Quantitative Aptitude PYQs

Shortcut Trick

Since the given ratios 17, 5, and 11 are pairwise co-prime numbers, their LCM is simply the product of their ratios multiplied by their HCF.

⇒ LCM = (Product of ratios) × HCF

⇒ 3740 = (17 × 5 × 11) × HCF

⇒ 3740 = 935 × HCF

⇒ HCF = 3740 ÷ 935 = 4

∴ The correct answer is 4.

Alternate Method

Given:

Ratio of the three numbers = 17 : 5 : 11

LCM of the three numbers = 3740

Formula Used:

Let the common factor (which is also the HCF) of the numbers be x.

The three numbers are 17x, 5x, and 11x.

Since 1

Shortcut Trick

According to the ratio 1 : 2, for every 1 part of the first ingredient, there are 2 parts of the second ingredient.

If 1 unit = 3 cups,

Then 2 units = 2 × 3 = 6 cups.

∴ The correct answer is 6 cups.

Alternate Method

Given: Ratio of ingredients = 1 : 2; Quantity of first ingredient = 3 cups.

⇒ Let the quantity of the second ingredient be x cups.

⇒ Ratio = (Ingredient 1) ÷ (Ingredient 2)

⇒ 1 ÷ 2 = 3 ÷ x

⇒ 1 × x = 3 × 2

⇒ x = 6 cups

∴ The correct answer is 6 cups.

Additional Information

Direct Proportion

Two quantities are in direct proportion if the ratio of their values remains constant (x/y = k). If one increases, the other increases accordingly.

Fourth Proportional

If four quantities a, b, c, d are such that a:b = c:d, then d is called the fourth proportional to a, b, and c, calculated as d = (b × c) ÷ a.

Scaling Ratios

To maintain the same ratio when one part is multiplied by a factor, all other parts of the ratio must be multiplied by the same scaling factor.

Shortcut Trick

Net Profit% = Mark-up% − Discount% − (Mark-up × Discount)/100

Net Profit% = 40 − 25 − (40 × 25)/100 = 15 − 10 = 5%

Selling Price = 105% of Cost Price ⇒ 21,000 = 1.05 × CP

CP = 21,000 ÷ 1.05 = 20,000

∴ The correct answer is ₹20,000.

Alternate Method

Given:

Mark-up Percentage = 40%

Discount Percentage = 25%

Selling Price (SP) = ₹21,000

⇒ Let Cost Price (CP) = 100 units

⇒ Marked Price (MP) = 100 + 40% of 100 = 140 units

⇒ Selling Price (SP) = 140 − 25% of 140

⇒ SP = 140 − (25 ÷ 100) × 140 = 140 − 35 = 105 units

⇒ Given, 105 units = ₹21,000

⇒ 1 unit = 21,000 ÷ 105 = 200

⇒ Cost Price (CP) = 100 units = 100 × 200 = ₹20,000

∴ The correct answer is ₹20,000.

Additional Information

Mark-up Percentage

The percentage by which the cost price is increased to set the marked price. Formula: Mark-up% = (MP − CP)/CP × 100.

Relationship between CP and MP

CP × (100 + Profit%) = MP × (100 − Discount%). This ratio can be used to directly find the relation between CP and MP.

Successive Percentage Change

If a value is increased by x% and then decreased by y%, the net change is [x − y − (xy/100)]%.

Given:

$(\frac{28}{8})^{17} \times (\frac{8}{28})^{20} \times (\frac{28}{8})^{3} = (\frac{8}{28})^{9m+14}$

Formula used:

1. (a/b)ⁿ = (b/a)⁻ⁿ
2. a^x × a^y × a^z = a^{x + y + z}
3. If a^x = a^y, then x = y

Calculations:

According to the given equation:

$(\frac{28}{8})^{17} \times (\frac{8}{28})^{20} \times (\frac{28}{8})^{3} = (\frac{8}{28})^{9m+14}$

To solve for m, express all terms with the same base (8/28):

⇒ (8/28)⁻¹⁷ × (8/28)²⁰ × (8/28)⁻³ = (8/28)^{9m + 14}

Applying the law of exponents on the left-hand side:

⇒ (8/28)^{(−17 + 20 − 3)} = (8/28)^{9m + 14}

⇒ (8/28)⁰ = (8/28)^{9m + 14}

Now, equating the exponents on both sides of the equation:

⇒ 0 = 9m + 14

⇒ 9m = −14

⇒ m = −14/9

∴ The correct answer is option (3).

Shortcut Trick

Apply quadrant rules: cos(90° − A) = sin A, cot(90° + A) = −tan A, and cos(180° + A) = −cos A.

Numerator = sin A × sin A = sin² A.

Denominator = (−tan A) × (−cos A) = tan A × cos A = (sin A / cos A) × cos A = sin A.

Expression = sin² A / sin A = sin A.

∴ The correct answer is sin A.

Alternate Method

Given: [sin A × cos(90° − A)] / [cot(90° + A) × cos(180° + A)]

Trigonometric Transformations:

Step-by-step Simplification:

⇒ Numerator: sin A × cos(90° − A)

⇒ In 1^st quadrant, cos(90° − A) = sin A

⇒ Numerator = sin A × sin A = sin² A

⇒ Denominator: cot(90° + A) × cos(180° + A)

⇒ In 2^nd quadrant, cot(90° + A) = −tan A

⇒ In 3^rd quadrant, cos(180° + A) = −cos A

⇒ Denominator = (−tan A) × (−cos A) = tan A × cos A

⇒ Denominator = (sin A / cos A) × cos A = sin A

⇒ Final Expression: sin² A / sin A

⇒ Expression = sin A

∴ The correct answer is sin A.

Additional Information

ASTC Rule

It determines the sign of trigonometric ratios: 1^st Quadrant (All positive), 2^nd (Sine/Cosec positive), 3^rd (Tan/Cot positive), and 4^th (Cos/Sec positive).

Complementary Ratio Changes

Ratios change to their co-ratios at (90° ± θ) or (270° ± θ): sin ↔ cos, tan ↔ cot, and sec ↔ cosec.

Tangent Identity

tan A = sin A / cos A and cot A = cos A / sin A. These are fundamental for simplifying product-form trigonometric expressions.

Shortcut Trick

Ratio of Initial Price to New Price = 400 : 500 = 4 : 5.

Increase = 5 − 4 = 1 unit.

Percentage Increase = (Increase ÷ Initial Ratio) × 100 = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Original Price of the shirt = ₹400

Increased Price of the shirt = ₹500

Formula Used:

Percentage Increase = [(New Price − Old Price) ÷ Old Price] × 100

Calculations:

⇒ Total price increase = 500 − 400 = ₹100

⇒ Percentage increase = (100 ÷ 400) × 100

⇒ Percentage increase = (1 ÷ 4) × 100

⇒ Percentage increase = 25%

Therefore, Option (3) is the correct answer.

∴ The correct answer is 25%.

Additional Information

Percentage Change Formula

Percentage Increase or Decrease = (Difference ÷ Original Value) × 100. Always use the starting (initial) value as the base.

Successive Percentage Change

If a value increases by a% and then by b%, the net percentage change is given by [a + b + (a × b ÷ 100)]%.

Fraction to Percentage Conversion

Common fractions help in fast calculations: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, and 1/8 = 12.5%.

Given:

(3/4) + 0.25 × 2

Formula used:

BODMAS rule: Perform Multiplication before Addition.

Fraction conversion: 3/4 = 0.75

Calculations:

(3/4) + 0.25 × 2

⇒ 0.75 + 0.25 × 2

⇒ 0.75 + 0.50

⇒ 1.25

∴ The correct answer is option (3).

Shortcut Trick

Total Distance = 20 km (Going) + 20 km (Returning) = 40 km.

Total Time = (20 ÷ 10) + 1 (Stay) + (20 ÷ 20) = 2 + 1 + 1 = 4 hours.

Average Speed = Total Distance ÷ Total Time = 40 ÷ 4 = 10 km/h.

∴ The correct answer is 10 km/h.

