Option 3: Effective Discount = 22%.
⇒ Option 4: Effective Discount = 24 + 25 − (24 × 25 ÷ 100) = 49 − 6 = 43%.
⇒ Comparing all values, 54.08% is the maximum discount.

∴ The correct answer is Two successive discounts of 44% and 18%.

Additional Information

Successive Discounts

Applying two or more discounts one after another on the remaining balance. Total discount is always less than the simple sum of the rates.

Customer Profit

A customer gains maximum advantage when the Net Effective Discount is the highest, resulting in the lowest Selling Price.

Shortcut Trick

Let the total work be LCM of 3, 12, and 4 = 12 units.

Efficiency of A = 4, B = 1, and C = 3.

Total efficiency of A + B + C = 4 + 1 + 3 = 8 units/day.

Twice the work = 2 × 12 = 24 units.

Time taken = 24 ÷ 8 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given:

Time taken by A = 3 days

Time taken by B = 12 days

Time taken by C = 4 days

Formula Used:

Efficiency = Total Work ÷ Time

Time = Total Work ÷ Efficiency

Calculations:

⇒ Total Efficiency of A + B + C = 4 + 1 + 3 = 8 units/day

⇒ New Total Work = 2 × 12 = 24 units

⇒ Time required for twice the work = 24 ÷ 8 = 3 days

∴ The correct answer is 3 days.

Additional Information

LCM Method in Time and Work

Taking the LCM of the individual days to represent the total work helps avoid working with complex fractions, making computations much faster.

Work and Efficiency Relation

Efficiency is inversely proportional to time taken when the work is constant. If work is multiplied by a factor k, the time taken with the same efficiency also gets multiplied by k.

Shortcut Trick

Income ratio = 5 : 8 | Expenditure ratio = 1 : 4

Difference in Income parts = 8 − 5 = 3 | Difference in Expenditure parts = 4 − 1 = 3

Since the differences are equal, the saving of 81,000 corresponds to the difference between Income and Expenditure units.

Saving = 5 − 1 = 4 units (or 8 − 4 = 4 units) → 4 units = ₹81,000

1 unit = ₹20,250 ∴ Shivam income (5 units) = 5 × ₹20,250 = ₹1,01,250.

∴ The correct answer is ₹1,01,250.

Alternate Method

Given: Ratio of Incomes = 5 : 8 | Ratio of Expenditure = 1 : 4 | Savings = ₹81,000 each.

Formula Used: Income − Savings = Expenditure

⇒ Let Shivam income = 5x and Rupesh income = 8x.

⇒ (5x − 81000) / (8x − 81000) = 1 / 4

⇒ 4 × (5x − 81000) = 1 × (8x − 81000)

⇒ 20x − 3,24,000 = 8x − 81,000

⇒ 12x = 2,43,000

⇒ x = 20,250

⇒ Shivam Income = 5 × 20,250 = ₹1,01,250.

∴ The correct answer is ₹1,01,250.

Additional Information

Income-Expenditure Concept

Basic relation is Income = Expenditure + Savings. For two people with equal savings, the difference in income units should ideally equal the difference in expenditure units to use direct subtraction.

Ratio Balancing

If the differences between parts are not equal, multiply the ratios by the difference of the other ratio to balance them (e.g., multiply 5:8 by (4-1) and 1:4 by (8-5)).

Cross-Multiplication Method

A faster way to solve ratio problems with fixed additions/subtractions: (Initial Ratio 1 × Final Ratio 2) − (Initial Ratio 2 × Final Ratio 1) units = Change × (Final Ratio 2 − Final Ratio 1).

Shortcut Trick

Apply the standard algebraic identity: (a + b)² − (a − b)² = 4ab

By comparing, let a = x and b = 5:

⇒ 4 × x × 5 = 80

⇒ 20x = 80 → x = 4

∴ The correct answer is 4.

Alternate Method

Given:

(x + 5)² − (x − 5)² = 80

Formula Used:

Expansion of square of a binomial:

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

Calculations:

Expanding the terms on the left-hand side of the equation:

⇒ (x² + 2 × x × 5 + 5²) − (x² − 2 × x × 5 + 5²) = 80

⇒ (x² + 10x + 25) − (x² − 10x + 25) = 80

⇒ x² + 10x + 25 − x² + 10x − 25 = 80

Grouping the like terms together:

⇒ (x² − x²) + (10x + 10x) + (25 − 25) = 80

⇒ 20x = 80

⇒ x = 80 ÷ 20

⇒ x = 4

∴ The correct answer is 4.

