Quantitative Aptitude PYQs

Shortcut Trick

Savings = 10%, therefore Expenditure = 90%.

90% of Income = ₹360

100% of Income = (360 ÷ 90) × 100 = 4 × 100 = ₹400.

∴ The correct answer is ₹400.

Alternate Method

Given:

Savings percentage = 10%

Expenditure = ₹360

Formula Used:

Income = Expenditure + Savings

⇒ Let total Income be x.

⇒ Savings = 10% of x = 0.1x

⇒ Expenditure = Income − Savings

⇒ 360 = x − 0.1x

⇒ 360 = 0.9x

⇒ x = 360 ÷ 0.9

⇒ x = 3600 ÷ 9 = 400

∴ The correct answer is ₹400.

Additional Information

Percentage Change in Income

If income increases by a% and expenditure increases by b%, the new values can be calculated using the base multiplier (1 + r/100).

Savings Ratio

The ratio of Income : Expenditure : Savings is vital for solving complex arithmetic problems involving multiple people.

Base Value Concept

In income-based problems, the Total Income is always considered as the 100% base value for calculations.

Shortcut Trick

Identify that 0.011 = 1.1 × 10⁻² and 0.009 = 0.9 × 10⁻².

Denominator = (10⁻²)² × (1.1² − 0.9²) = 10⁻⁴ × Numerator.

The expression becomes (1 / 10⁻⁴) × 0.1² = 10⁴ × 0.01.

Calculation: 10000 × 0.01 = 100.

∴ The correct answer is 100.

Alternate Method

Given: [(1.1² − 0.9²) / (0.011² − 0.009²)] × 0.1²

Formula Used: a² − b² = (a + b)(a − b)

⇒ Numerator = (1.1 + 0.9) × (1.1 − 0.9) = 2.0 × 0.2 = 0.4

⇒ Denominator = (0.011 + 0.009) × (0.011 − 0.009) = 0.020 × 0.002 = 0.00004

⇒ Expression = (0.4 / 0.00004) × (0.1 × 0.1)

⇒ Expression = (40000 / 4) × 0.01

⇒ Expression = 10000 × 0.01 = 100

∴ The correct answer is 100.

Additional Information

Difference of Squares Identity

The algebraic identity a² − b² = (a + b)(a − b) is essential for simplifying expressions involving squares of decimals or large numbers.

Powers of Ten in Decimals

When a number is shifted by n decimal places, its square is shifted by 2n decimal places. For example, (0.1)² = 0.01 and (0.01)² = 0.0001.

Decimal Division Rule

To divide decimals, multiply both the numerator and denominator by a power of 10 that makes the denominator an integer, maintaining the ratio.

Shortcut Trick

Total distance to cross each other = Sum of lengths = 444 + 444 = 888 m

Relative speed (opposite direction) = Distance ÷ Time = 888 ÷ 37 = 24 m/s

Since speeds are equal, individual speed = Relative speed ÷ 2 = 24 ÷ 2 = 12 m/s

∴ The correct answer is 12 m/s.

Alternate Method

Given:

Length of Train 1 (L₁) = 444 m

Length of Train 2 (L₂) = 444 m

Time taken to cross (t) = 37 seconds

Speed of each train = v (as they are equal)

Formula Used:

Relative Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = L₁ + L₂ = 444 + 444 = 888 m

⇒ Relative Speed = v + v = 2v (since they move in opposite directions)

⇒ 2v = 888 ÷ 37

⇒ 2v = 24

⇒ v = 24 ÷ 2

⇒ v = 12 m/s

∴ The correct answer is 12 m/s.

Additional Information

Relative Speed in Opposite Direction

When two objects move toward each other, their relative speed is the sum of their individual speeds: S_rel = S₁ + S₂.

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the difference of their individual speeds: S_rel = |S₁ − S₂|.

Conversion of Speed Units

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

Shortcut Trick

Effective Price percentage = 100 × (1 − d₁/100) × (1 − d₂/100) × (1 − d₃/100)

⇒ Effective Price = 100 × (1 − 11/100) × (1 − 18/100) × (1 − 20/100)

⇒ Effective Price = 100 × 0.89 × 0.82 × 0.80

⇒ Effective Price = 89 × 0.656 = 58.384%

Rounding to two decimal places, we get 58.38%.

∴ The correct answer is 58.38%.

Alternate Method

Given → Marked Price (MP) = 100, Discount 1 = 11%, Discount 2 = 18%, Discount 3 = 20%.
Formulas: Price after Discount = MP × (100 − D₁%)/100 × (100 − D₂%)/100 ...

⇒ Let Marked Price be 100 units.
⇒ Price after 11% discount = 100 − 11 = 89.
⇒ Price after 18% discount on 89 = 89 − (18% of 89) = 89 − 16.02 = 72.98.
⇒ Price after 20% discount on 72.98 = 72.98 − (20% of 72.98) = 72.98 − 14.596 = 58.384.
⇒ Effective Price percentage = (58.384 / 100) × 100 = 58.384%.
∴ The correct answer is 58.38%.

Additional Information

Successive Discounts

When multiple discounts are applied one after another, the single equivalent discount is calculated using the formula: a + b − (ab/100). For three discounts, apply the formula twice.

Effective Price vs Discount

The effective price percentage is (100 − Net Discount%). It represents the final amount the customer pays relative to the original marked price.

Shortcut Trick

The Mode of a data set is simply the observation with the highest frequency.

Scan the list and count occurrences: 22 appears 4 times, 23 appears 2 times, and 29 appears 2 times.

All other numbers (20, 21, 26, 27, 28, 31, 32, 33, 34) appear only once.

Since 22 has the maximum frequency (4), it is the mode.

∴ The correct answer is 22.

Alternate Method

Given Data: 22, 21, 22, 20, 33, 34, 22, 29, 29, 31, 23, 23, 28, 32, 22, 26, 27

Frequency calculation for each unique observation:

⇒ Count of 22: 4 times

⇒ Count of 23: 2 times

⇒ Count of 29: 2 times

⇒ Count of {20, 21, 26, 27, 28, 31, 32, 33, 34}: 1 time each

⇒ Mode = Value with Maximum Frequency

⇒ Maximum Frequency = 4 (for value 22)

∴ The correct answer is 22.

Additional Information

Mean (Arithmetic Average)

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

Median

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

Empirical Relationship

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Range

The difference between the maximum value and the minimum value in a data set: Range = Max − Min.

Shortcut Trick

The average of the first n odd natural numbers is always equal to n.

Here, n = 6 (as we need the average of the first 6 odd numbers).

∴ Average = 6.

∴ The correct answer is 6.

Alternate Method

Given: The first 6 odd natural numbers are 1, 3, 5, 7, 9, and 11.

Formula: Average = (Sum of observations) ÷ (Total number of observations)

⇒ Sum = 1 + 3 + 5 + 7 + 9 + 11

⇒ Sum = 36

⇒ Total Count = 6

⇒ Average = 36 ÷ 6

⇒ Average = 6

∴ The correct answer is 6.

Additional Information

Sum of first n odd numbers

The Sum of the first n consecutive odd natural numbers is always n².

Average of first n even numbers

The Average of the first n consecutive even natural numbers is (n + 1).

Average of first n natural numbers

The Average of the first n consecutive natural numbers is (n + 1) ÷ 2.

Shortcut Trick

Convert to bases: 25=5², 125=5³, 4=2², 10=2×5, 6=2×3

First bracket ⇒ (5²×2⁻⁸) ÷ (5⁻²×2⁻¹⁵) = 5⁴×2⁷

Second bracket ⇒ simplifies to 5⁵

Whole ⇒ (5⁴×2⁷) ÷ 5⁵ = 2⁷×5⁻¹

∴ Answer = 128 ÷ 5

Alternate Method

Given:

{ (25 × 4⁻⁴) ÷ (5⁻³ × 10 × 4⁻⁸) } ÷ { (3⁻⁵ × 10⁻⁵ × 125) ÷ (5⁻⁷ × 6⁻⁵) }

Concept Used:

Laws of indices

Calculation:

⇒ 25 = 5², 125 = 5³, 4 = 2², 10 = 2 × 5, 6 = 2 × 3

⇒ First bracket = (5² × 2⁻⁸) ÷ (5⁻³ × 2 × 5 × 2⁻¹⁶)