Alternate Method

Given:

Distance from Home to Market = 20 km

Speed while walking = 10 km/h

Stay time at market = 1 hour

Speed while returning by e-rickshaw = 20 km/h

Formula used:

Time = Distance ÷ Speed

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Time taken to reach the market = 20 ÷ 10 = 2 hours

⇒ Time taken for returning = 20 ÷ 20 = 1 hour

⇒ Total time taken for the entire trip = 2 (walking) + 1 (stay) + 1 (returning) = 4 hours

⇒ Total distance covered = 20 (going) + 20 (returning) = 40 km

⇒ Average Speed = 40 ÷ 4 = 10 km/h

∴ The correct answer is 10 km/h.

Additional Information

Average Speed (Equal Distances)

If a person covers equal distances at speeds x and y, the average speed (excluding stay time) is given by 2xy ÷ (x + y).

Stoppage Time Effect

When stay time or stoppage time is included, the total time of the journey increases, which subsequently reduces the average speed of the trip.

Unit Conversion

To convert speed from km/h to m/s, multiply the value by 5 ÷ 18. To convert m/s to km/h, multiply by 18 ÷ 5.

Shortcut Trick

Identify the numbers as a = 22, b = −13, and c = −9.

Check the sum: a + b + c = 22 + (−13) + (−9) = 0.

Using the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculation: 3 × 22 × (−13) × (−9) = 66 × 117 = 7722.

∴ The correct answer is 7722.

Alternate Method

Given:

The expression to evaluate is 22³ + (−13)³ + (−9)³.

Formula Used:

The conditional identity states that if a + b + c = 0, then a³ + b³ + c³ = 3abc.

⇒ Let a = 22, b = −13, and c = −9.

⇒ a + b + c = 22 + (−13) + (−9)

⇒ a + b + c = 22 − 22 = 0.

⇒ Since the sum of the terms is zero, apply the identity:

⇒ Value = 3 × (22) × (−13) × (−9)

⇒ Value = 66 × 117

⇒ Value = 7722

Therefore, the calculated value is 7722, which matches option (2).

∴ The correct answer is 7722.

Additional Information

General Algebraic Identity

The complete identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Negative Number Cubes

The cube of a negative number is always negative, for example, (−13)³ = −2197.

Applications of Identity

This identity is primarily used to simplify expressions involving cubes without performing large multiplications manually.

Shortcut Trick

Total units initially sold = 120 + 150 + 110 + 130 + 140 = 650 units

Initial Arithmetic Mean = 650 ÷ 5 = 130 units

A return of 10 units reduces the total sum by 10.

Reduction in mean = Total Reduction ÷ Number of items = 10 ÷ 5 = 2 units

New Arithmetic Mean = 130 − 2 = 128 units

∴ The correct answer is 128 units.

Alternate Method

Given:

Units sold for 5 items = 120, 150, 110, 130, 140

Return from the first item = 10 units

Formula Used:

Arithmetic Mean = Sum of all observations ÷ Total number of observations

Calculations:

⇒ New value of the first item = 120 − 10 = 110 units

⇒ Sum of new observations = 110 + 150 + 110 + 130 + 140

⇒ Total Sum = 640 units

⇒ New Arithmetic Mean = 640 ÷ 5

⇒ New Mean = 128 units

∴ The correct answer is 128 units.

Additional Information

Arithmetic Mean (Average)

The average of a set of numbers is the sum of the numbers divided by the count. Formula: x̄ = (Σx_i) ÷ n.

Property of Change in Mean

If every observation in a data set is decreased by a constant value k, then the mean of the new observations is decreased by k. If only one observation is decreased by k, the mean decreases by k/n.

Sum of Deviations

The sum of the deviations of all observations from their arithmetic mean is always equal to zero, i.e., Σ(x_i − x̄) = 0.

Given:

9² + √4² − 10 × √81 − 15

Formula used:

BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

9² + (√4)² − 10 × √81 − 15

⇒ 81 + 4 − 10 × 9 − 15

⇒ 81 + 4 − 90 − 15

⇒ 85 − 105

⇒ −20

∴ The correct answer is option (3).

Shortcut Trick

Total work = LCM of (12, 36) = 36 units.

Efficiency: A = 3 units/day, B = 1 unit/day.

Work done by (A + B) in 2 days = (3 + 1) × 2 = 8 units.

Remaining work = 36 − 8 = 28 units.

