Quantitative Aptitude PYQs

Shortcut Trick

Age of new member (teacher) = Old Average + (New Strength × Increase in Average)

Old average age = 35 years

Total people (after adding teacher) = 31 + 1 = 32

Increase in average = 36 − 35 = 1 year

Age of teacher = 35 + (32 × 1) = 35 + 32 = 67 years

∴ The correct answer is 67 years.

Alternate Method

Given:

Initial number of students = 31

Initial average age = 35 years

New average age after including teacher = 36 years

Formula Used:

Average = Total Sum of Ages ÷ Total Number of Persons

Total Sum = Average × Number of Persons

Calculations:

⇒ Total age of 31 students = 31 × 35 = 1085 years

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

⇒ Total age of 32 people = 32 × 36 = 1152 years

⇒ Age of teacher = Total age of 32 people − Total age of 31 students

⇒ Age of teacher = 1152 − 1085 = 67 years

∴ The correct answer is 67 years.

Additional Information

Effect of Adding a Member

When a new member is added, if the average increases, the new member's value is greater than the old average. Formula: Value = Old Avg + (New Total × Increase).

Effect of Removing a Member

If a member is excluded and the average decreases, the excluded member's value = Old Average + (Remaining Count × Decrease in Average).

Weighted Average

When two groups are combined, the resulting average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average.

Shortcut Trick

Initial Men = 104, Total Men = 104 + 52 = 156.

Using Inverse Variation: M₁ × D₁ = M₂ × D₂

104 × 12 = 156 × D₂

D₂ = (104 × 12) ÷ 156 = 8 weeks.

∴ The correct answer is 8 weeks.

Alternate Method

Given:

Initial number of men (M₁) = 104

Initial duration (D₁) = 12 weeks

Additional men = 52

Formula Used:

Total Work = Men × Time

M₁D₁ = M₂D₂

Calculations:

⇒ Total new men (M₂) = 104 + 52 = 156 men

⇒ 104 × 12 = 156 × D₂

⇒ D₂ = (104 × 12) ÷ 156

⇒ D₂ = 1248 ÷ 156

⇒ D₂ = 8 weeks

∴ The correct answer is 8 weeks.

Additional Information

Chain Rule Concept

The number of men is inversely proportional to the time taken to consume a fixed quantity of provision. Formula: M₁D₁H₁/W₁ = M₂D₂H₂/W₂.

Inverse Variation

If one quantity increases and the other decreases such that their product remains constant, it is called inverse variation (x × y = k).

Shortcut Trick

Effective rate of Compound Interest for 2 years at 20% = 20 + 20 + (20 × 20)/100 = 44%.

Total Amount = 100% (Principal) + 44% (Interest) = 144% of Principal.

Amount = 2800 × 144/100 = 28 × 144.

Amount = ₹4,032.

∴ The correct answer is ₹4,032.

Alternate Method

Given:

Principal (P) = ₹2,800

Rate (R) = 20% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 2800(1 + 20/100)²

⇒ A = 2800(1 + 1/5)²

⇒ A = 2800(6/5)²

⇒ A = 2800 × (36/25)

⇒ A = 112 × 36

⇒ A = ₹4,032

∴ The correct answer is ₹4,032.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate is calculated using the successive percentage formula: R + R + (R × R)/100.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is given by: CI − SI = P(R/100)².

Compounding Periods

If interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2n) in the standard formula.

Shortcut Trick

Apply Complementary Angles property: sec θ = cosec(90° − θ) and cot θ = tan(90° − θ).

Term 1: sec 33° ÷ cosec 57° = sec 33° ÷ sec 33° = 1.

Term 2: √3 × (cot 13° cot 77°) × (cot 21° cot 69°) × cot 60° = √3 × 1 × 1 × (1/√3) = 1.

Term 3: −2(sin² 41° + cos² 41°) = −2(1) = −2.

Net Result: 1 + 1 − 2 = 0.

∴ The correct answer is 0.

