Quantitative Aptitude PYQs

Shortcut Trick

Total amount to be deducted = ₹(7 + 41 + 12) = ₹60.

Remaining amount to divide = ₹1386 − ₹60 = ₹1326.

Sum of ratio parts = 4 + 10 + 12 = 26 units.

Value of 1 unit = 1326 ÷ 26 = ₹51.

Kajal's share (after deduction) = 10 × 51 = ₹510.

Kajal's actual share = ₹510 + ₹41 = ₹551.

∴ The correct answer is ₹551.

Alternate Method

Given: Total sum = ₹1386, Deductions = ₹7, ₹41, ₹12, Ratio = 4 : 10 : 12.

Formula: Actual Share = (Ratio Part × Value per unit) + Deducted Amount.

⇒ Total Deducted Amount = 7 + 41 + 12 = ₹60.

⇒ New Sum = 1386 − 60 = ₹1326.

⇒ Let the parts be 4x, 10x, and 12x.

⇒ 4x + 10x + 12x = 1326 ⇒ 26x = 1326.

⇒ x = 1326 ÷ 26 = 51.

⇒ Kajal's share after deduction = 10x = 10 × 51 = ₹510.

⇒ Kajal's actual share = 510 + 41 = ₹551.

∴ The correct answer is ₹551.

Additional Information

Ratio and Proportion

If a sum S is divided in ratio a:b:c, the share of b is [b ÷ (a+b+c)] × S.

Deduction Rule

When fixed amounts are subtracted from shares, subtract the total deduction from the original sum before applying ratio division.

Compound Ratio

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

Shortcut Trick

Convert the mixed fraction to decimal: 24 3/5 = 24.6

Perform the addition inside the brackets: 24.6 + 3.2 = 27.8

Simplify the expression: 4.05 × (27.8 ÷ 8.34)

Observe that 4.05 ÷ 8.34 = 135 ÷ 278 (after dividing both by 0.03).

Calculation: (135 ÷ 278) × 27.8 = 135 × 0.1 = 13.5

∴ The correct answer is 13.5.

Alternate Method

Given: Expression = 4.05 × ( (24 3/5 + 3.2) ÷ 8.34 )

Formula Used: BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ Step 1: Solve the mixed fraction 24 3/5 = (24 × 5 + 3) ÷ 5 = 123 ÷ 5 = 24.6
⇒ Step 2: Addition inside the parentheses = 24.6 + 3.2 = 27.8
⇒ Step 3: Expression becomes 4.05 × (27.8 ÷ 8.34)
⇒ Step 4: Divide the terms = 27.8 ÷ 8.34 = 2780 ÷ 834 ≈ 3.333
⇒ Step 5: Final multiplication = 4.05 × (27.8 ÷ 8.34) = (4.05 × 27.8) ÷ 8.34
⇒ Step 6: 112.59 ÷ 8.34 = 13.5

∴ The correct answer is 13.5.

Additional Information

BODMAS Rule

Always solve expressions in the order of Brackets, Of/Orders, Division, Multiplication, Addition, and Subtraction to avoid errors.

Mixed Fraction Conversion

A mixed fraction a b/c is converted to an improper fraction using the formula (a × c + b) ÷ c.

Decimal Point Removal

To simplify division of decimals, multiply both the numerator and denominator by 10ⁿ, where n is the maximum number of decimal places.

Shortcut Trick

Total Surface Area (TSA) of 1 Big Box = 2 × (30×25 + 25×10 + 10×30) = 2600 cm²

TSA of 1 Small Box = 2 × (18×12 + 12×10 + 10×18) = 1032 cm²

Total Cardboard required = 1.06 × (400 × 2600 + 500 × 1032) = 1.06 × 1,556,000 = 1,649,360 cm²

Total Cost = (1,649,360 ÷ 10,000) × 45 = ₹7,422.12 ≈ ₹7,400

∴ The correct answer is ₹7,400.