⇒ = (5² × 2⁻⁸) ÷ (5⁻² × 2⁻¹⁵)

⇒ = 5²⁻⁽⁻²⁾ × 2^{−8−(−15)} = 5⁴ × 2⁷

⇒ Second bracket = (3⁻⁵ × (2 × 5)⁻⁵ × 5³) ÷ (5⁻⁷ × (2 × 3)⁻⁵)

⇒ = (3⁻⁵ × 2⁻⁵ × 5⁻⁵ × 5³) ÷ (5⁻⁷ × 2⁻⁵ × 3⁻⁵)

⇒ = (5⁻²) ÷ (5⁻⁷) = 5⁵

⇒ Whole expression = (5⁴ × 2⁷) ÷ 5⁵

⇒ = 2⁷ × 5⁻¹ = 128 ÷ 5

∴ The correct answer is 128/5

Additional Information

Laws of Indices

a^m ÷ aⁿ = a^m−n, (a^m)ⁿ = a^mn

Negative Powers

a⁻ⁿ = 1 / aⁿ

Prime Factorization

Convert numbers into primes for easy cancellation

Simplification Tip

Cancel same bases early to reduce calculation

Shortcut Trick

CP of 1 orange = 300 ÷ 10 = ₹ 30.

SP of 1 orange = 175 ÷ 5 = ₹ 35.

Profit = 35 − 30 = ₹ 5.

Profit % = (5 ÷ 30) × 100 = 100 ÷ 6 = 16.666...%.

Rounding to two decimal places, we get 16.67%.

∴ The correct answer is 16.67%.

Alternate Method

Given: CP of 10 oranges = ₹ 300, SP of 5 oranges = ₹ 175.

Formulas: Profit = SP − CP | Profit % = (Profit ÷ CP) × 100

⇒ To compare easily, make the quantity equal.

⇒ CP of 10 oranges = ₹ 300.

⇒ SP of 10 oranges = 2 × (SP of 5 oranges) = 2 × 175 = ₹ 350.

⇒ Profit = 350 − 300 = ₹ 50.

⇒ Profit % = (50 ÷ 300) × 100 = 16.67%.

∴ The correct answer is 16.67%.

Additional Information

Unitary Method in P&L

Calculating the price of a single unit (Unitary Method) makes it easier to compare buying and selling rates when quantities differ.

Profit and Loss Percentage

Always calculate profit or loss percentage on the Cost Price (CP) unless stated otherwise in the problem.

Shortcut Trick

In ΔBDC, ∠BDC = 90° and ∠DBC = 66° → ∠C = 180° − (90° + 66°) = 24°.

In ΔAEC, ∠AEB is the exterior angle at vertex E.

Exterior angle ∠AEB = Sum of interior opposite angles = ∠CAE + ∠C.

∠AEB = 35° + 24° = 59°.

∴ The correct answer is 59°.

Alternate Method

Given: BD ⊥ AC, ∠DBC = 66°, ∠CAE = 35°.

Formula Used: Sum of angles in a triangle = 180°; Exterior Angle Theorem.

Step-by-step calculation:

⇒ In ΔBDC, the sum of angles is 180°.

⇒ ∠BDC + ∠DBC + ∠BCD = 180°

⇒ 90° + 66° + ∠C = 180°

⇒ ∠C = 180° − 156° = 24°

⇒ Now, consider ΔAEC.

⇒ ∠AEB is an exterior angle to ΔAEC at point E.

⇒ By Exterior Angle Property: ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 35° + 24° = 59°

∴ The correct answer is 59°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of all interior angles in any triangle is always 180°.

Right Angled Triangle

In a right-angled triangle, the two acute angles are complementary (their sum is 90°).

Shortcut Trick

Assume Total Work = LCM(8, 24, 36) = 72 units.

Efficiency of X + Y + Z = 72 ÷ 8 = 9 units/day.

Efficiency of X = 72 ÷ 24 = 3 units/day.

Efficiency of Y = 72 ÷ 36 = 2 units/day.

Efficiency of Z = (X + Y + Z) − (X + Y) = 9 − (3 + 2) = 4 units/day.

Time taken by Z = 72 ÷ 4 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given: Time taken by (X+Y+Z) = 8 days, X = 24 days, Y = 36 days.