Additional Information

Difference of Squares Identity

The expression can also be factored using the identity a² − b² = (a − b)(a + b), where a = (x + 5) and b = (x − 5).

Sum of Squares Identity

Adding two squared binomials results in another standard identity: (a + b)² + (a − b)² = 2(a² + b²).

Shortcut Trick

Identify Time using the direct ratio of Distance and Speed.

Time = 180 ÷ 72

Time = 90 ÷ 36 = 10 ÷ 4 = 2.5 hours.

∴ The correct answer is 2.5 hr.

Alternate Method

Given:

Speed of the train = 72 km/hr

Total Distance to cover = 180 km

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Time = 180 / 72

⇒ Time = (180 ÷ 12) / (72 ÷ 12)

⇒ Time = 15 / 6

⇒ Time = 2.5 hours

∴ The correct answer is 2.5 hr.

Additional Information

Unit Conversion (km/hr to m/s)

To convert speed from km/hr to m/s, multiply by 5/18. Conversely, multiply by 18/5 to convert m/s to km/hr.

Average Speed

For a journey with two equal distances at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

Relative Speed

When two objects move in the same direction, relative speed is the difference (S₁ − S₂); in opposite directions, it is the sum (S₁ + S₂).

Shortcut Trick

Ratio of Curved Surface Area (CSA) of Cylinder to sum of CSA of Cones = 4 : 1

2πrh : (πr₁l₁ + πr₂l₂) = 4 : 1

24r : (3 × 5 + 6 × 10) = 4 : 1

24r : 75 = 4 : 1 ⇒ 24r = 300 ⇒ r = 12.5

Calculating 2r = 2 × 12.5 = 25

∴ The correct answer is 25.

Alternate Method

Given:

Cylinder: Height (H) = 12 cm, Radius = r cm

Cone 1: Radius (r₁) = 3 cm, Height (h₁) = 4 cm

Cone 2: Radius (r₂) = 6 cm, Height (h₂) = 8 cm

Formula Used:

Curved Surface Area of Cylinder = 2πrH

Curved Surface Area of Cone = πrl, where slant height l = √(r² + h²)

Calculations:

⇒ Slant height of Cone 1 (l₁) = √(3² + 4²) = √(9 + 16) = 5 cm

⇒ Slant height of Cone 2 (l₂) = √(6² + 8²) = √(36 + 64) = 10 cm

⇒ CSA of Cone 1 = π × 3 × 5 = 15π cm²

⇒ CSA of Cone 2 = π × 6 × 10 = 60π cm²

⇒ Sum of CSA of two cones = 15π + 60π = 75π cm²

⇒ CSA of Cylinder = 2 × π × r × 12 = 24πr cm²

⇒ Given ratio: 24πr / 75π = 4 / 1

⇒ 24r / 75 = 4

⇒ 24r = 300

⇒ r = 300 / 24 = 12.5 cm

⇒ Value of 2r = 2 × 12.5 = 25

∴ The correct answer is 25.

Additional Information

Total Surface Area of Cylinder

It includes the curved area and the areas of the two circular bases: TSA = 2πr(r + h).

Volume of Solids

Volume of Cylinder = πr²H and Volume of Cone = (1/3)πr²h.

Pythagorean Triplets

Slant height calculation often involves common triplets like (3, 4, 5) and (6, 8, 10).

Shortcut Trick

Ratio of Meena : Sita is 5 : 4 (Gap = 1 unit) and becomes 9 : 8 (Gap = 1 unit) after 16 years.

The increase in units for both is 9 − 5 = 4 units or 8 − 4 = 4 units.

4 units → 16 years ⇒ 1 unit → 4 years.

Meena's present age = 5 units × 4 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given:

Present ages of Meena and Sita are in ratio 5 : 4. Let ages be 5x and 4x.

Calculations:

⇒ (5x + 16) / (4x + 16) = 9 / 8

⇒ 8 × (5x + 16) = 9 × (4x + 16)

⇒ 40x + 128 = 36x + 144

⇒ 40x − 36x = 144 − 128

⇒ 4x = 16

⇒ x = 4

⇒ Meena's present age = 5x = 5 × 4 = 20 years.

∴ The correct answer is 20 years.

