Quantitative Aptitude PYQs

Shortcut Trick

Simplify the expression by summing the exponents of common bases x and y.

Power of x → 3 + 1 + (−4) = 0 ⇒ x⁰ = 1.

Power of y → −2 − (−2) = 0 ⇒ y⁰ = 1.

The total value of the expression is 1 × 1 = 1.

The square of the expression is 1² = 1.

∴ The correct answer is 1.

Alternate Method

Given:

Expression: (x³/y²) × (1/x⁻¹) ÷ y⁻² × (1/x²)²

Formula Used:

1. a⁻ⁿ = 1/aⁿ | 2. (a^m)ⁿ = a^mn | 3. a^m × aⁿ = a^m+n | 4. a^m ÷ aