Quantitative Aptitude PYQs

Shortcut Trick

Effective capital ratio (Investment × Time):

A : B : C = (1,21,500 × 12) : (1,02,060 × 9) : (97,200 × 6 + 72,700 × 6)

A : B : C = 14,58,000 : 9,18,540 : 10,19,400 = 24,300 : 15,309 : 16,990

Sum of ratio parts = 24,300 + 15,309 + 16,990 = 56,599 units

Total profit = ₹1,69,797 ⇒ 1 unit = ₹3

Difference (A − B) = (24,300 − 15,309) × 3 = 8,991 × 3 = ₹26,973.

∴ The correct answer is ₹26,973.

Alternate Method

Given:

Investment of A = ₹1,21,500 for 12 months

Investment of B = ₹1,02,060 for 9 months

Investment of C = ₹97,200 for 6 months and (₹97,200 − ₹24,500) = ₹72,700 for 6 months

Total Profit = ₹1,69,797

Formula: Profit Share Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Equivalent Capital of A = 1,21,500 × 12 = 14,58,000

⇒ Equivalent Capital of B = 1,02,060 × 9 = 9,18,540

⇒ Equivalent Capital of C = (97,200 × 6) + (72,700 × 6) = 5,83,200 + 4,36,200 = 10,19,400

⇒ Ratio A : B : C = 14,58,000 : 9,18,540 : 10,19,400

⇒ Dividing by common factor 60: 24,300 : 15,309 : 16,990

⇒ Sum of ratios = 24,300 + 15,309 + 16,990 = 56,599

⇒ Total Profit = ₹1,69,797

⇒ Value of 1 unit = 1,69,797 ÷ 56,599 = ₹3

⇒ Difference between share of A and B = (24,300 − 15,309) units

⇒ Difference = 8,991 × 3 = ₹26,973

∴ The correct answer is ₹26,973.

Additional Information

Basic Partnership Law

Profit is always distributed in the ratio of the product of Capital (C) and Time (T), i.e., P₁ : P₂ = C₁×T₁ : C₂×T₂.

Effective Capital for Multiple Transactions

When a partner changes investment mid-term, effective capital = (C₁×T₁ + C₂×T₂ + ...), where T is the duration for each capital amount.

Calculating Unknown Shares

Individual Profit = (Individual Ratio Part ÷ Total Ratio Sum) × Total Profit.

Shortcut Trick

Total Simple Interest for 3 years = 20% × 3 = 60%.

Total Amount = 100% (Principal) + 60% (Interest) = 160% = ₹5,280.

Simple Interest for 2 years = 20% × 2 = 40%.

Required SI = (5,280 ÷ 160%) × 40% = 5,280 ÷ 4 = ₹1,320.

∴ The correct answer is ₹1,320.

Alternate Method

Given:

Amount (A) = ₹5,280, Rate (R) = 20% p.a., Time (T₁) = 3 years.

Formula Used:

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

Amount (A) = Principal (P) + SI = P × (1 + (R × T) ÷ 100)

Calculations:

⇒ 5,280 = P × (1 + (20 × 3) ÷ 100)

⇒ 5,280 = P × (1 + 0.6)

⇒ 5,280 = 1.6P

⇒ Principal (P) = 5,280 ÷ 1.6 = ₹3,300

⇒ Simple Interest for 2 years (T₂ = 2) = (3,300 × 20 × 2) ÷ 100

⇒ SI = 33 × 40 = ₹1,320

∴ The correct answer is ₹1,320.

Additional Information

Simple Interest Properties

In Simple Interest, the interest remains constant for every year as it is always calculated on the original Principal amount.

Principal Formula

The Principal can be calculated if the Amount, Rate, and Time are known: P = (A × 100) ÷ (100 + RT).

SI vs CI

For the first year (if compounded annually), Simple Interest and Compound Interest are equal for the same principal and rate.

Shortcut Trick

The largest 6-digit number is 999999.