Formula Used: Total Work = Efficiency × Time | 1/Z = 1/(X+Y+Z) − 1/X − 1/Y

⇒ Efficiency of X + Y + Z = 9

⇒ Efficiency of X = 3

⇒ Efficiency of Y = 2

⇒ Efficiency of Z = 9 − 3 − 2 = 4 units/day

⇒ Time for Z alone = Total Work ÷ Efficiency of Z

⇒ Time = 72 ÷ 4 = 18 days

∴ The correct answer is 18 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Fraction Method

Work done by Z in 1 day = (1/Total days) − (1/X's days) − (1/Y's days).

Combined Work Formula

If A and B work together, total time = (A × B) ÷ (A + B).

Shortcut Trick

For a Sphere, the ratio of Volume (V) to Surface Area (S) is given by r/3.

Since the Volume and Surface Area are numerically equal, we have V/S = 1.

⇒ r/3 = 1

⇒ r = 3 units.

∴ The correct answer is 3 units.

Alternate Method

Given:

The Volume of a sphere is numerically equal to its surface area.

Formula Used:

1. Volume of Sphere = (4/3) × π × r³
2. Surface Area of Sphere = 4 × π × r²

Calculations:

⇒ Volume = Surface Area

⇒ (4/3) × π × r³ = 4 × π × r²

⇒ (r³) / (r²) = (4 × π × 3) / (4 × π)

⇒ r = 3 units.

∴ The correct answer is 3 units.

Additional Information

Volume and Area of Hemisphere

The volume of a hemisphere is (2/3)πr³ and its Total Surface Area (TSA) is 3πr².

Great Circle

A great circle of a sphere is a circle that has the same radius and center as the sphere itself.

Archimedes' Principle for Sphere

The volume of a sphere is 2/3 of the volume of its circumscribed cylinder.

Shortcut Trick

Identify the Unitary Multiplier: 36 words ÷ 4 lines = 9 words per line.

Apply the multiplier to the new total: 50 lines × 9 words/line = 450 words.

∴ The correct answer is 450 words.

Alternate Method

Given:

Number of lines (L₁) = 4

Number of words (W₁) = 36

Target lines (L₂) = 50

Formula Used:

Direct Proportion: (W₁ / L₁) = (W₂ / L₂)

Calculations:

⇒ 36 / 4 = W₂ / 50

⇒ 9 = W₂ / 50

⇒ W₂ = 9 × 50

⇒ W₂ = 450

∴ The correct answer is 450 words.

Additional Information

Direct Proportion

When two quantities increase or decrease at the same rate, they are in direct proportion (y = kx).

Unitary Method

Finding the value of a single unit first to calculate the value of multiple units.

Ratio Equality

The ratio of lines to words remains constant (1 line : 9 words).

Shortcut Trick

Net percentage change = a + b + (a × b) ÷ 100

Here, a = +25% (Markup) and b = −28% (Discount).

Net change = 25 − 28 + (25 × −28) ÷ 100 = −3 + (−700) ÷ 100

Net change = −3 − 7 = −10%.

Negative sign indicates a 10% loss.

∴ The correct answer is 10% loss.

Alternate Method

Given: Markup = 25%, Discount = 28%.

Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = (CP − SP) ÷ CP × 100

⇒ Let Cost Price (CP) be 100.

⇒ Marked Price (MP) = 100 + 25% of 100 = 125.

⇒ Selling Price (SP) = 125 − 28% of 125 = 125 − (28 ÷ 100 × 125) = 125 − 35 = 90.

⇒ Loss = CP − SP = 100 − 90 = 10.

⇒ Loss Percentage = (10 ÷ 100) × 100 = 10%.

∴ The correct answer is 10% loss.

Additional Information

Markup and Discount Relation

Profit or Loss percentage is the result of successive percentage changes on the Cost Price: (1 + Markup%) × (1 − Discount%) − 1.

Marked Price

The price listed on the label of an item, also known as the List Price or MRP.

Shortcut Trick

Original Distance = 225 km, Original Time = 15 hours → Original Speed = 15 kmph.

New Distance (40% less) = 225 × 0.6 = 135 km.

New Speed (20% faster) = 15 × 1.2 = 18 kmph.