Alternate Method

Given:

Big Box: Length (l₁) = 30 cm, Breadth (b₁) = 25 cm, Height (h₁) = 10 cm | Quantity = 400

Small Box: Length (l₂) = 18 cm, Breadth (b₂) = 12 cm, Height (h₂) = 10 cm | Quantity = 500

Cardboard Cost: ₹45 per 10,000 cm² | Overlap: 6% extra

Formula Used: Total Surface Area (TSA) of Cuboid = 2(lb + bh + lh)

⇒ TSA of 400 big boxes = 400 × [2 × (30×25 + 25×10 + 30×10)]

⇒ 400 × [2 × (750 + 250 + 300)] = 400 × 2600 = 1,040,000 cm²

⇒ TSA of 500 small boxes = 500 × [2 × (18×12 + 12×10 + 18×10)]

⇒ 500 × [2 × (216 + 120 + 180)] = 500 × 1032 = 516,000 cm²

⇒ Total TSA = 1,040,000 + 516,000 = 1,556,000 cm²

⇒ Area with 6% overlap = 1,556,000 × (106 ÷ 100) = 1,649,360 cm²

⇒ Total Cost = (1,649,360 ÷ 10,000) × 45

⇒ Total Cost = 164.936 × 45 = ₹7,422.12

⇒ Rounding off to the nearest hundreds, Cost = ₹7,400.

∴ The correct answer is ��7,400.

Additional Information

Total Surface Area of Cuboid

The total area covering all six faces: TSA = 2(lb + bh + hl).

Lateral Surface Area of Cuboid

The area of the four side walls, excluding top and bottom: LSA = 2h(l + b).

Volume and Diagonal

Volume = l × b × h | Diagonal = √(l² + b² + h²).

Unit Conversion

1 m² = 10,000 cm². In cost calculations, always ensure the area and rate units match.

Shortcut Trick

When distance is constant, Speed is inversely proportional to Time.

Speed ratio = 5 : 6 → Time ratio = 6 : 5

Difference in time ratio = 6 − 5 = 1 unit

1 unit → 16 minutes

Usual time (5 units) = 5 × 16 = 80 minutes

∴ The correct answer is 80 min.

Alternate Method

Given: New speed = 5/6 of usual speed, Time delay = 16 minutes.

Formula Used: Distance = Speed × Time

⇒ Let Usual Speed = s and Usual Time = t

⇒ Distance (D) = s × t

⇒ New Speed = 5/6s

⇒ New Time = t + 16

⇒ D = (5/6s) × (t + 16)

⇒ st = 5/6s(t + 16)

⇒ t = 5/6(t + 16)

⇒ 6t = 5t + 80

⇒ t = 80 minutes

∴ The correct answer is 80 min.

Additional Information

Inverse Proportionality Rule

If speed changes in the ratio a:b, then time taken for the same distance changes in the ratio b:a.

Late/Early Logic

Usual Time = (Late Time × Numerator) ÷ (Denominator − Numerator) when speed becomes a/b of original.

Average Speed

For constant distance, Average Speed = (2 × s₁ × s₂) ÷ (s₁ + s₂).

Shortcut Trick

Income ratio = 5 : 9, Expenditure ratio = 2 : 4 (or 1 : 2).

Since savings are equal (₹45,000), make the difference between income and expenditure parts equal.

Income parts: 5, 9 (diff = 4). Expenditure parts: 2, 4 (diff = 2). Multiply Expenditure by 2 → 4 : 8.

Now, Saving = Income − Expenditure = 5 − 4 = 1 unit or 9 − 8 = 1 unit.

1 unit = ₹45,000. Chetan income (5 units) = 5 × 45,000 = ₹2,25,000.

∴ The correct answer is ₹2,25,000.

Alternate Method

Given: Ratio of Income = 5 : 9 | Ratio of Expenditure = 2 : 4 | Savings = ₹45,000 each.

Formula Used: Income − Savings = Expenditure

⇒ Let Chetan income be 5x and Vipul income be 9x.

⇒ (5x − 45000) / (9x − 45000) = 2 / 4

⇒ (5x − 45000) / (9x − 45000) = 1 / 2

⇒ 2 × (5x − 45000) = 1 × (9x − 45000)

⇒ 10x − 90000 = 9x − 45000

⇒ 10x − 9x = 90000 − 45000

⇒ x = 45000

⇒ Monthly income of Chetan = 5x = 5 × 45000 = ₹2,25,000.

∴ The correct answer is ₹2,25,000.