Time taken by B to complete it = 28 ÷ 1 = 28 days.

∴ The correct answer is 28 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 36 days.

Formula Used: Efficiency = Total Work ÷ Time; Time = Work ÷ Efficiency

⇒ Total efficiency of A and B = 3 + 1 = 4 units/day.

⇒ Work completed by both in 2 days = 4 × 2 = 8 units.

⇒ Remaining work to be done by B = 36 − 8 = 28 units.

⇒ Days taken by B = 28 ÷ 1 = 28 days.

∴ The correct answer is 28 days.

Additional Information

Work and Wages

Wages are always distributed in the ratio of work done by the individuals, which is usually the ratio of their efficiencies if they work for the same duration.

Pipes and Cisterns

The concept is identical to Time and Work, where an inlet pipe does positive work and an outlet pipe (leak) does negative work.

Combined Efficiency

If multiple people work together, their net efficiency is the sum of their individual efficiencies (E_total = E₁ + E₂ + ...).

Shortcut Trick

Remainder = 42

Divisor = 3 × 42 = 126

Quotient = 126 ÷ 9 = 14

Dividend = (126 × 14) + 42 = 1764 + 42 = 1806

∴ The correct answer is 1806.

Alternate Method

Given: The remainder is 42. The divisor is 9 times the quotient and 3 times the remainder.

Formula Used: Dividend = (Divisor × Quotient) + Remainder

⇒ Remainder = 42

⇒ Divisor = 3 × 42 = 126

⇒ Quotient = 126 ÷ 9 = 14

⇒ Dividend = (126 × 14) + 42

⇒ Dividend = 1764 + 42 = 1806

∴ The correct answer is 1806.

Additional Information

Division Algorithm

For any two integers a and b (b ≠ 0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < |b|.

Remainder Rule

The remainder must always be non-negative and strictly smaller than the divisor.

Verification of Division

To verify a division result, compute (Divisor × Quotient) + Remainder; the sum must equal the dividend.

Shortcut Trick

Ratio of quantities sold: 50 : (30% of 50) : (50 − 15) ⇒ 50 : 15 : 35 ⇒ 10 : 3 : 7.

Net Profit x% = [(10 × 37) + (3 × 26) − (7 × 24)] ÷ (10 + 3 + 7)

x% = [370 + 78 − 168] ÷ 20 ⇒ 280 ÷ 20 = 14%.

∴ The correct answer is 14%.

Alternate Method

Given: 50% goods at 37% profit, 30% of remaining at 26% profit, rest at 24% loss.

Assume total quantity = 100 units and Cost Price (CP) of each unit = ₹1.

⇒ Total Cost Price = ₹100.

⇒ Case 1: 50 units sold at 37% profit ⇒ Profit₁ = 50 × 37/100 = ₹18.5.

⇒ Remaining units = 100 − 50 = 50 units.

⇒ Case 2: 30% of 50 = 15 units sold at 26% profit ⇒ Profit₂ = 15 × 26/100 = ₹3.9.

⇒ Case 3: Remaining units = 50 − 15 = 35 units sold at 24% loss ⇒ Loss₃ = 35 × 24/100 = ₹8.4.

⇒ Total Net Profit = Profit₁ + Profit₂ − Loss₃

⇒ Total Net Profit = 18.5 + 3.9 − 8.4 = ₹14.

⇒ Overall Profit Percentage (x) = (Total Profit ÷ Total CP) × 100

⇒ x = (14 ÷ 100) × 100 = 14%.

∴ The correct answer is 14%.

Additional Information

Weighted Average Formula

Net Profit/Loss% = (n₁p₁ + n₂p₂ + ... + n_kp_k) ÷ (n₁ + n₂ + ... + n_k), where n is quantity and p is profit(+)/loss(−) percentage.

Net Profit Percentage

Overall Profit % = [(Total Selling Price − Total Cost Price) ÷ Total Cost Price] × 100.

Successive Percentage

If a value is changed by a% and then by b%, the net change is [a + b + (a × b)/100] %.

Shortcut Trick

For consecutive years in Compound Interest, the Amount of the previous year acts as the Principal for the subsequent year.