Additional Information

Ratio Adjustment Method

If the difference between ratios is not same, multiply the ratios by the difference of the other ratio to make the age gap constant.

Age Difference Property

The difference between the ages of two persons remains constant regardless of the number of years passed.

Cross-Multiplication Method

For age problems, (Initial Ratio Product Diff) × x = (Final Ratio Diff) × (Years Passed).

Shortcut Trick

Effective interest rate for 2 years at 8% = 8 + 8 + (8 × 8)/100 = 16.64%

Net Amount percentage = 100% (Principal) + 16.64% (Interest) = 116.64%

Amount = 7500 × 116.64/100 = 75 × 116.64

Amount = ₹8,748

∴ The correct answer is ₹8,748.

Alternate Method

Given:

Principal (P) = ₹7,500

Rate of Interest (R) = 8% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 7500 × (1 + 8/100)²

⇒ A = 7500 × (108/100) × (108/100)

⇒ A = 7500 × (27/25) × (27/25)

⇒ A = 300 × (27/25) × 27

⇒ A = 12 × 27 × 27

⇒ A = 12 × 729

⇒ A = 8748

∴ The correct answer is ₹8,748.

Additional Information

Difference between CI and SI

For a period of 2 years, the difference between Compound Interest and Simple Interest is given by P × (R/100)².

Effective Rate Formula

If interest is compounded annually for 2 years at r%, the effective rate is [r + r + (r × r)/100]%.

Compounding Periods

If interest is compounded half-yearly, the rate becomes R/2 and time becomes 2n. If quarterly, rate becomes R/4 and time becomes 4n.

Shortcut Trick

Required percentage = (Value ÷ Base Value) × 100

Here, Value = 1/20 and Base Value = 10/30 = 1/3

Required percentage = (1/20 ÷ 1/3) × 100 = (3/20) × 100

⇒ 3 × 5 = 15%

∴ The correct answer is 15%.

Alternate Method

Given:

First fraction = 1/20

Second fraction = 10/30

Formula Used:

Percentage = (A / B) × 100

Calculations:

⇒ First fraction = 1/20

⇒ Second fraction = 10/30 = 1/3

⇒ Required % = [(1/20) / (1/3)] × 100

⇒ Required % = (1/20 × 3/1) × 100

⇒ Required % = (3/20) × 100

⇒ Required % = 3 × 5

⇒ Required % = 15%

∴ The correct answer is 15%.

Additional Information

Percentage Change

Percentage change = [(New value − Old value) / Old value] × 100.

Fraction to Percentage

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

Base Change Rule

If A is x% more than B, then B is [x / (100 + x)] × 100% less than A.

Shortcut Trick

Let the total eligible voters be 100 units.

Voters aged 18−21 who voted = 5% × 80% = 4 units.

Voters aged 21−25 who voted = 12% × 60% = 7.2 units.

Difference in voters = 7.2 − 4 = 3.2 units → 320.

Difference in eligible voters = 12% − 5% = 7 units → 7 × 100 = 700.

∴ The correct answer is 700.

Alternate Method

Given:

Total eligible voters = X

Voters aged 18−21 = 5% of X = 0.05X

Voters aged 21−25 = 12% of X = 0.12X

⇒ Actual voters (18−21) = 80% × 0.05X = 0.04X

⇒ Actual voters (21−25) = 60% × 0.12X = 0.072X

⇒ Difference in actual voters = 0.072X − 0.04X = 0.032X

⇒ 0.032X = 320

⇒ X = 320 ÷ 0.032 = 10,000

⇒ Difference in eligible voters (21-25 and 18-21) = (12% − 5%) of X

⇒ 7% of 10,000 = 0.07 × 10,000 = 700

∴ The correct answer is 700.

Additional Information

Percentage of a Percentage

To find the effective percentage of a subgroup, multiply the two percentages: Effective % = (Subgroup % ÷ 100) × (Total Group % ÷ 100) × 100.

Successive Percentage Change

When a value is increased/decreased by a% and then b%, the net change is given by [a + b + (ab/100)]%.

Base Value Identification

Always identify the base (100%) correctly; in election problems, the base is usually the total number of eligible voters unless specified otherwise.

Shortcut Trick

Teacher Age = New Average + (Initial Number of Students × Increase in Average)

Increase in average = 35 − 34 = 1 year

Initial students = 31

Teacher Age = 35 + (31 × 1) = 35 + 31 = 66 years

∴ The correct answer is 66 years.