New Time = 135 ÷ 18 = 7.5 hours.

∴ The correct answer is 7.5 hours.

Alternate Method

Given:

Original Distance (D₁) = 225 km

Original Time (T₁) = 15 hours

Formula Used:

Speed = Distance ÷ Time

Time = Distance ÷ Speed

Calculations:

⇒ Original Speed (S₁) = 225 ÷ 15 = 15 kmph

⇒ New Distance (D₂) = 225 − (40% of 225)

⇒ D₂ = 225 − (0.40 × 225) = 225 − 90 = 135 km

⇒ New Speed (S₂) = 15 + (20% of 15)

⇒ S₂ = 15 + (0.20 × 15) = 15 + 3 = 18 kmph

⇒ New Time (T₂) = D₂ ÷ S₂ = 135 ÷ 18

⇒ T₂ = 7.5 hours

∴ The correct answer is 7.5 hours.

Additional Information

Speed-Distance Relationship

When time is constant, distance is directly proportional to speed (D₁/D₂ = S₁/S₂).

Average Speed

If a person covers two equal distances at speeds x and y, the Average Speed = 2xy ÷ (x + y).

Unit Conversion

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

Shortcut Trick

Total Days = 7 (Feb) + 31 (Mar) + 23 (Apr) = 61 days.

As 2024 is a leap year, total days = 366.

Simple Interest (SI) = (4000 × 6 × 61) ÷ (100 × 366)

SI = 240 × (61 ÷ 366) = 240 × (1 ÷ 6) = ₹40.

∴ The correct answer is ₹40.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate (R) = 6% per annum

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

⇒ Number of days calculation:

⇒ 2024 is a leap year (divisible by 4), so February has 29 days.

⇒ Days in February = 29 − 22 = 7 days

⇒ Days in March = 31 days

⇒ Days in April = 23 days

⇒ Total Time (T) = (7 + 31 + 23) = 61 days

⇒ In years, T = 61 ÷ 366 (since 2024 is a leap year)

⇒ SI = (4000 × 6 × 61) ÷ (100 × 366)

⇒ SI = (40 × 6 × 61) ÷ 366

⇒ SI = 240 × (61 ÷ 366) = 240 × (1 ÷ 6)

⇒ SI = ₹40

∴ The correct answer is ₹40.

Additional Information

Leap Year Rule

A year is a leap year if it is divisible by 4. For century years, they must be divisible by 400. Leap years have 366 days instead of 365.

Simple Interest (SI) Formula

SI = (P × R × T) ÷ 100. Always ensure the Time (T) is in years; if given in days, divide by 365 (normal year) or 366 (leap year).

Day Counting Rule

While calculating the period, the day on which money is deposited is generally excluded, and the day on which money is withdrawn is included.

Shortcut Trick

Convert 6 hours into minutes: 6 × 60 = 360 minutes.

Total lines printed = 1,76,400.

Lines per minute = 1,76,400 ÷ 360 = 17,640 ÷ 36 = 490.

∴ The correct answer is 490 lines.

Alternate Method

Given:

Total lines printed = 1,76,400

Total time of operation = 6 hours

Formula Used:

Rate of printing = Total lines ÷ Total time (in minutes)

Calculations:

⇒ Total time in minutes = 6 × 60 minutes

⇒ Total time in minutes = 360 minutes

⇒ Lines per minute = 1,76,400 ÷ 360

⇒ Lines per minute = 17,640 ÷ 36

⇒ Lines per minute = 490

∴ The correct answer is 490 lines.

Additional Information

Unit Conversion

Always ensure the units of time are consistent with the required answer format (1 hour = 60 minutes = 3600 seconds).

Rate and Efficiency

Rate = Total Output ÷ Total Time. This concept is similar to speed (Distance ÷ Time) or efficiency in Time & Work problems.

Shortcut Trick

Let the total work be the LCM of 3, 9, and 18, which is 18 units.

Efficiencies of A, B, and C are 6, 2, and 1 units/day, respectively.

Combined efficiency of A + B + C = 6 + 2 + 1 = 9 units/day.