Additional Information

Income, Expenditure and Savings Relationship

The basic accounting identity used is: Income = Expenditure + Savings. If savings are constant, the difference between income and expenditure ratios remains proportional.

Ratio and Proportion Property

When a constant value is added or subtracted from both terms of a ratio (a:b), the new ratio (c:d) can be solved using the cross-multiplication method: x = [K × (c − d)] / [ad − bc].

Base Change in Percentage

Savings as a percentage of income is calculated as (Savings / Income) × 100. This is a common variation in bank exams.

Shortcut Trick

Ratio of Father : Son now is 4 : 1 (difference = 3 units).

Ratio of Father : Son after 18 years is 2 : 1 (difference = 1 unit).

Make the difference equal: New ratio after 18 years = 6 : 3 (difference = 3 units).

Increase in units = 6 − 4 = 2 units. These 2 units = 18 years ⇒ 1 unit = 9 years.

Present age of father = 4 units × 9 = 36 years.

∴ The correct answer is 36 years.

Alternate Method

Given:

Present age of Son = x years

Present age of Father = 4x years

According to the question, after 18 years:

⇒ (4x + 18) = 2 × (x + 18)

⇒ 4x + 18 = 2x + 36

⇒ 4x − 2x = 36 − 18

⇒ 2x = 18

⇒ x = 9

⇒ Father's present age = 4x = 4 × 9 = 36 years.

∴ The correct answer is 36 years.

Additional Information

Age Ratio Constant Difference

The numerical difference between the ages of two people always remains constant, regardless of how many years pass.

Future and Past Calculations

To find age 'n' years hence, add n to the present age. To find age 'n' years ago, subtract n from the present age.

Ratio Balancing

When comparing ratios at different times, multiply the terms of one ratio by the difference of the other ratio to balance the units.

Shortcut Trick

For a number to be divisible by 3, the sum of its digits must be divisible by 3.

Sum of given digits = a + 3 + 7 + 2 + 4 = a + 16.

Nearest multiple of 3 greater than 16 is 18.

⇒ a + 16 = 18 ⇒ a = 2.

∴ The correct answer is 2.

Alternate Method

Given: The number is a3724.

Condition: The number must be divisible by 3.

Divisibility Rule of 3: Sum of all digits ÷ 3 should leave remainder 0.

⇒ Sum = a + 3 + 7 + 2 + 4

⇒ Sum = a + 16

⇒ For (a + 16) to be divisible by 3, the possible values for (a + 16) are 18, 21, 24, ...

⇒ If a + 16 = 18, then a = 18 − 16 = 2

⇒ If a + 16 = 21, then a = 21 − 16 = 5

⇒ If a + 16 = 24, then a = 24 − 16 = 8

The smallest value among these is 2.

∴ The correct answer is 2.

Additional Information

Divisibility Rule of 9

Similar to the rule for 3, a number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility Rule of 4

A number is divisible by 4 if the last two digits of the number are divisible by 4 or are both zero.

Divisibility Rule of 6

A number is divisible by 6 if it is simultaneously divisible by both 2 (even number) and 3 (sum of digits rule).

Shortcut Trick

Ratio of cost of TV to Washing Machine (WM) = 100 : 74.

Total cost ratio for 8 WM and 4 TV = (8 × 74) : (4 × 100) = 592 : 400 = 148 : 100.

Net change % = [(148 × 86%) + (100 × −38%)] ÷ (148 + 100).

Net change % = [127.28 − 38] ÷ 248 = 89.28 ÷ 248 = 0.36 or 36%.

Since the result is positive, there is an increase.

∴ The correct answer is Increase by 36%.

Alternate Method

Given: Cost of TV = 100, Cost of WM = 100 − 26 = 74. Quantity: 8 WM and 4 TV.

Formulas: Total Cost = (Price₁ × Qty₁) + (Price₂ × Qty₂) | % Change = (New − Old) ÷ Old × 100.

⇒ Initial Total Cost = (8 × 74) + (4 × 100) = 592 + 400 = 992.

⇒ New WM Cost = 74 × 1.86 = 137.64; New TV Cost = 100 × 0.62 = 62.

⇒ Final Total Cost = (8 × 137.64) + (4 × 62) = 1101.12 + 248 = 1349.12.

⇒ Change = 1349.12 − 992 = 357.12.