Alternate Method

Given:

Number of students = 31

Initial average age = 34 years

New average age (including teacher) = 35 years

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total age of 31 students = 31 × 34 = 1054 years

⇒ Total number of people including teacher = 31 + 1 = 32

⇒ Total age of 32 people = 32 × 35 = 1120 years

⇒ Teacher age = Total age (32 people) − Total age (31 students)

⇒ Teacher age = 1120 − 1054 = 66 years

∴ The correct answer is 66 years.

Additional Information

Basic Average Formula

The average of a set of numbers is the sum of the numbers divided by the count of numbers: Average = Σx ÷ n.

Net Deviation Concept

In any group, the sum of deviations from the average is always zero. This is the basis for the Shortcut Trick used above.

Inclusion and Exclusion

If a new person joins and the average increases, the new person age must be greater than the initial average. If the average decreases, the age is less than the initial average.

Shortcut Trick

Net Discount % = a + b − (a × b) ÷ 100

Net Discount = 36 + 25 − (36 × 25) ÷ 100 = 61 − 9 = 52%

Selling Price (SP) = 100% − 52% = 48% of Marked Price (MP)

Given SP = ₹3,300 ⇒ 48% = 3,300

MP = (3,300 ÷ 48) × 100 = 6,875

∴ The correct answer is ₹6,875.

Alternate Method

Given: SP = ₹3,300, d₁ = 36%, d₂ = 25%

Formulas: SP = MP × (100 − d₁)% × (100 − d₂)%

⇒ Let Marked Price be x.

⇒ x × (64 ÷ 100) × (75 ÷ 100) = 3,300

⇒ x × (16 ÷ 25) × (3 ÷ 4) = 3,300

⇒ x × (12 ÷ 25) = 3,300

⇒ x = (3,300 × 25) ÷ 12

⇒ x = 275 × 25 = 6,875

∴ The correct answer is ₹6,875.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP. SP = MP − Discount.

Successive Discounts

When multiple discounts are applied one after another, the first discount is on the MP, and the subsequent discount is on the reduced price.

Shortcut Trick

List the first 11 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.

Group them for easy addition: (2 + 3 + 5 + 7 + 13 + 17) + (11 + 19) + (23 + 29) + 31

⇒ 47 + 30 + 52 + 31 = 160

∴ The correct answer is 160.

Alternate Method

Given: We need to find the sum of the first 11 prime numbers.

Formula: Sum = P₁ + P₂ + P₃ + ... + P₁₁

⇒ First 11 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31

⇒ Sum = (2 + 3 + 5 + 7) + (11 + 13 + 17 + 19) + (23 + 29 + 31)

⇒ Sum = 17 + 60 + 83

⇒ Sum = 160

∴ The correct answer is 160.

Additional Information

Definition of Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Properties of Prime Numbers

The number 2 is the only even prime number. All other prime numbers are odd. The number 1 is neither prime nor composite.

Sum of First N Prime Numbers

There is no simple algebraic formula to find the sum of the first N prime numbers; they must be listed and added manually.

Shortcut Trick

In any triangle, the angle formed by the internal angle bisectors of two angles at the incenter (O) is given by the formula:

∠RON = 90° + (∠M / 2)

Given ∠M = 60°:

∠RON = 90° + (60° / 2) = 90° + 30° = 120°

∴ The correct answer is 120°.

Alternate Method

Given: ∠M = 60°, ∠MRN = 66°, NO and RO are bisectors of ∠MNR and ∠MRN respectively.

Formula Used: Sum of all interior angles of a triangle is 180°.

Calculations:

⇒ In ΔMNR, ∠MNR + ∠MRN + ∠M = 180°

⇒ ∠MNR + 66° + 60° = 180°

⇒ ∠MNR = 180° − 126° = 54°

⇒ Since NO is the bisector of ∠MNR, ∠ONR = 54° / 2 = 27°

⇒ Since RO is the bisector of ∠MRN, ∠ORN = 66° / 2 = 33°

⇒ In ΔRON, ∠RON + ∠ONR + ∠ORN = 180°

⇒ ∠RON + 27° + 33° = 180°

⇒ ∠RON = 180° − 60° = 120°

∴ The correct answer is 120°.

Additional Information

Incenter

The point of intersection of the internal angle bisectors of a triangle is called the Incenter. It is equidistant from all sides of the triangle.