Alternate Method

Given: sec 33°/cosec 57° + √3 cot 13° cot 21° cot 60° cot 69° cot 77° − 2(sin² 41° + sin² 49°)

Formula Used:

1. sec θ = cosec(90° − θ)
2. cot θ₁ cot θ₂ = 1 if θ₁ + θ₂ = 90°
3. sin² θ + cos² θ = 1

Calculations:

⇒ [sec 33° / cosec(90° − 33°)] + √3 [cot 13° cot(90° − 13°)] [cot 21° cot(90° − 21°)] cot 60° − 2[sin² 41° + sin²(90° − 41°)]

⇒ [sec 33° / sec 33°] + √3 [cot 13° tan 13°] [cot 21° tan 21°] cot 60° − 2[sin² 41° + cos² 41°]

⇒ 1 + √3 × 1 × 1 × (1/√3) − 2(1)

⇒ 1 + 1 − 2

⇒ 0

∴ The correct answer is 0.

Additional Information

Complementary Angle Identities

sin θ = cos(90° − θ), tan θ = cot(90° − θ), and sec θ = cosec(90° − θ). These are useful for simplifying ratios where angles sum to 90°.

Standard Trigonometric Values

cot 60° = 1/√3, tan 60° = √3, sin 30° = 1/2, cos 30° = √3/2. Memorizing these is essential for quick simplification.

Trigonometric Unitary Rule

The product cot A × cot B = 1 and tan A × tan B = 1 if A + B = 90°. Similarly, sin²A + sin²B = 1 if A + B = 90°.

Shortcut Trick

Discount = Marked Price − Selling Price

Discount = 3,050 − 2,220 = 830

Discount% = (830 ÷ 3,050) × 100

Discount% = 8300 ÷ 305 ≈ 27.2131...

Rounding to two decimal places, we get 27.21%.

∴ The correct answer is 27.21%.

Alternate Method

Given: Marked Price (MP) = ₹3,050 | Selling Price (SP) = ₹2,220
Formulas: Discount = MP − SP | Discount% = (Discount ÷ MP) × 100

⇒ Calculate Discount: 3,050 − 2,220 = 830
⇒ Apply Discount% formula: (830 ÷ 3,050) × 100
⇒ 83000 ÷ 3050 = 8300 ÷ 305
⇒ 27.2131...% ≈ 27.21%

∴ The correct answer is 27.21%.

Additional Information

Marked Price and Discount

The Marked Price (MP) is the price printed on the product. Discount is always calculated on the MP unless stated otherwise.

Rounding Decimals

To round to two decimal places, look at the third decimal digit. If it is 5 or greater, round up; otherwise, keep the second digit as is.

Shortcut Trick

Ratio of Arman : Riya (Present) = 4 : 1 (Difference = 3 units)

Ratio after 8 years = 3 : 1 (Difference = 2 units)

Equalize difference: Present (4:1)×2 = 8:2 | After 8 yrs (3:1)×3 = 9:3

1 unit increase (8 to 9 or 2 to 3) = 8 years

Present age of Arman = 8 units = 8 × 8 = 64 years

∴ The correct answer is 64 years.

Alternate Method

Given:

Present age of Riya = x years

Present age of Arman = 4x years

Calculations:

⇒ According to the question: (4x + 8) = 3 × (x + 8)

⇒ 4x + 8 = 3x + 24

⇒ 4x − 3x = 24 − 8

⇒ x = 16

⇒ Present age of Arman = 4x = 4 × 16 = 64 years

∴ The correct answer is 64 years.

Additional Information

Ratio Difference Method

When a constant number of years is added to two people, the absolute difference between their ages remains constant over time.

Successive Age Calculation

If current age is A and the ratio after n years is x:y, then (A + n) / (B + n) = x / y can be used to solve for unknown variables.

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

Required percentage = (1/30) ÷ (2/50) × 100

Required percentage = (1/30) × (50/2) × 100 = 5/6 × 100

Required percentage = 500/6 ≈ 83.33%

∴ The correct answer is 83.33%.

Alternate Method

Given values:

First fraction = 1/30

Second fraction = 2/50

Formula: Percentage = (Value / Base) × 100

⇒ Required Percentage = [(1/30) / (2/50)] × 100

⇒ Required Percentage = (1/30) × (50/2) × 100

⇒ Required Percentage = (50 / 60) × 100

⇒ Required Percentage = (5 / 6) × 100

⇒ Required Percentage = 83.3333...

∴ The correct answer is 83.33%.

Additional Information

Fraction to Percentage Conversion

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

Reciprocal Percentages

Standard values: 1/6 = 16.67%, therefore 5/6 = 5 × 16.67% = 83.33%.

