Quantitative Aptitude PYQs

Shortcut Trick

Apply the BODMAS rule: solve Brackets, then Of, then Division, followed by Addition and Subtraction.

Solve Bracket and Of: 91 − 42 + (33 × 1) + 9 + 27 − 13.

Combine positive and negative values: (91 + 33 + 9 + 27) − (42 + 13).

Final calculation: 160 − 55 = 105.

∴ The correct answer is 105.

Alternate Method

Given: 91 − 42 + 33 of 1 + (18 ÷ 2) + 27 − 13

Follow the BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) priority:

⇒ 91 − 42 + 33 of 1 + 9 + 27 − 13 (Solving Bracket: 18 ÷ 2 = 9)

⇒ 91 − 42 + 33 + 9 + 27 − 13 (Solving Of: 33 × 1 = 33)

⇒ 124 − 42 + 9 + 27 − 13 (Performing Addition: 91 + 33)

⇒ 82 + 9 + 27 − 13 (Performing Subtraction: 124 − 42)

⇒ 91 + 27 − 13 (Performing Addition: 82 + 9)

⇒ 118 − 13 (Performing Addition: 91 + 27)

⇒ 105

∴ The correct answer is 105.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets (), Of (multiplication), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Of Operation

The 'Of' operator represents Multiplication but holds a higher priority than Division and standard multiplication in the calculation sequence.

Types of Brackets

Always solve in the order: Circular Brackets ( ), then Curly Brackets { }, and finally Square Brackets [ ].

Shortcut Trick

Ratio sum (x + y) = Sum ÷ HCF = 156 ÷ 12 = 13.

Ratio product (xy) = LCM ÷ HCF = 480 ÷ 12 = 40.

Find two coprime factors of 40 that add up to 13: These are 5 and 8.

Required numbers = 12 × 5 and 12 × 8.

∴ The correct answer is 60 and 96.

Alternate Method

Given: HCF = 12, LCM = 480, Sum = 156.

Formula Used: Let the numbers be 12x and 12y (where x and y are coprime).

Sum = 12x + 12y = 12(x + y)

Product = 12x × 12y = HCF × LCM

⇒ 12(x + y) = 156

⇒ x + y = 13

⇒ 12 × x × 12 × y = 12 × 480

⇒ 12xy = 480 ⇒ xy = 40

⇒ Using (x − y)² = (x + y)² − 4xy

⇒ (x − y)² = (13)² − 4(40) = 169 − 160 = 9

⇒ x − y = √9 = 3

⇒ Solving x + y = 13 and x − y = 3:

⇒ 2x = 16 ⇒ x = 8

⇒ 2y = 10 ⇒ y = 5

⇒ Numbers are 12 × 8 = 96 and 12 × 5 = 60.

∴ The correct answer is 60 and 96.

Additional Information

Fundamental Relation of HCF and LCM

For any two numbers A and B: HCF(A, B) × LCM(A, B) = A × B. This rule applies only to two numbers.

HCF as a Factor of LCM

The Highest Common Factor (HCF) is always a factor of the Least Common Multiple (LCM).

Coprime Numbers

If the HCF of two numbers is H, the numbers can be represented as Hx and Hy, where x and y are coprime (HCF of x and y is 1).

Shortcut Trick

Separate the whole numbers and the fractions: (6 + 4 − 2 − 1) + (1/3 + 1/2 − 1/4 − 1/6)

Calculate whole numbers: 6 + 4 − 2 − 1 = 7

Calculate fractions using LCM of 12: (4 + 6 − 3 − 2) / 12 = 5/12

Combine the results: 7 + 5/12 = 7(5/12)

∴ The correct answer is 7(5/12).

Alternate Method

Given: 6(1/3) + 4(1/2) − 2(1/4) − 1(1/6)

Method: Convert mixed fractions into improper fractions and simplify.