Angle at Incenter

The angle formed by the bisectors of two angles at the incenter is always 90° + (half of the third angle).

Angle at Excenter

The angle formed by the external bisectors of two angles at the excenter is 90° − (half of the third angle).

Shortcut Trick

Efficiency ratio (Square of diameters) → 2² : 3² : 4² = 4 : 9 : 16

Total Capacity = Efficiency of largest pipe × Time = 16 × 87

Combined Efficiency = 4 + 9 + 16 = 29

Required Time = (16 × 87) ÷ 29 = 16 × 3 = 48 minutes

∴ The correct answer is 48 minutes.

Alternate Method

Given: Diameters of pipes are 2 cm, 3 cm, and 4 cm. Largest pipe time = 87 minutes.

Formula Used: Rate of flow (Efficiency) = k × (Diameter)²; Total Work = Efficiency × Time

⇒ Efficiency of Pipe 1 (E₁) = 2² = 4 units/min

⇒ Efficiency of Pipe 2 (E₂) = 3² = 9 units/min

⇒ Efficiency of Pipe 3 (E₃) = 4² = 16 units/min (Largest Pipe)

⇒ Total capacity of cistern = E₃ × 87 = 16 × 87 = 1392 units

⇒ Total Efficiency when all open = 4 + 9 + 16 = 29 units/min

⇒ Time taken = 1392 ÷ 29 = 48 minutes

∴ The correct answer is 48 minutes.

Additional Information

Pipes and Cisterns Efficiency

The rate of filling or emptying a tank is inversely proportional to the time taken. Efficiency = Work ÷ Time.

Flow Rate and Diameter

In fluid dynamics, the volume flow rate is proportional to the area of cross-section, which is proportional to the square of the diameter (d²).

Combined Work Formula

If multiple pipes operate together, their individual efficiencies are added: E_total = E₁ + E₂ + ... + E_n.

Shortcut Trick

Convert bases: 256=2⁸, 64=2⁶; 125=5³, 25=5²; 81=3⁴, 27=3³

Combine powers ⇒ 2³²/2³⁰=2², 5⁹/5⁸=5, 3²⁰/3¹⁸=3²

Multiply ⇒ 2² × 5 × 3² = 4 × 5 × 9

= 180

∴ Answer = 180

Alternate Method

Given:

(256)⁴ × (125)³ × (81)⁵ ÷ [(64)⁵ × (25)⁴ × (27)⁶]

Concept Used:

BODMAS rule and laws of exponents

Calculation:

⇒ = (2⁸)⁴ × (5³)³ × (3⁴)⁵ ÷ [(2⁶)⁵ × (5²)⁴ × (3³)⁶]

⇒ = 2³² × 5⁹ × 3²⁰ ÷ [2³⁰ × 5⁸ × 3¹⁸]

⇒ = 2^(32−30) × 5⁽⁹⁻⁸⁾ × 3^(20−18)

⇒ = 2² × 5 × 3²

⇒ = 4 × 5 × 9

⇒ = 180

∴ The correct answer is 180.

Additional Information

Exponent Law

a^m ÷ aⁿ = a^(m−n)

Power Rule

(a^m)ⁿ = a^mn

Prime Factorization

Convert numbers into primes for easy simplification.

Shortcut Trick

Since Volume ∝ r³, use successive percentage formula for 15% three times.

Successive of 15% and 15% = 15 + 15 + (15 × 15)/100 = 32.25%.

Successive of 32.25% and 15% = 32.25 + 15 + (32.25 × 15)/100 = 47.25 + 4.8375 = 52.0875%.

Rounding up to 2 decimal places = 52.09%.

∴ The correct answer is 52.09%.

Alternate Method

Given:

⇒ Increase in Radius (r) = 15%

⇒ Volume of Sphere (V) = (4/3) × π × r³

Calculations:

⇒ Let the Initial Radius (r₁) = 100 units

⇒ New Radius (r₂) = 100 + 15% of 100 = 115 units

⇒ Initial Volume (V₁) = (4/3) × π × (100)³ = 1,000,000 × (4/3)π

⇒ New Volume (V₂) = (4/3) × π × (115)³ = 1,520,875 × (4/3)π

⇒ Increase in Volume = V₂ − V₁ = 520,875 × (4/3)π

⇒ Percentage Increase = (Increase ÷ V₁) × 100

⇒ Percentage Increase = (520,875 ÷ 1,000,000) × 100 = 52.0875%

⇒ Rounded up to 2 decimal places = 52.09%

∴ The correct answer is 52.09%.