Percentage Change

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

Shortcut Trick

Convert percentages and fractions: 20% = 1/5 and 50% = 1/2.

Net multiplier = 1/5 × 1/2 × 5/9 = 1/18.

Let the number be N.

N × (1/18) = 409 ⇒ N = 409 × 18 = 7362.

∴ The correct answer is 7362.

Alternate Method

Given: Result = 409, Operations = 20%, 50%, 5/9

Let the unknown number be x.

⇒ 20% × 50% × (5/9) × x = 409

⇒ (20/100) × (50/100) × (5/9) × x = 409

⇒ (1/5) × (1/2) × (5/9) × x = 409

⇒ (5 / 90) × x = 409

⇒ (1 / 18) × x = 409

⇒ x = 409 × 18

⇒ x = 7362

∴ The correct answer is 7362.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, remember common fractions: 20% = 1/5, 25% = 1/4, 50% = 1/2, and 75% = 3/4.

Successive Percentage Application

When taking percentages of percentages, multiply the fractions together: a% of b% of C = (a/100) × (b/100) × C.

Of Operator in Math

The word of in mathematical word problems almost always signifies the multiplication operation.

Shortcut Trick

Combined Ratio calculation: First : Second : Third

Ratio 1 (S:F) = 3 : 13 → F:S = 13 : 3

Ratio 2 (T:S) = 8 : 12 → S:T = 12 : 8 = 3 : 2

Since Second number is '3' in both ratios, Combined Ratio F : S : T = 13 : 3 : 2

Total units = 13 + 3 + 2 = 18 units → 840. Value of 3 units = (840 ÷ 18) × 3 = 840 ÷ 6 = 140.

∴ The correct answer is 140.

Alternate Method

Given:

Sum of numbers (A + B + C) = 840

Ratio B : A = 3 : 13 ⇒ A : B = 13 : 3

Ratio C : B = 8 : 12 ⇒ B : C = 12 : 8 = 3 : 2

Calculations:

⇒ Combined Ratio A : B : C = 13 : 3 : 2

⇒ Let the numbers be 13x, 3x, and 2x

⇒ 13x + 3x + 2x = 840

⇒ 18x = 840

⇒ x = 840 ÷ 18 = 140 ÷ 3

⇒ Second number (B) = 3x = 3 × (140 ÷ 3)

⇒ Second number = 140

∴ The correct answer is 140.

Additional Information

Combined Ratio Rule

If A : B = m : n and B : C = p : q, then A : B : C = (m×p) : (n×p) : (n×q).

Proportion Fundamentals

In any ratio a : b, 'a' is the antecedent and 'b' is the consequent. For a sum S divided in ratio a:b:c, the second part is [b ÷ (a+b+c)] × S.

Shortcut Trick

Relative Speed in the same direction = Difference of speeds.

Relative Speed = 82 − 68 = 14 km/h.

Distance = Relative Speed × Time.

Distance = 14 × 2.5 = 35 km.

∴ The correct answer is 35 km.

Alternate Method

Given:

Speed of Train 1 (S₁) = 68 km/h

Speed of Train 2 (S₂) = 82 km/h

Time (T) = 2.5 hours

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Distance covered by Train 1 = 68 × 2.5

⇒ Distance₁ = 170 km

⇒ Distance covered by Train 2 = 82 × 2.5

⇒ Distance₂ = 205 km

⇒ Distance between the two trains = Distance₂ − Distance₁

⇒ Distance = 205 − 170 = 35 km

∴ The correct answer is 35 km.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the difference: (S₁ − S₂).

Relative Speed in Opposite Direction

When two objects move in opposite directions, their relative speed is the sum: (S₁ + S₂).

Unit Conversion

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

Shortcut Trick

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

Let a = 3x and b = 4y.

Numerator = 4 × 3x × 4y = 48xy

Given expression = 48xy ÷ 24xy = 2

∴ The correct answer is 2.