⇒ Percentage Change = (357.12 ÷ 992) × 100 = 36%.

∴ The correct answer is Increase by 36%.

Additional Information

Weighted Average Method

When components of a total change by different percentages, the net change is the weighted average of the individual percentage changes based on their initial values.

Percentage Change

Percentage change = [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive result indicates an increase, while a negative result indicates a decrease.

Shortcut Trick

For Compound Interest at 10% for 2 years, the effective interest rate is 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = 100% (Principal) + 21% (Interest) = 121% of Principal.

⇒ Amount = 3600 × 121% = 36 × 121 = ₹4356.

∴ The correct answer is ₹4356.

Alternate Method

Given:

Principal (P) = ₹3,600

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 3600 × (1 + 10/100)²

⇒ A = 3600 × (110/100)²

⇒ A = 3600 × (1.1)²

⇒ A = 3600 × 1.21

⇒ A = ₹4356

∴ The correct answer is ₹4356.

Additional Information

Effective Rate Formula

For two successive percentage changes x% and y%, the net change is given by (x + y + xy/100)%.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is P(R/100)².

Compound Interest Formula

Amount A = P(1 + r/n)^nt, where n is the number of times interest is compounded per year.

Shortcut Trick

Using the standard trigonometric identity: sec²x − tan²x = 1, we get tan²x − sec²x = −1.

Substitute this directly into the given expression: 3(tan²x − sec²x) + 4 = 3(−1) + 4 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

The trigonometric expression is: 3tan²x − 3sec²x + 4

Formula Used:

⇒ sec²x = 1 + tan²x

Calculations:

⇒ 3tan²x − 3(1 + tan²x) + 4

⇒ 3tan²x − 3 − 3tan²x + 4

⇒ −3 + 4

⇒ 1

∴ The correct answer is 1.

Additional Information

Fundamental Trigonometric Identities

The three main Pythagorean trigonometric identities are: sin²x + cos²x = 1, 1 + tan²x = sec²x, and 1 + cot²x = cosec²x.

Value Substitution Trick

For expressions independent of the angle x, we can substitute a convenient value such as x = 0°, giving 3(0) − 3(1) + 4 = 1.

Shortcut Trick

Put k = 2, m = 1

Convert bases smartly:

8 = 2³, 4 = 2² → they cancel largely

10 = 2×5 → will cancel with 5^2k

18 = 2×3², 27 = 3³

Now track powers:

2’s → (2⁶ × 2 × 2³) / 2⁶ = 2⁴

3’s → 3² / 3³ = 1/3

5’s → 5³ / 5⁴ = 1/5

Final = 2⁴ / (3×5) = 16/15 = \( 1\frac{1}{15} \)

∴ Answer = \( 1\frac{1}{15} \)

Alternate Method

Given:

\( \frac{8^k \times 18^m \times 10^{k+m}}{4^{k+m} \times 27^m \times 5^{2k}} \)

Concept Used:

\( \text{Laws of exponents and prime factorization} \)

Calculation:

\( \Rightarrow \frac{8^2 \times 18^1 \times 10^{2+1}}{4^{2+1} \times 27^1 \times 5^{2\times2}} \)

\( \Rightarrow \frac{8^2 \times 18 \times 10^3}{4^3 \times 27 \times 5^4} \)

\( \Rightarrow \frac{(2^3)^2 \times (2 \times 3^2) \times (2^3 \times 5^3)}{(2^2)^3 \times (3^3) \times 5^4} \)

\( \Rightarrow \frac{2^6 \times 2 \times 3^2 \times 2^3 \times 5^3}{2^6 \times 3^3 \times 5^4} \)

\( \Rightarrow \frac{2^{10} \times 3^2 \times 5^3}{2^6 \times 3^3 \times 5^4} \)

\( \Rightarrow 2^{10-6} \times 3^{2-3} \times 5^{3-4} \)

\( \Rightarrow 2^4 \times 3^{-1} \times 5^{-1} \)

\( \Rightarrow \frac{16}{3 \times 5} \)

\( \Rightarrow \frac{16}{15} \) = \( 1\frac{1}{15} \)

∴ The correct answer is \( 1\frac{1}{15} \)

Shortcut Trick

Mode is defined as the observation that appears most frequently in a data set.