Divide 999999 by 96 to find the remainder.

999999 ÷ 96 gives a remainder of 63.

Subtract this remainder from 999999 to get the largest divisible number.

999999 − 63 = 999936.

∴ The correct answer is 999936.

Alternate Method

Given: Divisor = 96, Largest 6-digit number = 999999.

Formula: Required Number = (Largest n-digit number) − (Remainder when largest n-digit number is divided by divisor).

⇒ Largest 6-digit number = 999999

⇒ Dividing 999999 by 96:

⇒ 999999 = 10416 × 96 + 63

⇒ Remainder = 63

⇒ Required number = 999999 − 63

⇒ Required number = 999936

∴ The correct answer is 999936.

Additional Information

Divisibility Rule for 96

A number is divisible by 96 if it is divisible by both 3 (sum of digits) and 32 (last 5 digits divisible by 32).

Smallest n-digit number divisible by x

Required number = Smallest n-digit number + (x − remainder), where remainder is obtained by dividing the smallest n-digit number by x.

Division Algorithm

Dividend = (Divisor × Quotient) + Remainder. For perfect divisibility, the remainder must be zero.

Shortcut Trick

Assume the initial price of fuel is 100.

Final Price = 100 × (0.55) × (0.75) × (0.85) × (1.45)

Final Price = 100 × 0.50840625 = 50.840625

Net percentage decrease = 100 − 50.840625 = 49.159375%

Rounding to 2 decimal places, we get 49.16%.

∴ The correct answer is 49.16%.

Alternate Method

Given:

Initial changes: 45% decrease, 25% decrease, 15% decrease.

Fourth month change: 45% increase.

Formula Used:

Final Value = Initial Value × (1 − r₁/100) × (1 − r₂/100) × (1 − r₃/100) × (1 + r₄/100)

Calculations:

⇒ Let Initial Price = 100

⇒ Price after 1^st month = 100 − 45% of 100 = 55

⇒ Price after 2^nd month = 55 − 25% of 55 = 55 − 13.75 = 41.25

⇒ Price after 3^rd month = 41.25 − 15% of 41.25 = 41.25 − 6.1875 = 35.0625

⇒ Price after 4^th month = 35.0625 + 45% of 35.0625 = 35.0625 + 15.778125 = 50.840625

⇒ Net Percentage Decrease = 100 − 50.840625 = 49.159375%

⇒ Rounding to two decimal places = 49.16%

∴ The correct answer is 49.16%.

Additional Information

Successive Percentage Change

For two successive changes of a% and b%, the net change is given by (a + b + ab/100)%. Use negative values for decrease.

Multiplying Factor Concept

To find the final value after an x% change, multiply the base by (1 ± x/100). For multiple changes, multiply all factors sequentially.

Rounding Rules

When rounding to 2 decimal places, if the third decimal digit is 5 or greater, round up the second digit.

Shortcut Trick

Let Cost Price (CP) = 100.

Marked Price (MP) = 100 + 120 = 220.

To gain 43%, Selling Price (SP) must be 143.

Discount needed = MP − SP = 220 − 143 = 77.

Discount % = (77 ÷ 220) × 100 = 35%.

∴ The correct answer is 35%.

Alternate Method

Given:

Markup = 120% above Cost Price (CP)

Required Gain (Profit) = 43%

Formula Used:

1. Marked Price (MP) = CP × (100 + Markup%) ÷ 100
2. Selling Price (SP) = CP × (100 + Profit%) ÷ 100
3. Discount % = [(MP − SP) ÷ MP] × 100

Calculations:

⇒ Let CP = 100

⇒ MP = 100 + (120% of 100) = 100 + 120 = 220

⇒ SP = 100 + (43% of 100) = 100 + 43 = 143

⇒ Discount = MP − SP = 220 − 143 = 77

⇒ Discount % = (77 ÷ 220) × 100

⇒ Discount % = (7 ÷ 20) × 100 = 7 × 5 = 35%

∴ The correct answer is 35%.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is directly related to the profit and discount: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Net Percentage Change Formula

The overall profit or loss can be calculated using the successive percentage change formula: Net % = Markup − Discount − (Markup × Discount ÷ 100).