Additional Information

Surface Area Increase

If the radius of a sphere increases by x%, the surface area increases by [2x + (x²/100)]%.

Successive Percentage for Three Variables

For changes a%, b%, and c%, the net change is [a + b + c + (ab+bc+ca)/100 + (abc)/10000]%.

Volume and Radius Proportionality

For a sphere, the Volume is proportional to the cube of the radius (V ∝ r³), while for a circle, the Area is proportional to the square of the radius (A ∝ r²).

Shortcut Trick
Ratio of CP to SP is inversely proportional to the Quantity.
Quantity Ratio (CP : SP) = 30 : 40.
Price Ratio (CP : SP) = 40 : 30 = 4 : 3.
Loss = CP − SP = 4 − 3 = 1 unit.
Loss% = (Loss ÷ CP) × 100 = (1 ÷ 4) × 100 = 25%.
∴ The correct answer is 25%.

Alternate Method
Given: Cost Price (CP) of 30 chairs = Selling Price (SP) of 40 chairs.
Formulas: Loss = CP − SP | Loss% = (Loss ÷ CP) × 100

⇒ Let the CP of 1 chair be Rs.

1. ⇒ Total CP of 30 chairs = Rs.
2. ⇒ According to the question, SP of 40 chairs = CP of 30 chairs = Rs.
3. ⇒ CP of 40 chairs = Rs.
4. ⇒ Loss on 40 chairs = CP − SP = 40 − 30 = Rs. 10.
   ⇒ Loss Percentage = (10 ÷ 40) × 100 = 25%.
   ∴ The correct answer is 25%.

Additional Information
Cost Price and Selling Price Relation
When CP of x articles = SP of y articles, then Profit or Loss% = [(x − y) ÷ y] × 100. A positive result indicates profit, while a negative result indicates loss.
Loss Condition
Loss occurs when the Selling Price is less than the Cost Price (SP < CP).

Shortcut Trick

To find the smaller number when the sum (S) and difference (D) are given:

Smaller Number = (Sum − Difference) ÷ 2

⇒ (45 − 9) ÷ 2

⇒ 36 ÷ 2 = 18

∴ The correct answer is 18.

Alternate Method

Given:

Let the larger number be x and the smaller number be y.

Sum: x + y = 45 --- (Equation 1)

Difference: x − y = 9 --- (Equation 2)

Calculations:

Subtracting (Equation 2) from (Equation 1):

⇒ (x + y) − (x − y) = 45 − 9

⇒ x + y − x + y = 36

⇒ 2y = 36

⇒ y = 36 ÷ 2

⇒ y = 18

∴ The correct answer is 18.

Additional Information

Finding the Larger Number

The larger number can be found using the formula: (Sum + Difference) ÷ 2. Here, (45 + 9) ÷ 2 = 27.

Product of Two Numbers

If the sum (S) and difference (D) are known, the product is (S² − D²) ÷ 4.

Ratio of Two Numbers

The ratio of the larger number to the smaller number is (Sum + Difference) : (Sum − Difference).

Shortcut Trick

Convert 7 days to hours by multiplying with 24.

Ratio = 20 : (7 × 24) = 20 : 168.

Divide both sides by the HCF (4) → (20 ÷ 4) : (168 ÷ 4) = 5 : 42.

∴ The correct answer is 5:42.

Alternate Method

Given: First quantity = 20 hours; Second quantity = 7 days.

Formula Used: Ratio = Quantity 1 ÷ Quantity 2 (Units must be identical).

⇒ Quantity 1 = 20 hours

⇒ Quantity 2 = 7 × 24 hours = 168 hours

⇒ Required Ratio = 20 ÷ 168

⇒ Required Ratio = 5 ÷ 42

∴ The correct answer is 5:42.

Additional Information

Unit Homogeneity

Before calculating a ratio, ensure both quantities are in the same unit (e.g., both in hours or both in days).

Time Conversion

Standard conversion: 1 day = 24 hours; 1 hour = 60 minutes; 1 minute = 60 seconds.

Simplest Form

A ratio a:b is in its simplest form if the Greatest Common Divisor (GCD) of a and b is 1.