Alternate Method

Given:

Expression: [(3x + 4y)² − (3x − 4y)²] ÷ (24xy)

Formulas Used:

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

Calculations:

Expanding the binomials in the numerator:

⇒ (3x + 4y)² = 9x² + 24xy + 16y²

⇒ (3x − 4y)² = 9x² − 24xy + 16y²

Subtracting the second term from the first term:

⇒ (3x + 4y)² − (3x − 4y)² = (9x² + 24xy + 16y²) − (9x² − 24xy + 16y²)

⇒ (3x + 4y)² − (3x − 4y)² = 9x² + 24xy + 16y² − 9x² + 24xy − 16y²

⇒ (3x + 4y)² − (3x − 4y)² = 48xy

Dividing the simplified numerator by the denominator:

⇒ 48xy ÷ 24xy = 2

∴ The correct answer is 2.

Additional Information

Difference of Squares

The difference of squares identity is written as: (a + b)(a − b) = a² − b².

Sum of Squared Binomials

The sum of the squares of a sum and difference of two numbers is: (a + b)² + (a − b)² = 2(a² + b²).

Square of a Trinomial

The square of a trinomial is given by the formula: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Shortcut Trick

Scheme A: Net discount = 18 + 10 − (18 × 10 ÷ 100) = 28 − 1.8 = 26.2%.

Scheme B: Buy 2, get 8 means 8 is the total items and 6 are free. Discount = (6 ÷ 8) × 100 = 75%.

Scheme C: Flat discount = 24%.

Scheme D: Net discount = 42 + 45 − (42 × 45 ÷ 100) = 87 − 18.9 = 68.1%.

Comparing 26.2%, 75%, 24%, and 68.1%, the highest discount is 75%.

∴ The correct answer is Scheme B.

Alternate Method

Formulas: Successive Discount = a + b − (ab ÷ 100) | Buy X Get Y Discount% = [Free Items ÷ Total Items (X+Y)] × 100.

⇒ Scheme A: 18 + 10 − 1.8 = 26.2%.
⇒ Scheme B: Buy 2 get 8 means you pay for 2 and get 8. Free items = 8 − 2 = 6. Discount% = (6 ÷ 8) × 100 = 75%.
⇒ Scheme C: 24%.
⇒ Scheme D: 42 + 45 − 18.9 = 68.1%.
⇒ Comparing all values: 75% > 68.1% > 26.2% > 24%.

∴ The correct answer is Scheme B.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the effective discount is always less than (a + b)% due to the compounding effect.

Buy X Get Y Scheme

In 'Buy X Get Y' (where Y is total), free items = Y − X. In 'Buy X Get Y free', free items = Y and total = X + Y.

Shortcut Trick

Relative Speed in same direction = 102 − 66 = 36 km/h.

Convert to m/s: 36 × (5/18) = 10 m/s.

Total Distance to cover = Sum of lengths = 230 + 230 = 460 m.

Time taken = Total Distance ÷ Relative Speed = 460 ÷ 10 = 46 sec.

∴ The correct answer is 46 sec.

Alternate Method

Given:

Length of each train (L₁ = L₂) = 230 m

Speed of faster train (S₁) = 102 km/h

Speed of slower train (S₂) = 66 km/h

Formula Used:

Time = (Total Distance) ÷ (Relative Speed)

Calculations:

⇒ Total Distance = L₁ + L₂ = 230 + 230 = 460 m

⇒ Relative Speed (Same direction) = S₁ − S₂ = 102 − 66 = 36 km/h

⇒ Convert km/h to m/s: 36 × (5/18) = 10 m/s

⇒ Time taken = 460 ÷ 10

⇒ Time taken = 46 sec

∴ The correct answer is 46 sec.

Additional Information

Relative Speed Concepts

When moving in the same direction, Relative Speed = S₁ − S₂. When moving in opposite directions, Relative Speed = S₁ + S₂.

Speed Conversion

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

Total Crossing Distance

When one train passes another completely, the total distance covered is always the sum of the lengths of both trains.

Shortcut Trick

The surface area of a sphere is given by the formula 4πr².

Given radius (r) = 6 cm.

Surface Area = 4 × π × (6)² = 4 × 36 × π = 144π cm².

∴ The correct answer is 144 π cm².

Alternate Method

Given:

Radius (r) of the sphere = 6 cm

Formula Used:

Surface Area (A) of a sphere = 4πr²

Calculations:

⇒ Surface Area = 4 × π × r²

⇒ Surface Area = 4 × π × (6)²

⇒ Surface Area = 4 × π × 36

⇒ Surface Area = 144π cm²

∴ The correct answer is 144 π cm².