Frequency of 1 = 3 | Frequency of 2 = 2 | Frequency of 3 = 4 | Frequency of 4 = 2 | Frequency of 9 = 2 | Frequency of 6 and 8 = 1 each.

Since the number 3 repeats the maximum number of times (4 times), it is the mode.

∴ The correct answer is 3.

Alternate Method

Given: Observations = 1, 1, 8, 3, 1, 3, 4, 3, 9, 9, 4, 3, 2, 2, 6

Formula Used: Mode = Value with the highest frequency.

⇒ Count of 1: 3

⇒ Count of 2: 2

⇒ Count of 3: 4

⇒ Count of 4: 2

⇒ Count of 6: 1

⇒ Count of 8: 1

⇒ Count of 9: 2

⇒ Maximum frequency = 4 (for observation 3).

∴ The correct answer is 3.

Additional Information

Empirical Relationship

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

Median

The middle value of a data set when arranged in ascending or descending order. For n observations, if n is odd, Median = ((n+1)/2)^th term.

Types of Mode

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

Shortcut Trick

Volume of a cylinder is proportional to r²h.

Multiplying factor for radius (r) = 1 − 0.28 = 0.72

Multiplying factor for height (h) = 1 + 1.14 = 2.14

Net change factor in volume = 0.72 × 0.72 × 2.14 = 1.109376

Percentage increase = (1.109376 − 1) × 100 = 10.9376% ≈ 11%.

∴ The correct answer is 11%.

Alternate Method

Given: Base radius decreased by 28%, Height increased by 114%.

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

⇒ Let initial radius (r₁) = 100 units and initial height (h₁) = 100 units.

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

⇒ New radius (r₂) = 100 − 28 = 72 units.

⇒ New height (h₂) = 100 + 114 = 214 units.

⇒ New Volume (V₂) = π × (72)² × 214 = π × 5184 × 214 = 1,109,376π cubic units.

⇒ Increase in Volume = 1,109,376π − 1,000,000π = 109,376π.

⇒ Percentage Increase = (109,376π / 1,000,000π) × 100 = 10.9376%.

⇒ Rounded to the nearest integer, the increase is 11%.

∴ The correct answer is 11%.

Additional Information

Volume of Cylinder

The volume of a cylinder is given by V = πr²h, where r is the base radius and h is the height.

Curved Surface Area (CSA)

The area of the curved surface of a cylinder is 2πrh.

Total Surface Area (TSA)

The total surface area of a solid cylinder is 2πr(r + h).

Net Percentage Change

If a value changes by a% and then by b%, the effective change is [a + b + (ab/100)]%.

Shortcut Trick

Identify the expression as the Standard Algebraic Form: ((a + b)² + (a − b)²) / (a² + b²).

Apply the identity (a + b)² + (a − b)² = 2(a² + b²).

The expression simplifies to 2(a² + b²) / (a² + b²), which equals 2.

The final value is independent of the numerical values of a and b.

∴ The correct answer is 2.

Alternate Method

Given:

Expression: ((112 + 99)² + (112 − 99)²) / (112 × 112 + 99 × 99)

Formula Used:

(a + b)² + (a − b)² = 2(a² + b²)

Calculations:

⇒ Let a = 112 and b = 99.

⇒ The Numerator is (a + b)² + (a − b)².

⇒ Expanding the squares: (a² + 2ab + b²) + (a² − 2ab + b²).

⇒ Combining like terms: 2a² + 2b² = 2(a² + b²).

⇒ The Denominator is (a × a + b × b) = (a² + b²).

⇒ Total Expression = 2(a² + b²) / (a² + b²).

⇒ Canceling common terms: 2.

∴ The correct answer is 2.

Additional Information

Square of Sum and Difference

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

Difference of Squares

a² − b² = (a + b)(a − b) is used to factorize the difference between two squared terms.

4ab Identity

Subtracting the square of difference from the square of sum yields (a + b)² − (a − b)² = 4ab.

BODMAS Rule

Always solve terms within Brackets first, then Orders (powers), followed by Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

Income = 100%

Expenditure = 60%, Savings = 40%

40% of Income = ₹90,000

Total Income = 90,000 ÷ 40 × 100 = 2,25,000

∴ The correct answer is ₹2,25,000.