Interest for the 3^rd year = ₹28,800 − ₹24,000 = ₹4,800.

Rate of Interest (R) = (Interest ÷ Principal for the year) × 100

R = (4,800 ÷ 24,000) × 100 = 1/5 × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Amount after 2 years (A₂) = ₹24,000

Amount after 3 years (A₃) = ₹28,800

Formula Used:

A = P(1 + R ÷ 100)ⁿ

Calculations:

⇒ A₂ = P(1 + R/100)² = 24000

⇒ A₃ = P(1 + R/100)³ = 28800

⇒ A₃ ÷ A₂ = [P(1 + R/100)³] ÷ [P(1 + R/100)²]

⇒ 28800 ÷ 24000 = (1 + R/100)

⇒ 288 ÷ 240 = 1 + R/100

⇒ 1.2 = 1 + R/100

⇒ R/100 = 1.2 − 1 = 0.2

⇒ R = 0.2 × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Successive Growth in Compound Interest

When time periods are consecutive, the annual rate of interest can be calculated as [(A_n+1 − A_n) ÷ A_n] × 100.

Effective Interest Rate

If interest is compounded n times a year, the effective annual rate is given by R_eff = [1 + (R ÷ n)]ⁿ − 1.

Difference between SI and CI

For a period of 2 years, the difference between Simple Interest and Compound Interest is D = P × (R ÷ 100)².

Shortcut Trick

Identify the excess cost: 6 Naveen Pens cost ₹15 more each than Rajat Pens.

Total excess cost = 6 × ₹15 = ₹90.

Subtract excess from total: ₹540 − ₹90 = ₹450 (This represents the cost if all 10 pens were Rajat Pens).

Cost of one Rajat Pen = ₹450 ÷ 10 = ₹45.

∴ The correct answer is ₹45.

Alternate Method

Given:

Total pens = 4 Rajat + 6 Naveen

Total Cost Price = ₹540

Price difference = Naveen Pen − Rajat Pen = ₹15

Let the cost of one Rajat Pen be ₹R.

Then, the cost of one Naveen Pen = ₹(R + 15).

⇒ 4 × R + 6 × (R + 15) = 540

⇒ 4R + 6R + 90 = 540

⇒ 10R + 90 = 540

⇒ 10R = 540 − 90

⇒ 10R = 450

⇒ R = 450 ÷ 10

⇒ R = 45

∴ The correct answer is ₹45.

Additional Information

Linear Equation in One Variable

An equation of the form ax + b = c, where a, b, and c are constants. It is solved by isolating the variable x on one side.

Substitution Method

Expressing one variable in terms of another (e.g., N = R + 15) and substituting it into the main equation to reduce the number of unknowns.

Cost Price (CP)

The total amount paid by a buyer to acquire an item. Total CP = (Quantity 1 × Rate 1) + (Quantity 2 × Rate 2).

Shortcut Trick

Direct application of the Time-Speed-Distance relationship:

Time = Distance ÷ Speed

Time = 150 ÷ 50 = 3 hr

∴ The correct answer is 3 hr.

Alternate Method

Given:

Total Distance (D) = 150 km

Speed (S) = 50 km/h

Formula Used:

Time (T) = Distance ÷ Speed

Calculations:

⇒ Time taken = 150 ÷ 50 ⇒ Time taken = 3 hr

∴ The correct answer is 3 hr.

Additional Information

Fundamental Formulas

Distance = Speed × Time | Speed = Distance ÷ Time.

Unit Conversion

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Average Speed

For a trip with different speeds over equal distances, Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).

Shortcut Trick

Arrange the monthly incomes in ascending order: 35, 37, 38, 39, 40, 41, 42, 44, 46, 50.

Count of observations (n) = 10 (which is even).

Median = Average of the middle two terms (5^th and 6^th terms).

Median = (40 + 41) ÷ 2 = 40.5.

∴ The correct answer is 40.5.

Alternate Method

Given: Monthly incomes of 10 employees (in ₹ thousands) are 35, 40, 38, 50, 46, 42, 39, 37, 41, and 44.

Formula Used:

For an even number of observations (n):

Median = [(n/2)^th observation + (n/2 + 1)^th observation] ÷ 2

⇒ Total observations (n) = 10

⇒ Arrange incomes in ascending order: 35, 37, 38, 39, 40, 41, 42, 44, 46, 50

⇒ n/2 = 10/2 = 5^th term

⇒ n/2 + 1 = 10/2 + 1 = 6^th term

⇒ 5^th term = 40 and 6^th term = 41

⇒ Median = (40 + 41) ÷ 2

⇒ Median = 81 ÷ 2 = 40.5

∴ The correct answer is 40.5.

Additional Information

Median for Odd Number of Terms

If n is odd, the Median is simply the middle-most value, calculated as the ((n + 1) ÷ 2)^th term after sorting.

Empirical Relationship

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

Effect of Outliers

The Median is a measure of central tendency that is robust against outliers, unlike the arithmetic mean which is heavily influenced by extreme values.

Shortcut Trick

When a large sphere is melted into n identical small spheres, the radius ratio is R/r = n^1/3.

For n = 27, R/r = 27^1/3 = 3.

Ratio of Surface Area = R² : (n × r²) = (R/r)² : n.

Required Ratio = 3² : 27 = 9 : 27 = 1 : 3.

∴ The correct answer is 1:3.

Alternate Method

Given: Radius of the larger metallic sphere (R) = 6 units, Total number of smaller spheres (n) = 27.

Formula Used: Volume of sphere = (4/3)πr³; Surface area of sphere = 4πr².

⇒ Total Volume remains constant: Volume of Large Sphere = 27 × Volume of Small Sphere

⇒ (4/3)π × 6³ = 27 × (4/3)π × r³

⇒ 216 = 27 × r³

⇒ r³ = 216 / 27 = 8

⇒ r = √³8 = 2 units

⇒ Surface Area of Large Sphere (SA_L) = 4π × R² = 4π × 6² = 144π

⇒ Total Surface Area of 27 Small Spheres (SA_{S_total}) = 27 × (4π × r²)

⇒ SA_{S_total} = 27 × 4π × 2² = 27 × 16π = 432π

⇒ Required Ratio = SA_L : SA_{S_total} = 144π : 432π

⇒ Ratio = 144 / 432 = 1 / 3

∴ The correct answer is 1:3.

Additional Information

Volume Conservation

In melting and recasting problems, the initial volume of the object always equals the final total volume of all new objects formed.

Surface Area to Volume Relationship

If the radius of a sphere is scaled by a factor k, its surface area scales by k² and its volume scales by k³.

Surface Area of Hemisphere

For a sphere cut in half, the curved surface area is 2πr² and the total surface area is 3πr².

Shortcut Trick

Ratio 4 years ago (Teacher : Daughter) = 3 : 1 (Difference = 2 units)

Ratio 8 years hence (Teacher : Daughter) = 2 : 1 → 4 : 2 (Difference = 2 units)

Increase in ratio units = 4 − 3 = 1 unit ⇒ Total time gap = 4 + 8 = 12 years.

1 unit = 12 years ∴ Teacher's age 4 years ago = 3 × 12 = 36 years.

Present age of teacher = 36 + 4 = 40 years.

∴ The correct answer is 40 years.

Alternate Method

Given: 4 years ago, the teacher was 3 times her daughter's age. 8 years later, she will be twice her daughter's age.

Let the daughter's age 4 years ago be x.

Then, the teacher's age 4 years ago = 3x.

⇒ Present age of teacher = 3x + 4

⇒ Present age of daughter = x + 4

According to the second condition (8 years from now):

⇒ (3x + 4 + 8) = 2 × (x + 4 + 8)

⇒ 3x + 12 = 2 × (x + 12)

⇒ 3x + 12 = 2x + 24

⇒ 3x − 2x = 24 − 12

⇒ x = 12

Present age of teacher = 3 × 12 + 4 = 36 + 4 = 40 years.

∴ The correct answer is 40 years.

Additional Information

Constant Age Difference

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

Ratio Balancing Method

When solving with ratios, multiply the ratios by the difference of the other ratio's parts to make the unit difference uniform across time periods.

Linear Equation Application

For multiple conditions, use the present age as the base variable (x) to avoid confusion when moving backward or forward in time.