Additional Information

Volume of a Sphere

The amount of space occupied by a sphere is given by Volume = (4/3)πr³.

Total Surface Area of Hemisphere

For a solid hemisphere, the total surface area includes the curved part and the base: TSA = 3πr².

Curved Surface Area of Hemisphere

The surface area of just the curved part of a hemisphere is exactly half that of a sphere: CSA = 2πr².

Volume of Hemisphere

The volume of a hemisphere is half the volume of a sphere: Volume = (2/3)πr³.

Shortcut Trick

Total Capacity (LCM of 9 and 12) = 36 units

Efficiency of P + Q = 36 ÷ 9 = 4 units/hour

Efficiency of P = 36 ÷ 12 = 3 units/hour

Efficiency of Q = 4 − 3 = 1 unit/hour

Time for Q to fill 3/4^th work = (3/4 × 36) ÷ 1 = 27 hours

∴ The correct answer is 27 hours.

Alternate Method

Given:

Time taken by P and Q together = 9 hours

Time taken by P alone = 12 hours

Calculations:

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

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

⇒ 3/4^th of Total Cistern Capacity = 3/4 × 36 = 27 units

⇒ Time taken by Q alone = 3/4^th Capacity ÷ Efficiency of Q

⇒ Time = 27 ÷ 1 = 27 hours

∴ The correct answer is 27 hours.

Additional Information

Combined Work Formula

If two individuals fill a tank in x and y hours respectively, they together fill it in (xy) ÷ (x + y) hours.

Individual Work Formula

If together they take T hours and one takes x hours, the other takes (T × x) ÷ (x − T) hours.

Pipes and Cisterns

Inlet pipes have positive efficiency, while outlet (leakage) pipes have negative efficiency.

Shortcut Trick

Identify the volume of water to be removed: Volume = π × r² × drop in height.

Base Area = (22/7) × 70 × 70 = 15,400 cm².

Volume to pump = 15,400 × 50 = 770,000 cm³ = 770 Litres (as 1L = 1000 cm³).

Time required = Total Litres ÷ Flow Rate = 770 ÷ 5 = 154 seconds.

∴ The correct answer is 154 sec.

Alternate Method

Given: Internal radius (r) = 70 cm, Total height (H) = 1.5 m = 150 cm, Water level drop (h) = 50 cm, Rate = 5 litres/sec.

Formula Used: Volume of water = πr²h | Time = Volume ÷ Rate

⇒ Volume to be removed (V) = π × r² × h

⇒ V = (22/7) × 70 × 70 × 50 cm³

⇒ V = 22 × 10 × 70 × 50 = 770,000 cm³

⇒ Volume in litres = 770,000 ÷ 1000 = 770 litres

⇒ Time = 770 ÷ 5 = 154 seconds

∴ The correct answer is 154 sec.

Additional Information

Volume of a Cylinder

The volume (V) is given by V = πr²h, where r is the radius and h is the height/depth.

Unit Conversion

1 Litre = 1000 cm³ = 0.001 m³. Also, 1 m³ = 1000 Litres.

Flow Rate Relation

Time = Total Volume ÷ Rate of Flow. Ensure all units (volume and rate) are consistent before division.

Shortcut Trick

In a mean proportional relationship, P × 169 = 104².

P = (104 × 104) ÷ 169

P = (13 × 8 × 13 × 8) ÷ (13 × 13) = 8 × 8 = 64.

∴ The correct answer is 64.

Alternate Method

Given: First number = P, Second number = 169, Mean Proportional = 104.

Formula Used: If b is the mean proportional between a and c, then b² = a × c or b = √(a × c).

⇒ 104 = √(P × 169)

⇒ 104² = P × 169

⇒ 10816 = P × 169

⇒ P = 10816 ÷ 169

⇒ P = 64

∴ The correct answer is 64.

Additional Information

Mean Proportional

It is the geometric mean of two numbers. For a and c, the mean proportional b satisfies a/b = b/c.

Third Proportional

If a, b, and c are in continued proportion, then c is the third proportional to a and b, calculated as c = b² ÷ a.

Fourth Proportional

For numbers a, b, and c, the fourth proportional d satisfies a : b = c : d, calculated as d = (b × c) ÷ a.

Shortcut Trick

The given expression is in the standard form of the Algebraic Identity: (a³ + b³) ÷ (a² − ab + b²).