Alternate Method

Given:

Expenditure = 60% of Income

Savings = ₹90,000

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let the total income be x.

⇒ Expenditure = 60% of x = 0.60x

⇒ Savings = x − 0.60x = 0.40x

⇒ 0.40x = 90,000

⇒ x = 90,000 ÷ 0.40

⇒ x = 9,000,000 ÷ 40

⇒ x = 2,25,000

∴ The correct answer is ₹2,25,000.

Additional Information

Percentage Change

Percentage change = (Difference ÷ Original Value) × 100.

Base Change Rule

If income increases by r%, then to return to original, it must decrease by [r/(100+r)] × 100%.

Income-Expenditure Relation

Income × (100 − Expenditure%) = Savings.

Shortcut Trick

In any quadrilateral PQRS, the angle formed by the bisectors of two adjacent angles (say, ∠R and ∠S) meeting at point T is always equal to half the sum of the other two angles:

∠STR = (∠P + ∠Q) ÷ 2

⇒ 24° = (∠P + ∠Q) ÷ 2

⇒ ∠P + ∠Q = 48°

Given the ratio of ∠P : ∠Q = 5 : 7, the difference between them is 2 parts out of a total of 12 parts:

⇒ Difference = (2 ÷ 12) × 48° = 8°

∴ The correct answer is 8°.

Alternate Method

Given:

⇒ In quadrilateral PQRS, bisectors of ∠R and ∠S meet at T.

⇒ ∠STR = 24°

⇒ Ratio of ∠P : ∠Q = 5 : 7

Formula Used:

⇒ Sum of angles in a triangle = 180°

⇒ Sum of interior angles of a quadrilateral = 360°

Calculations:

In triangle STR:

⇒ ∠TSR + ∠TRS + ∠STR = 180°

Since ST and RT are angle bisectors of ∠S and ∠R respectively:

⇒ (1/2)∠S + (1/2)∠R + 24° = 180°

⇒ (1/2)(∠S + ∠R) = 180° − 24°

⇒ (1/2)(∠S + ∠R) = 156°

⇒ ∠S + ∠R = 312°

In quadrilateral PQRS:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

Substitute the value of (∠S + ∠R):

⇒ ∠P + ∠Q + 312° = 360°

⇒ ∠P + ∠Q = 360° − 312°

⇒ ∠P + ∠Q = 48°

Let ∠P = 5x and ∠Q = 7x:

⇒ 5x + 7x = 48°

⇒ 12x = 48°

⇒ x = 4°

Now, find the difference between the measures of ∠Q and ∠P:

⇒ Difference = ∠Q − ∠P

⇒ Difference = 7x − 5x = 2x

⇒ Difference = 2 × 4° = 8°

∴ The correct answer is 8°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the intersection of the bisectors of any two adjacent angles of a quadrilateral is equal to half the sum of the remaining two interior angles.

Sum of Interior Angles of a Polygon

For any n-sided convex polygon, the sum of all interior angles is given by the formula (n − 2) × 180°. For a quadrilateral, n = 4, making the sum 360°.

Shortcut Trick
Effective Cost Price (CP) = 3,00,000 + 30,000 = ₹3,30,000.
Multiplier for 5% loss = (100 − 5) ÷ 100 = 0.95.
Selling Price (SP) = CP × Multiplier = 3,30,000 × 0.95.
SP = 3,30,000 × (1 − 0.05) = 3,30,000 − 16,500 = ₹3,13,500.
∴ The correct answer is ₹3,13,500.

Alternate Method
Given → Purchase Price = ₹3,00,000, Overhead Expenses = ₹30,000, Loss% = 5%.
Formulas: Total CP = Purchase Price + Overhead Expenses | SP = CP × (100 − Loss%) ÷ 100.

→ Total CP = 3,00,000 + 30,000 = ₹3,30,000.
→ Loss Amount = 5% of 3,30,000 = (5 ÷ 100) × 3,30,000 = 16,500.
→ SP = Total CP − Loss Amount = 3,30,000 − 16,500 = ₹3,13,500.
∴ The correct answer is ₹3,13,500.