⇒ (6 × 3 + 1)/3 + (4 × 2 + 1)/2 − (2 × 4 + 1)/4 − (1 × 6 + 1)/6

⇒ 19/3 + 9/2 − 9/4 − 7/6

⇒ LCM of denominators (3, 2, 4, 6) = 12

⇒ (19 × 4 + 9 × 6 − 9 × 3 − 7 × 2) / 12

⇒ (76 + 54 − 27 − 14) / 12

⇒ 89 / 12

⇒ 7(5/12)

∴ The correct answer is 7(5/12).

Additional Information

BODMAS Rule

Always perform operations in the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Mixed Fractions

A mixed fraction a(b/c) can be converted to an improper fraction using the formula: (a × c + b) / c.

LCM for Fractions

When adding or subtracting fractions with different denominators, always find the Least Common Multiple (LCM) to make the denominators common.

Shortcut Trick

Directly calculate the square roots: √0.04 = 0.2, √0.16 = 0.4, and √0.36 = 0.6.

Substitute these into the expression: (0.2 ÷ 0.4) × (0.8 ÷ 0.4) × (1.2 ÷ 0.6).

This simplifies to: 0.5 × 2 × 2 = 2.

∴ The correct answer is 2.

Alternate Method

Given: (√0.04 / 0.4) × (0.8 / √0.16) × (1.2 / √0.36)

Step 1: Simplify the square root terms in the numerator and denominator.

⇒ √0.04 = √(4 / 100) = 2 / 10 = 0.2

⇒ √0.16 = √(16 / 100) = 4 / 10 = 0.4

⇒ √0.36 = √(36 / 100) = 6 / 10 = 0.6

Step 2: Replace the simplified values back into the expression.

⇒ (0.2 / 0.4) × (0.8 / 0.4) × (1.2 / 0.6)

Step 3: Perform the division for each fraction.

⇒ (1 / 2) × 2 × 2

⇒ 0.5 × 2 × 2

Step 4: Multiply the remaining values to find the final result.

⇒ 1 × 2 = 2

∴ The correct answer is 2.

Additional Information

Square Roots of Decimals

When finding the square root of a decimal, the number of decimal places in the result is always half of those in the original number (e.g., √0.0001 = 0.01).

BODMAS Rule

Always follow the BODMAS order (Brackets, Orders, Division, Multiplication, Addition, Subtraction) to solve complex numerical expressions correctly.

Shortcut Trick

To find x, square the result on the right side and subtract the constant term inside the square root.

Step: 5² − 1

⇒ 25 − 1 = 24

∴ The correct answer is 24.

Alternate Method

Given:

√(x + 1) = 5

Method Used:

Squaring both sides to eliminate the radical sign.

Calculations:

⇒ (√(x + 1))² = 5²

⇒ x + 1 = 25

⇒ x = 25 − 1

⇒ x = 24

∴ The correct answer is 24.

Additional Information

Radical Equations

An equation in which a variable is under a radical is called a radical equation. To solve it, we must isolate the radical and square both sides.

Properties of Equality

If a = b, then a² = b². This property is used to simplify equations involving square roots.

Transposition Rule

When moving a positive constant to the other side of the equals sign, it becomes negative: x + a = b → x = b − a.

Shortcut Trick

For an odd number of terms in an Arithmetic Progression, the average is the middle term.

Average = Total Sum ÷ Count ⇒ 72 ÷ 3 = 24.

The numbers are consecutive even numbers, so the set is: (24 − 2), 24, (24 + 2).

The numbers are 22, 24, and 26.

The largest number is 26.

∴ The correct answer is 26.

Alternate Method

Given: Sum of three consecutive even numbers = 72.

Formula Used: If the first even number is x, then the next numbers are (x + 2) and (x + 4).

⇒ x + (x + 2) + (x + 4) = 72

⇒ 3x + 6 = 72

⇒ 3x = 72 − 6

⇒ 3x = 66

⇒ x = 66 ÷ 3

⇒ First number (x) = 22

⇒ Largest number (x + 4) = 22 + 4 = 26

∴ The correct answer is 26.

Additional Information

Consecutive Even/Odd Numbers

The difference between any two consecutive even or consecutive odd numbers is always 2. For consecutive integers, the difference is 1.

Middle Term Shortcut

For an Arithmetic Progression with n terms (where n is odd), the middle term is always equal to (Total Sum ÷ n).

Sum of AP Formula

The sum of an AP is given by S_n = (n ÷ 2) × [2a + (n − 1)d], where a is the first term and d is the common difference.

Shortcut Trick

Apply the Algebraic Identity: (a + b)² + (a − b)² = 2(a² + b²)

Here, a = √11 and b = √4. Numerator = 2(11 + 4) = 30.

Expression = 30 ÷ 10√15 = 3 / √15.

On Rationalizing: (3 × √15) / 15 = √15 / 5.

∴ The correct answer is √15 / 5.

Alternate Method

Given: ((√11 + √4)² + (√11 − √4)²) ÷ 10√15

Formula Used: (a + b)² + (a − b)² = 2(a² + b²)

⇒ Numerator = (√11 + √4)² + (√11 − √4)²

⇒ Numerator = 2((√11)² + (√4)²)

⇒ Numerator = 2(11 + 4) = 2 × 15 = 30

⇒ Expression = 30 / 10√15

⇒ Expression = 3 / √15

⇒ Multiply numerator and denominator by √15 (Rationalization):

⇒ (3 × √15) / (√15 × √15) = 3√15 / 15

⇒ Result = √15 / 5

∴ The correct answer is √15 / 5.

Additional Information

Algebraic Identity (a + b)² − (a − b)²

The difference of squares of sum and difference is 4ab: (a + b)² − (a − b)² = 4ab.

Rationalization of Denominator

To remove a radical from the denominator, multiply the fraction by the conjugate or the radical itself to make the denominator a rational number.

Surds Property

For any positive number x, (√x)² = x. Also, √a × √b = √(a × b).

Shortcut Trick

To find the unit digit of a product, simply multiply the unit digits of the individual numbers.

Unit digits are 2, 7, and 7.

Calculation: 2 × 7 × 7 = 14 × 7 = 98.

The last digit is 8.

∴ The correct answer is 8.

Alternate Method

Given: The numbers are 242, 367, and 847.

To find the unit digit of 242 × 367 × 847:

⇒ Unit digit of 242 = 2

⇒ Unit digit of 367 = 7

⇒ Unit digit of 847 = 7

⇒ Unit digit of the entire product = (2 × 7 × 7) mod 10

⇒ 2 × 7 = 14 (unit digit is 4)

⇒ 4 × 7 = 28

⇒ The unit digit of 28 is 8.

∴ The correct answer is 8.

Additional Information

Unit Digit of Power

The unit digit of a number raised to a power can be found using the cyclicity of the last digit. For example, the cyclicity of 2, 3, 7, and 8 is 4.

Cyclicity of 0, 1, 5, 6

Numbers ending in 0, 1, 5, or 6 will always have the same unit digit (0, 1, 5, and 6 respectively) regardless of the power n^k (where k > 0).

Cyclicity of 4 and 9

The digit 4 has a cyclicity of 2 (4¹=4, 4²=6) and 9 also has a cyclicity of 2 (9¹=9, 9²=1).

Shortcut Trick

By using the Pythagoras Triplet (5, 12, 13), where 13 is the diagonal and 12 is the length.

The breadth of the rectangle is 5 cm.

Area of the rectangle = Length × Breadth = 12 × 5 = 60 cm².

∴ The correct answer is 60 cm².

Alternate Method

Given: Length = 12 cm, Diagonal = 13 cm

Formula Used: Diagonal² = Length² + Breadth² and Area = Length × Breadth

⇒ (13)² = (12)² + (Breadth)²

⇒ 169 = 144 + (Breadth)²

⇒ (Breadth)² = 169 − 144 = 25

⇒ Breadth = √25 = 5 cm

⇒ Area of rectangle = 12 cm × 5 cm = 60 cm²

∴ The correct answer is 60 cm².

Additional Information

Pythagoras Triplets

A set of three integers a, b, c satisfying a² + b² = c². Commonly used triplets in exams are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Rectangle Formulas

Perimeter = 2 × (Length + Breadth) | Diagonal = √(Length² + Breadth²). In a rectangle, diagonals are equal and bisect each other.

Shortcut Trick

Recognize the algebraic identity: (a − b)² + 4ab = (a + b)².

Here, a = 540 and b = 460.

The sum (a + b) = 540 + 460 = 1000.

The result is 1000² = 1000000.

∴ The correct answer is 1000000.

Alternate Method

Given:

(540 − 460)² + 4 × 540 × 460

Formula Used:

Expansion of (a − b)² = a² − 2ab + b²

Combined identity: (a − b)² + 4ab = (a + b)²

Calculations:

⇒ Let a = 540 and b = 460.

⇒ Expression = (a − b)² + 4ab

⇒ a² − 2ab + b² + 4ab

⇒ a² + 2ab + b²

⇒ (a + b)²

⇒ (540 + 460)²

⇒ 1000²

⇒ 1000000

∴ The correct answer is 1000000.

Additional Information

Square of a Sum and Difference

The difference between the square of a sum and the square of a difference is given by (a + b)² − (a − b)² = 4ab.

Sum of Squares Identity

Another useful form is (a + b)² + (a − b)² = 2(a² + b²), which simplifies calculations involving addition of squared binomials.

BODMAS Rule

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

Shortcut Trick

Total sum of ages of E, F, and G = Average × Total persons

Total sum = 40 × 3 = 120 years.

Sum of ages of E and F = 50 years.

Age of G = Total sum − Sum of (E + F) = 120 − 50 = 70 years.

∴ The correct answer is 70 years.

Alternate Method

Given:

Sum of ages of E and F = 50 years.

Average age of E, F, and G = 40 years.

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Sum of ages of (E + F + G) = 40 × 3

⇒ E + F + G = 120

⇒ We know, E + F = 50

⇒ 50 + G = 120

⇒ G = 120 − 50

⇒ G = 70 years.

∴ The correct answer is 70 years.

Additional Information

Average Formula

Average = (Sum of all observations) ÷ (Total number of observations).

Sum of N Natural Numbers

The sum of the first n natural numbers is given by [n(n + 1)] ÷ 2.

Weighted Average

When groups have different sizes, the combined average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Shortcut Trick

To find the value, multiply the non-zero digits and adjust: X = 4 × 6 = 24.

Similarly for Y: 5 × 4 = 20.

The sum of X and Y is 24 + 20 = 44.

∴ The correct answer is 44.

Alternate Method

Given:

X = 40 percent of 60

Y = 50 percent of 40

Formula Used:

Value = (Percentage ÷ 100) × Total

⇒ X = (40 ÷ 100) × 60

⇒ X = 0.4 × 60 = 24

⇒ Y = (50 ÷ 100) × 40

⇒ Y = 0.5 × 40 = 20

⇒ Sum = 24 + 20 = 44

∴ The correct answer is 44.

Additional Information

Fractional Conversions

Expressing percentages as fractions simplifies calculation. For example, 50% = 1/2 and 40% = 2/5.

Commutative Property

In percentage calculations, a% of b is equal to b% of a. Thus, 40% of 60 is the same as 60% of 40.

Percentage Change Formula

Percentage change = [(New Value − Original Value) ÷ Original Value] × 100.

Shortcut Trick

Loss percentage of 28% corresponds to a Loss of Rs. 98. Unitary method: 1% of the Cost Price (CP) = 98 ÷ 28 = 3.5.

Selling Price (SP) percentage = 100% − 28% = 72%.

Selling Price = 72 × 3.5 = 252.

∴ The correct answer is Rs. 252.

Alternate Method

Given:

Loss = Rs. 98

Loss percentage = 28%

Formula Used:

Loss = (Loss% × Cost Price) ÷ 100

Selling Price (SP) = Cost Price (CP) − Loss

Calculations:

⇒ 28% of CP = 98

⇒ CP = (98 × 100) ÷ 28

⇒ CP = 3.5 × 100 = Rs. 350

⇒ SP = CP − Loss

⇒ SP = 350 − 98 = Rs. 252

∴ The correct answer is Rs. 252.

Additional Information

Profit and Loss Percentage

Profit or loss is always calculated on the Cost Price (CP). Profit% = (Profit ÷ CP) × 100 and Loss% = (Loss ÷ CP) × 100.

Selling Price Formula

SP = CP × (100 + Profit%) ÷ 100 when there is a profit, and SP = CP × (100 − Loss%) ÷ 100 when there is a loss.

Marked Price and Discount

Discount is a reduction offered on the Marked Price (MP). SP = MP − Discount.

Shortcut Trick

Difference between the two Selling Prices = 1200 − 1000 = 200.

Ratio of Profit to Loss is given as 9 :

1. Total units = 9 + 1 = 10 units ⇒ 10 units = 200.

Therefore, 1 unit (Loss) = 20. Cost Price (CP) = 1000 + 20 = 1020.

Selling Price for 20% Profit = 1020 × 1.2 = 1224.

∴ The correct answer is ₹1224.

Alternate Method

Given:

Selling Price 1 (SP₁) = ₹1200 (at Profit)

Selling Price 2 (SP₂) = ₹1000 (at Loss)

Profit = 9 × Loss

Formula Used:

Profit = SP − CP

Loss = CP − SP

New SP = CP × (100 + Profit %)/100

Calculations:

Let the Cost Price of the article be CP.

⇒ Profit = 1200 − CP

⇒ Loss = CP − 1000

According to the question:

⇒ (1200 − CP) = 9 × (CP − 1000)

⇒ 1200 − CP = 9CP − 9000

⇒ 1200 + 9000 = 9CP + CP

⇒ 10200 = 10CP

⇒ CP = 1020

To earn a 20% profit:

⇒ Required SP = 1020 × (120/100)

⇒ Required SP = 102 × 12 = 1224

∴ The correct answer is ₹1224.

Additional Information

Profit and Loss Percentage

Profit % = (Profit/CP) × 100 and Loss % = (Loss/CP) × 100. Both are always calculated on the Cost Price unless stated otherwise.

Selling Price Formula

When profit % is given: SP = CP × (100 + P%)/100. When loss % is given: SP = CP × (100 − L%)/100.

Relation between SP, CP and MP

Marked Price (MP) − Discount = Selling Price (SP). CP × (100 + Profit %) = MP × (100 − Discount %).

Shortcut Trick

Post-graduate employees percentage = (100% − 30%) × (100% − 60%) = 70% × 40% = 28%.

Given that 28% corresponds to 84 post-graduates.

Then, 1% → 84 ÷ 28 = 3.

Number of matriculates = 30% → 30 × 3 = 90.

∴ The correct answer is 90.

Alternate Method

Given:

Percentage of Matriculates = 30%

Percentage of Graduates = 60% of the remaining employees

Number of Post-graduates = 84

Formula Used:

Remaining Value = Total Value × (100 − used percentage)/100

Calculations:

⇒ Let the total number of employees be X.

⇒ Employees remaining after matriculates = X − 30% of X = 0.70X.

⇒ Employees who are graduates = 60% of 0.70X = 0.42X.

⇒ Employees who are post-graduates = 0.70X − 0.42X = 0.28X.

⇒ According to the question, 0.28X = 84.

⇒ X = 84 ÷ 0.28 = 300.

⇒ Number of matriculate employees = 30% of 300 = 0.30 × 300 = 90.

∴ The correct answer is 90.

Additional Information

Successive Percentage Remaining

When percentages are taken from the remaining amount, the final remaining fraction is (1 − r₁) × (1 − r₂) × ... × (1 − r_n).

Percentage to Fraction Conversion

Converting percentages to fractions can simplify calculations: 30% = 3/10, 60% = 3/5, 40% = 2/5.

Concept of Base Change

Always identify the base for each percentage. In this problem, 30% is based on the total, while 60% is based on the 70% remainder.

Shortcut Trick

Total Work (LCM of 8, 12, 24) = 24 units.

Combined Efficiency of A, B and C = (24/8) + (24/12) + (24/24) = 3 + 2 + 1 = 6 units/day.

Required Time = Total Work ÷ Combined Efficiency = 24 ÷ 6 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given:

Time taken by A = 8 days

Time taken by B = 12 days

Time taken by C = 24 days

Formula Used:

Total Work = LCM of time taken by individuals

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Total efficiency of A + B + C = 3 + 2 + 1 = 6 units/day.

⇒ Time taken by all three together = Total Work ÷ Total Efficiency

⇒ Required Time = 24 ÷ 6 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Reciprocal Method

If A, B, and C can do a work in x, y, and z days, then together they take 1 ÷ (1/x + 1/y + 1/z) days.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken. If the ratio of time is a:b, the ratio of efficiency is b:a.

Work and Wages

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

Shortcut Trick

Given, (Q + R) is 2 times as efficient as P ⇒ (Q + R) = 2P.

Since Total Work = Efficiency × Time, and work is the same:

Efficiency_(Q+R) × Time_(Q+R) = Efficiency_P × Time_P

⇒ 2P × 90 = P × Time_P

⇒ Time_P = 180 days.

∴ The correct answer is 180 days.

Alternate Method

Given:

Ratio of Efficiency (Q + R) : P = 2 : 1 (Sum of parts = 3)

Ratio of Efficiency (P + R) : Q = 1.5 : 1 = 3 : 2 (Sum of parts = 5)

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Equalize the total efficiency (LCM of 3 and 5 = 15):

⇒ New (Q + R) : P = (2 : 1) × 5 = 10 : 5

⇒ New (P + R) : Q = (3 : 2) × 3 = 9 : 6

⇒ Efficiency of P = 5 units/day, Efficiency of Q = 6 units/day.

⇒ Efficiency of R = 15 − (5 + 6) = 4 units/day.

⇒ Total Work = Efficiency of (Q + R) × Time = 10 × 90 = 900 units.

⇒ Days taken by P = 900 ÷ 5 = 180 days.

∴ The correct answer is 180 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken. If the ratio of efficiency is a : b, then the ratio of time taken is b : a.

Combined Efficiency

When people work together, their individual efficiencies are added to find the total efficiency of the group.

Ratio Balancing

To compare multiple ratios involving the same total sum, multiply each ratio by a factor such that the sum of the parts becomes equal to the LCM of all sums.

Shortcut Trick

In opposite direction, Relative Speed = (Sum of Lengths) ÷ Time

Since speeds are equal, 2 × Speed = (165 + 165) ÷ 15

2 × Speed = 330 ÷ 15

2 × Speed = 22 m/s

Speed = 11 m/s

∴ The correct answer is 11 m/s.

Alternate Method

Given:

Length of first train (L₁) = 165 m

Length of second train (L₂) = 165 m

Time taken to cross each other (T) = 15 sec

Let the speed of each train be 'v' m/s.

Formula Used:

Relative Speed (Opposite Direction) = Speed₁ + Speed₂

Relative Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = L₁ + L₂ = 165 + 165 = 330 m

⇒ Relative Speed = v + v = 2v m/s

⇒ Relative Speed = Total Distance ÷ Time

⇒ 2v = 330 ÷ 15

⇒ 2v = 22

⇒ v = 22 ÷ 2

⇒ v = 11 m/s

∴ The correct answer is 11 m/s.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, the relative speed is the difference of their individual speeds: |v₁ − v₂|.

Unit Conversion

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

Crossing Rule

When a train crosses another train (stationary or moving), the total distance covered is always the sum of the lengths of both trains.

Shortcut Trick

The distance covered in each hour forms an Arithmetic Progression (AP): 20, 30, 40, ...

Check the sum for 6 hours by adding consecutive terms:

20 + 30 + 40 + 50 + 60 + 70 = 270 km

Since the sum of distances over 6 hours equals the total distance, the time required is 6 hours.

∴ The correct answer is 6 hr.

Alternate Method

Given:

Initial Speed (a) = 20 km/hr

Speed Increase (d) = 10 km/hr every hour

Total Distance (S_n) = 270 km

Formula Used:

Sum of n terms of an AP: S_n = (n ÷ 2) × [2a + (n − 1)d]

Calculations:

⇒ 270 = (n ÷ 2) × [2 × 20 + (n − 1) × 10]

⇒ 270 = (n ÷ 2) × [40 + 10n − 10]

⇒ 270 = (n ÷ 2) × [30 + 10n]

⇒ 270 = 15n + 5n²

⇒ 5n² + 15n − 270 = 0

⇒ n² + 3n − 54 = 0

⇒ n² + 9n − 6n − 54 = 0

⇒ n(n + 9) − 6(n + 9) = 0

⇒ (n − 6)(n + 9) = 0

⇒ n = 6 or n = −9

⇒ Since time cannot be negative, n = 6 hours.

∴ The correct answer is 6 hr.

Additional Information

Arithmetic Progression Sum

The sum of the first n terms of an AP is given by S_n = (n/2)[2a + (n−1)d] or S_n = (n/2)[a + l], where l is the last term.

Average Speed in AP

When speeds are in AP over equal intervals of time, the Average Speed is the arithmetic mean of the initial and final speeds: (a + l)/2.

General Term of AP

The n^th term (speed at the start of the n^th hour) is calculated as a_n = a + (n − 1)d.

Shortcut Trick

Use the direct ratio relation: CP / MP = (100 − Discount%) / (100 + Profit%).

CP / MP = (100 − 15) / (100 + 30) = 85 / 130 = 17 / 26.

If no discount is given, the Marked Price becomes the Selling Price.

New Profit = 26 − 17 = 9 units.

Profit % = (9 / 17) × 100 = 52.94%.

∴ The correct answer is 52.94 percent.

Alternate Method

Given:

Selling Price (SP) of one article = ₹60

Discount = 15%

Profit = 30%

Formula Used:

Marked Price (MP) = SP / (1 − Discount%)

Cost Price (CP) = SP / (1 + Profit%)

Profit % = [(SP − CP) / CP] × 100

Calculations:

⇒ MP = 60 / 0.85 = 6000 / 85 = 1200 / 17

⇒ CP = 60 / 1.3 = 600 / 13

⇒ If no Discount is given, New SP = MP = 1200 / 17

⇒ New Profit = (1200 / 17) − (600 / 13)

⇒ New Profit % = [((1200 / 17) − (600 / 13)) / (600 / 13)] × 100

⇒ New Profit % = [((1200 / 17) / (600 / 13)) − 1] × 100

⇒ New Profit % = [(2 × 13 / 17) − 1] × 100

⇒ New Profit % = [26 / 17 − 1] × 100 = (9 / 17) × 100

⇒ New Profit % = 52.9411...% ≈ 52.94%

∴ The correct answer is 52.94 percent.

Additional Information

CP and MP Relation

The ratio of Cost Price to Marked Price is always given by CP / MP = (100 − D%) / (100 + P%).

Marked Price (List Price)

It is the price displayed on the product. Discount is always calculated on the Marked Price.

Successive Discounts

If two discounts d₁ and d₂ are given, the Effective Discount is d₁ + d₂ − (d₁ × d₂ / 100).