To complete twice the work (18 × 2 = 36 units), the time taken = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given:

Time taken by A = 3 days

Time taken by B = 9 days

Time taken by C = 18 days

Formula Used:

Efficiency = Total Work ÷ Time Taken

Time Taken = Total Work ÷ Total Efficiency

Calculations:

⇒ Total combined efficiency of A, B and C = 6 + 2 + 1 = 9 units/day

⇒ Twice the original work = 2 × 18 = 36 units

⇒ Time taken to complete twice the work = 36 ÷ 9 = 4 days

∴ The correct answer is 4 days.

Additional Information

Total Work LCM Method

Taking the LCM of individual days as the total work simplifies calculation by converting fractional work rates into whole number efficiencies.

Work and Efficiency Relation

Efficiency is inversely proportional to the time taken to complete a task. If a person takes more days, their daily work efficiency is lower.

Shortcut Trick

Let the younger sister be x and elder sister be x + 3.

Equation: (x + 3) + 2 = 2 × (x − 2)

x + 5 = 2x − 4 ⇒ x = 9.

Present ages: 9 and 12 years (Sum = 21).

Sum after 8 years = 21 + (2 × 8) = 37 years.

∴ The correct answer is 37 years.

Alternate Method

Given: Difference in ages = 3 years. Relation after 2 years provided.

⇒ Let the present age of the younger sister = y

⇒ Present age of the elder sister = y + 3

⇒ According to the question: (y + 3) + 2 = 2 × (y − 2)

⇒ y + 5 = 2y − 4

⇒ 2y − y = 5 + 4

⇒ y = 9 years

⇒ Elder sister age = 9 + 3 = 12 years

⇒ Sum of their ages after 8 years = (9 + 8) + (12 + 8)

⇒ 17 + 20 = 37 years

∴ The correct answer is 37 years.

Additional Information

Age Difference Property

The difference between the ages of two people remains constant regardless of the number of years passed or preceding.

Sum of Ages in Future

To find the sum of ages of 'n' people after 't' years, add (n × t) to the present sum of their ages.

Ratio Method for Ages

When age ratios at different time intervals are given, equate the unit difference of the parts to the actual year difference.

Shortcut Trick

Since the time is the same, the ratio of height to shadow remains constant.

Ratio = 15 ÷ 5 = 3.

Height of tower = 3 × Shadow length = 3 × 46.5.

Height of tower = 139.5 m.

∴ The correct answer is 139.5 m.

Alternate Method

Given:

Height of stick (h) = 15 m

Shadow of stick (s) = 5 m

Shadow of tower (S) = 46.5 m

Formula Used:

For similar triangles formed by sun rays: Height₁/Shadow₁ = Height₂/Shadow₂

Calculations:

⇒ 15 / 5 = Height of tower / 46.5

⇒ 3 = Height of tower / 46.5

⇒ Height of tower = 3 × 46.5

⇒ Height of tower = 139.5 m

∴ The correct answer is 139.5 m.

Additional Information

Similar Triangles in Shadows

Objects and their shadows form similar right-angled triangles because the angle of elevation of the sun is the same for both at any given moment.

Height and Distance Basics

Tan θ = Opposite / Adjacent. Here, Tan θ = Height / Shadow. Since θ (sun angle) is constant, the ratio H/S is always constant.

Shortcut Trick

Ratio of First : Second : Third numbers = 12 : 21 : 5.

Given: 50% of 1^st number (12 units) = 78 → 6 units = 78 → 1 unit = 13.

Absolute difference between 2^nd (21 units) and 3^rd (5 units) = 16 units.

Required: 50% of 16 units = 8 units = 8 × 13 = 104.

∴ The correct answer is 104.

Alternate Method

Given: Ratio = 12 : 21 : 5. Let numbers be 12x, 21x, and 5x.

⇒ 50% of 12x = 78

⇒ 6x = 78

⇒ x = 13

⇒ Absolute difference of 3^rd and 2^nd = |5x − 21x| = 16x

⇒ 50% of 16x = 8x

⇒ 8 × 13 = 104

∴ The correct answer is 104.

Additional Information

Ratio and Proportion

If a : b : c is the ratio, the values can be taken as ax, bx, and cx where x is the common multiplier.

Percentage Calculation

50% of a number is simply half of that number (Value × 1/2).

Absolute Difference

The absolute difference |a − b| is always a non-negative value regardless of which number is larger.