Additional Information
Effective Cost Price
When a person spends money on repairs, transportation, or modifications, these overhead expenses are always added to the original purchase price to calculate the Total Cost Price.
Loss Percentage
Loss occurs when Selling Price < Cost Price. Formula: Loss% = (Loss ÷ CP) × 100.

Shortcut Trick

Total discount multiplier = (100 − 22)% × (100 − 25)% = 0.78 × 0.75

Selling Price (SP) = 3,800 × (78/100) × (75/100)

SP = 3,800 × 0.78 × 0.75 = 2,223

∴ The correct answer is ₹2,223.

Alternate Method

Given: Marked Price (MP) = ₹3,800, Successive Discounts = 22% and 25%.

Formulas: Equivalent Discount = a + b − (a × b) ÷ 100 | SP = MP × (100 − d1)% × (100 − d2)%

⇒ Price after first 22% discount = 3,800 − (22% of 3,800)

⇒ Price = 3,800 − (22 × 38) = 3,800 − 836 = 2,964

⇒ Price after second 25% discount = 2,964 − (25% of 2,964)

⇒ SP = 2,964 − (2,964 ÷ 4) = 2,964 − 741 = 2,223

∴ The correct answer is ₹2,223.

Additional Information

Successive Discounts

Applying two successive discounts of a% and b% is the same as a single discount of [a + b − (ab/100)]%.

Marked Price vs Selling Price

The Marked Price is the price printed on the label, while the Selling Price is the actual cost paid by the customer after all deductions.

Shortcut Trick

Ratio of Length (L) to Breadth (B) is 1.5 : 1 ⇒ 3 : 2.

Sum of ratio parts (L + B) = 3 + 2 = 5 units.

As Perimeter = 2(L + B) = 500 cm, then (L + B) = 250 cm.

5 units = 250 cm ⇒ 1 unit = 50 cm.

L = 3 × 50 = 150 cm and B = 2 × 50 = 100 cm.

Area = 150 × 100 = 15,000 cm².

∴ The correct answer is 15,000 cm².

Alternate Method

Given:

Perimeter of rectangle = 500 cm

Length (L) = 1.5 × Breadth (B)

Formula Used:

Perimeter = 2 × (Length + Breadth)

Area = Length × Breadth

Calculations:

⇒ 2 × (1.5B + B) = 500

⇒ 2 × 2.5B = 500

⇒ 5B = 500

⇒ B = 100 cm

⇒ L = 1.5 × 100 = 150 cm

⇒ Area = 150 × 100

⇒ Area = 15,000 cm²

∴ The correct answer is 15,000 cm².

Additional Information

Properties of a Rectangle

Opposite sides are equal and parallel, and each internal angle is 90°.

Diagonal of a Rectangle

The length of the diagonal (d) is given by d = √(L² + B²).

Square as a Special Case

A square is a rectangle where Length = Breadth. Its Area = side² and Perimeter = 4 × side.

Shortcut Trick

Let the total capacity of the cistern be the LCM of 6 and 24, which is 24 units.

Combined efficiency of P and Q = 24 ÷ 6 = 4 units/hour.

Efficiency of P alone = 24 ÷ 24 = 1 unit/hour.

Efficiency of Q alone = 4 − 1 = 3 units/hour.

Time taken by Q to fill one-fourth of the cistern = (24 ÷ 4) ÷ 3 = 6 ÷ 3 = 2 hours.

∴ The correct answer is 2 hours.

Alternate Method

Given:

Time taken by P and Q together = 6 hours

Time taken by P alone = 24 hours

Formula Used:

1/Q = 1/(P+Q) − 1/P

Calculations:

⇒ Efficiency of Q = Efficiency of (P + Q) − Efficiency of P

⇒ Efficiency of Q = 4 − 1 = 3 units/hour

⇒ One-fourth of the total work = 24 × (1/4) = 6 units

⇒ Time taken by Q to complete 6 units of work = 6 ÷ 3 = 2 hours

∴ The correct answer is 2 hours.

Additional Information

Pipes & Cisterns Efficiency Concept

The rate of work done by an inlet pipe is positive, whereas the rate of work done by an outlet pipe (leakage) is negative. Total Work = Time Taken × Efficiency.

Combined Work Formula

If two entities P and Q can complete a task in 'p' and 'q' hours respectively, then together they can complete the work in (p × q) ÷ (p + q) hours.