According to the identity, (a³ + b³) ÷ (a² − ab + b²) = a + b.

Identify the values from the expression: a = 5.9 and b = 0.1.

Applying the shortcut: 5.9 + 0.1 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

Numerator = (5.9)³ + (0.1)³

Denominator = (5.9)² + (0.1)² − (5.9 × 0.1)

Formula Used:

The Sum of Two Cubes identity is: a³ + b³ = (a + b)(a² − ab + b²).

Calculations:

⇒ Let a = 5.9 and b = 0.1.

⇒ The denominator is already in the form (a² + b² − ab).

⇒ Expression = [(a + b)(a² − ab + b²)] ÷ (a² + b² − ab)

⇒ The term (a² − ab + b²) cancels out from both the numerator and denominator.

⇒ Expression = a + b

⇒ Expression = 5.9 + 0.1

⇒ Expression = 6.0

∴ The correct answer is 6.

Additional Information

Difference of Cubes

The identity for the difference of cubes is a³ − b³ = (a − b)(a² + ab + b²).

Algebraic Squares

The square of a sum is (a + b)² = a² + 2ab + b² and the square of a difference is (a − b)² = a² − 2ab + b².

Difference of Squares

The expression a² − b² factors into (a − b)(a + b).

Shortcut Trick

Total Work (LCM of 18, 24) = 72 units.

Efficiency: A = 4, B = 3. Combined = 7 units/day.

Work done in 5 days = 7 × 5 = 35 units.

Remaining work = 72 − 35 = 37 units.

B completes remaining = 37 ÷ 3 = 12(1/3) days.

∴ The correct answer is 12(1/3) days.

Alternate Method

Given: Time taken by A = 18 days, Time taken by B = 24 days. Working together for 5 days.

Formula Used: Efficiency = Total Work ÷ Time; Remaining Work = Total Work − Work Done.

⇒ Combined Efficiency of A and B = 4 + 3 = 7 units/day.

⇒ Work completed in 5 days = 7 × 5 = 35 units.

⇒ Remaining Work = 72 − 35 = 37 units.

⇒ Time taken by B to finish remaining work = 37 ÷ 3 days.

⇒ Time = 12(1/3) days.

∴ The correct answer is 12(1/3) days.

Additional Information

Total Work and Efficiency

The LCM method is the fastest way to calculate total work. Efficiency is inversely proportional to the time taken.

Work and Wages

Wages are distributed based on the ratio of work done by each person, which is Efficiency × Time.

Fractional Work

If a person completes a job in 'D' days, their 1-day work is 1/D. This can be used for fractional calculations.

Shortcut Trick

Let A be the cost of an apple and B be the cost of a banana.

Equation 1: 3A + 5B = 75 (Multiply by 4 → 12A + 20B = 300)

Equation 2: 4A + 2B = 86 (Multiply by 3 → 12A + 6B = 258)

Difference: (20B − 6B) = (300 − 258) ⇒ 14B = 42

Cost of one banana (B) = 42 ÷ 14 = ₹3.

∴ The correct answer is ₹3.

Alternate Method

Given:

Cost of 3 apples and 5 bananas = ₹75

Cost of 4 apples and 2 bananas = ₹86

Formula Used:

Elimination Method for Simultaneous Linear Equations.

Let the cost of one apple be x and one banana be y.

⇒ 3x + 5y = 75 --- (Equation i)

⇒ 4x + 2y = 86 --- (Equation ii)

To eliminate x, multiply Equation (i) by 4 and Equation (ii) by 3:

⇒ 4 × (3x + 5y) = 4 × 75 ⇒ 12x + 20y = 300 --- (Equation iii)

⇒ 3 × (4x + 2y) = 3 × 86 ⇒ 12x + 6y = 258 --- (Equation iv)

Subtracting Equation (iv) from Equation (iii):

⇒ (12x + 20y) − (12x + 6y) = 300 − 258

⇒ 14y = 42

⇒ y = 42 ÷ 14

⇒ y = 3

∴ The correct answer is ₹3.

Additional Information

Elimination Method

A process of solving a system of equations by adding or subtracting them to remove one variable.

Substitution Method

Solve for one variable in one equation and substitute its value into the other equation.

Cost-Price Consistency

In linear problems, the individual price per unit remains constant regardless of the quantity purchased in different sets.