Discount Calculation

Discount is always calculated on the Marked Price. Discount = MP − SP. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

The effective price after successive discounts is: MP × 70% × 88% = SP.

MP × 0.7 × 0.88 = ₹2,464

MP × 0.616 = ₹2,464

MP = 2,464 ÷ 0.616 = ₹4,000.

∴ The correct answer is ₹4,000.

Alternate Method

Given:

Trade Discount = 30%

Scheme Discount = 12%

Selling Price (SP) = ₹2,464

Formula Used:

Final SP = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ Let the Marked Price (MP) of the electric kettle be M.

⇒ Price after 30% trade discount = M × (100 − 30) ÷ 100 = 0.70M.

⇒ Final Selling Price after 12% scheme discount = 0.70M × (100 − 12) ÷ 100.

⇒ 2,464 = 0.70M × 0.88

⇒ 2,464 = 0.616 × M

⇒ M = 2,464 ÷ 0.616

⇒ M = 4,000

∴ The correct answer is ₹4,000.

Additional Information

Single Equivalent Discount

For two successive discounts of a% and b%, the single equivalent discount is [a + b − (a × b)/100]%.

Relation between MP and CP

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

Basis of Discount

A discount (whether trade or scheme) is always calculated on the current Marked Price or the preceding balance price in a chain.

Shortcut Trick

Break the expression into three parts: Part 1 is √36 = 6.

Part 2 is √(324 ÷ 81) = √4 = 2.

Part 3: Count decimal places. Top has 6, Bottom has 8. Net adjustment is 10² in the numerator.

Simplify: √[(95 ÷ 19) × (85 ÷ 17) × (189 ÷ 21) × 100] = √(5 × 5 × 9 × 100) = 150.

Total Sum = 6 + 2 + 150 = 158.

∴ The correct answer is 158.

Alternate Method

Given Expression: √36 + √(324 ÷ 81) + √((9.5 × 0.0085 × 18.9) ÷ (0.021 × 0.0017 × 1.9))

The expression involves square roots and decimal simplification. We solve each term step-by-step:

⇒ Term 1: √36 = 6

⇒ Term 2: √(324 ÷ 81) = √4 = 2

⇒ Term 3: To simplify √((9.5 × 0.0085 × 18.9) ÷ (0.021 × 0.0017 × 1.9)), remove decimals by multiplying both numerator and denominator by 10⁸.

⇒ √((95 × 85 × 189 × 100) ÷ (21 × 17 × 19))

⇒ √((95 ÷ 19) × (85 ÷ 17) × (189 ÷ 21) × 100)

⇒ √(5 × 5 × 9 × 100)

⇒ √(22500) = 150

Now, add all the simplified values obtained from the steps above:

⇒ 6 + 2 + 150 = 158

∴ The correct answer is 158.

Additional Information

BODMAS Rule

Always prioritize Orders (roots and exponents) before addition and subtraction to ensure correct arithmetic evaluation.

Decimal Shift in Fractions

When dividing decimals, shift the decimal point to the right in both the numerator and denominator by the same number of places to simplify into whole numbers.

Square Root of Products

The property √(a × b × c) = √a × √b × √c allows for the factorization of large numbers under a root for easier calculation.

Shortcut Trick

Ratio 5 years ago (M:S) = 7 : 1 (Difference = 6 units)

Current Ratio (M:S) = 4 : 1 (Difference = 3 units)

Equalize differences by multiplying current ratio by 2 → 8 : 2

Change in son's age units from 5 years ago to now = 2 − 1 = 1 unit → 5 years

Son's present age = 2 units = 2 × 5 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given:

Current age of man is four times the age of his son.

Five years ago, the man was seven times older than his son.

Formula Used:

Age difference and ratio calculation using linear equations.

Calculations:

⇒ Let the son's present age be S years.

⇒ Therefore, the man's present age is 4S years.

⇒ Five years ago, son's age = (S − 5) years.

⇒ Five years ago, man's age = (4S − 5) years.

⇒ According to the condition: (4S − 5) = 7 × (S − 5)

⇒ 4S − 5 = 7S − 35

⇒ 35 − 5 = 7S − 4S

⇒ 30 = 3S

⇒ S = 10

∴ The correct answer is 10 years.

Additional Information

Age Difference Constant Rule

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

Ratio Scaling

When solving age problems using ratios, always ensure the unit difference between the two ratios is equalized relative to the time gap.

General Age Formula

If the present age is X, then age 'n' years ago is (X − n) and age after 'n' years is (X + n).

Shortcut Trick

Ratio of Cost Price (CP) for P to Q = 100 : 125 = 4 : 5

Ratio of CP for Q to R = 100 : 40 = 5 : 2

Combined ratio P : Q : R = 4 : 5 : 2

Given R paid 2 units = ₹5,166

1 unit = ₹2,583 ⇒ P purchased for 4 units = 4 × 2,583 = ₹10,332

∴ The correct answer is ₹10,332.

Alternate Method

Given:

Price paid by R = ₹5,166

Profit percentage of P = 25%

Loss percentage of Q = 60%

Formula Used:

Selling Price (SP) = Cost Price (CP) × (100 ± Profit/Loss%)/100

Calculations:

⇒ Let the purchase price of the item for P be x.

⇒ P sells it to Q at 25% profit, so CP of Q = x × (125/100) = 1.25x

⇒ Q sells it to R at 60% loss, so CP of R = 1.25x × (1 − 60/100)

⇒ CP of R = 1.25x × 0.40 = 0.5x

⇒ Given that R paid ₹5,166, therefore: 0.5x = 5,166

⇒ x = 5,166 ÷ 0.5 = 5,166 × 2

⇒ x = 10,332

∴ The correct answer is ₹10,332.

Additional Information

Successive Percentage Change

When a value changes by a% and then by b%, the effective percentage change is given by (a + b + ab/100)%.

Fractional Conversion

Profit of 25% is equivalent to a multiplier of 5/4 (1 + 1/4), and a loss of 60% is equivalent to a multiplier of 2/5 (1 − 3/5).

Final Price Formula

Final Price = Initial Price × (100 ± r₁)/100 × (100 ± r₂)/100 × ... × (100 ± r_n)/100.

Shortcut Trick

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

The side opposite to 30° (building height) corresponds to 1 unit = 100√3 m.

The side opposite to 60° (horizontal distance) corresponds to √3 units.

⇒ Horizontal distance = 100√3 × √3 = 300 m.

∴ The correct answer is 300 m.

Alternate Method

Given:

Height of the building (AB) = 100√3 m

Angle of depression from the top of the building to the bicycle = 30°

Since the horizontal ground is parallel to the horizontal line of sight from the top, the angle of elevation of the top of the building from the bicycle (Angle ACB) = 30° (alternate interior angles).

Let the horizontal distance from the base of the building to the bicycle (BC) = d

Formula Used:

In a right-angled triangle ABC:

tan(θ) = Opposite Side ÷ Adjacent Side

Calculations:

In right-angled triangle ABC:

⇒ tan(30°) = AB ÷ BC

⇒ 1 ÷ √3 = 100√3 ÷ d

⇒ d = 100√3 × √3

⇒ d = 100 × 3

⇒ d = 300 m

∴ The correct answer is 300 m.

Additional Information

Angle of Elevation and Depression

The angle of elevation is measured upwards from the horizontal line, while the angle of depression is measured downwards. They are equal in magnitude because of alternate interior angles.

Standard Trigonometric Ratios for Height & Distance

Common ratio values to remember: tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) =

Given:

$\sqrt{\dfrac{169^2 \times 2.56^2 \times 0.25^2}{13^4 \times 1.6^4 \times 0.5^2}}$ = ?

Formula used:

Perfect Squares: 169 = 13², 2.56 = 1.6², and 0.25 = 0.5²

Laws of Indices: (a^m)ⁿ = a^mn and a^m ÷ aⁿ = a^m−n

Calculations:

Substitute the perfect square values into the expression:

⇒ $\sqrt{\dfrac{(13^2)^2 \times (1.6^2)^2 \times (0.5^2)^2}{13^4 \times 1.6^4 \times 0.5^2}}$

⇒ $\sqrt{\dfrac{13^4 \times 1.6^4 \times 0.5^4}{13^4 \times 1.6^4 \times 0.5^2}}$

Cancel common terms 13⁴ and 1.6⁴ from the numerator and denominator:

⇒ √0.5^{(4 − 2)}

⇒ √0.5²

⇒ 0.5

∴ The correct answer is option (1).

Shortcut Trick

Difference in percentage = 40% − 25% = 15%

Given that 15% of the number corresponds to 42.

Therefore, 1% of the number = 42 ÷ 15 = 2.8

The total number (100%) = 2.8 × 100 = 280.

∴ The correct answer is 280.

Alternate Method

Given: Let the number be x. The value of 40% of x is 42 more than 25% of x.

Formula Used: Percentage Value = (Percentage ÷ 100) × Number

⇒ (40 ÷ 100) × x − (25 ÷ 100) × x = 42

⇒ 0.40x − 0.25x = 42

⇒ 0.15x = 42

⇒ x = 42 ÷ 0.15

⇒ x = (42 × 100) ÷ 15

⇒ x = 280

∴ The correct answer is 280.

Additional Information

Basic Percentage Formula

To find p% of a number (N), use the relation: Value = (p × N) ÷ 100.

Percentage Difference Principle

When comparing two percentages of the same base, the difference in their percentage values is directly proportional to the difference in their absolute values.

Fraction Equivalents

Using fractions can speed up calculations: 25% = 1/4, 40% = 2/5, 50% = 1/2, and 75% = 3/4.

Shortcut Trick

The required number must be of the form: LCM(11, 7, 14) × k + 2.

LCM of 11, 7, and 14 = 154.

Largest 4-digit multiple of 154: 154 × 64 = 9856.

Required number = 9856 + 2 = 9858.

∴ The correct answer is 9858.

Alternate Method

Given: Divisors = 11, 7, 14; Remainder = 2 in each case; Target = Largest 4-digit number.

Step 1: Calculate the LCM of the given divisors.

⇒ Prime factorization: 11 = 11¹, 7 = 7¹, 14 = 2 × 7 ⇒ LCM(11, 7, 14) = 11 × 2 × 7 = 154

Step 2: Find the largest 4-digit multiple of 154.

⇒ Largest 4-digit number = 9999 ⇒ 9999 ÷ 154 → Quotient = 64, Remainder = 143 ⇒ Largest multiple = 9999 − 143 = 9856

Step 3: Add the common remainder to the multiple.

⇒ Required number = 9856 + 2 = 9858

∴ The correct answer is 9858.

Additional Information

General Remainder Theorem

For a number that leaves the same remainder 'r' when divided by a, b, and c, the general form is LCM(a, b, c) × k + r.

Divisibility Rule of 7

Double the last digit and subtract it from the remaining number; if the result is divisible by 7, the original number is also divisible by 7.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Given:

Amount of sugar = 0.75 cups

Amount of flour = 1 1/4 cups

Formula used:

Total amount = Amount of sugar + Amount of flour

Calculations:

First, convert the mixed fraction representing the amount of flour into a decimal:

1 1/4 = 1 + (1 ÷ 4)

⇒ 1 + 0.25 = 1.25 cups

Now, add the quantity of sugar and flour to find the total ingredients used in the recipe:

Total = 0.75 + 1.25

⇒ Total = 2.0 cups

∴ The correct answer is option (4).

Shortcut Trick

The age difference between two people remains constant over time (8 years).

7 years ago, the ratio of Ramu's age to Keeran's age was 3 : 1.

Difference in ratio units = 3 − 1 = 2 units → 8 years.

1 ratio unit = 8 ÷ 2 = 4 years.

Ramu's age 7 years ago = 3 units × 4 = 12 years.

Ramu's present age = 12 + 7 = 19 years.

∴ The correct answer is 19 years.

Alternate Method

Given:

Difference between Ramu and Keeran = 8 years

Condition: 7 years ago, Ramu was three times as old as Keeran.

Formula Used:

Present Age = Age (n years ago) + n; Age difference is always constant.

Calculations:

⇒ Let Keeran's present age be K years.

⇒ Ramu's present age = (K + 8) years.

⇒ 7 years ago, Keeran's age = (K − 7) and Ramu's age = (K + 8 − 7) = (K + 1).

⇒ According to the question: (K + 1) = 3 × (K − 7)

⇒ K + 1 = 3K − 21

⇒ 3K − K = 21 + 1

⇒ 2K = 22

⇒ K = 11 years

⇒ Ramu's present age = 11 + 8 = 19 years.

∴ The correct answer is 19 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals never changes, regardless of how many years pass or how many years ago the comparison was made.

Ratio Method for Ages

When a ratio is given for a specific time, calculate the difference in ratio units and equate it to the actual constant age difference to find the value of one unit.

Past and Future age Calculation

If the present age is A, then the age 'n' years ago was (A − n) and the age after 'n' years will be (A + n).

Shortcut Trick

Ratio of investments A : B = 12,000 : 8,000 = 3 : 2.

Total units = 3 + 2 = 5 units.

5 units = ₹10,000 → 1 unit = ₹2,000.

A's share = 3 units = 3 × ₹2,000 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Investment of Partner A = ₹12,000

Investment of Partner B = ₹8,000

Total profit = ₹10,000

Formula Used:

In a partnership, if the time period is constant, the profit is distributed in the ratio of the capitals invested: Profit Ratio = Capital Ratio.

⇒ Profit ratio (A : B) = 12,000 : 8,000

⇒ Profit ratio (A : B) = 3 : 2

⇒ Sum of ratio parts = 3 + 2 = 5 units

⇒ 5 units = ₹10,000

⇒ A's share of profit = (3 ÷ 5) × 10,000 = ₹6,000

∴ The correct answer is ₹6,000.

Additional Information

Basic Partnership

When investments are made for the same duration, the profit share is directly proportional to the capital: P₁ : P₂ = C₁ : C₂.

Compound Partnership

When investments are for different durations, profit share depends on both capital and time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs. Sleeping Partner

A working partner manages the business and may receive a salary or fixed percentage of profit before the remaining profit is divided.

Shortcut Trick

Check if a + b + c = 0 for the terms 23, −19, and −4.

Since 23 + (−19) + (−4) = 0, apply the Algebraic Identity: a³ + b³ + c³ = 3abc.

Calculation → 3 × 23 × (−19) × (−4) = 3 × 23 × 76.

Product → 69 × 76 = 5244.

∴ The correct answer is 5244.

Alternate Method

Given: a = 23, b = −19, c = −4

Formula Used: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Step-by-step Calculation:

⇒ First, check the sum: 23 − 19 − 4 = 0.

⇒ When the sum of three numbers is zero, the sum of their cubes equals three times their product.

⇒ Result = 3 × (23) × (−19) × (−4)

⇒ Result = 3 × 23 × 76

⇒ Result = 69 × 76

⇒ Result = 5244.

∴ The correct answer is 5244.

Additional Information

Special Algebraic Identity

The general identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If (a + b + c) = 0, the entire right-hand side becomes zero.

Alternative Form

The term (a² + b² + c² − ab − bc − ca) can also be written as 1/2[(a − b)² + (b − c)² + (c − a)²].

Product Calculation Hint

Multiplying 69 × 76 can be simplified as (70 − 1) × 76 = 5320 − 76 = 5244.

Shortcut Trick

Assume Total Work = LCM(12, 30) = 60 units.

Efficiency of A = 5 units/day and Efficiency of B = 2 units/day.

Work done in 4 days by both = (5 + 2) × 4 = 28 units.

Remaining work = 60 − 28 = 32 units.

Time taken by B = 32 ÷ 2 = 16 days.

∴ The correct answer is 16 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 30 days, Working together time = 4 days.

Formula: Work = Efficiency × Time

⇒ Total work to be done = LCM of 12 and 30 = 60 units.

⇒ Daily work (Efficiency) of A = 60 ÷ 12 = 5 units/day.

⇒ Daily work (Efficiency) of B = 60 ÷ 30 = 2 units/day.

⇒ Work completed by A and B together in 4 days = (5 + 2) × 4 = 7 × 4 = 28 units.

⇒ Remaining work = 60 − 28 = 32 units.

⇒ Time taken by B to complete the remaining work = 32 ÷ 2 = 16 days.

∴ The correct answer is 16 days.

Additional Information

Total Work Concept

The Total Work is usually taken as the LCM of the individual days to avoid working with complex fractions.

Efficiency and Time

Efficiency is the amount of work done in a single unit of time. It is inversely proportional to the Time taken to complete the task.

Work and Wages

When workers are paid for a job, the wages are distributed in the ratio of the work done by each person, which is the product of Efficiency × Time.

Given:

Mathematical expression = 0.5 × 0.125 ÷ 8⁻²

Formula used:

Using laws of indices: a⁻ⁿ = 1 / aⁿ

Calculation:

Convert the decimal values into standard fractions:

⇒ 0.5 = 1/2

⇒ 0.125 = 1/8

Substitute these values into the given expression:

⇒ (1/2) × (1/8) ÷ (1 / 8²)

⇒ (1/16) ÷ (1/64)

Applying the division of fractions rule:

⇒ (1/16) × 64

⇒ 4

∴ The correct answer is option (3).

Shortcut Trick

Use the ratio CP : MP = (100 − Discount%) : (100 + Profit%)

Initial Ratio: CP : MP = (100 − 25) : (100 + 10) = 75 : 110

New Selling Price (SP) at 10% discount = 110 − 11 = 99 units

New Profit % = [(99 − 75) ÷ 75] × 100 = 32%

∴ The correct answer is 32%.

Alternate Method

Given:

Initial Discount = 25% | Initial Profit = 10% | New Discount = 10%

Formula Used:

1. SP = CP × (1 + P/100) | 2. SP = MP × (1 − D/100) | 3. CP/MP = (100 − D)/(100 + P)

Calculations:

⇒ Let Cost Price (CP) = 100 − 25 = 75 units

⇒ Marked Price (MP) = 100 + 10 = 110 units

⇒ New Discount = 10% of MP = 0.10 × 110 = 11 units

⇒ New Selling Price (SP) = MP − Discount = 110 − 11 = 99 units

⇒ Profit = SP − CP = 99 − 75 = 24 units

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = (24 ÷ 75) × 100 = 32%

∴ The correct answer is 32%.

Additional Information

Relationship between CP and MP

The ratio MP ÷ CP = (100 + P%) ÷ (100 − D%) is a direct result of equate SP in both profit and discount scenarios.

Markup Percentage

It is the percentage by which the Cost Price is increased to set the Marked Price. Formula: Markup % = [(MP − CP) ÷ CP] × 100.

Successive Discounts

If two successive discounts of a% and b% are given, the Effective Discount = (a + b − ab/100)%.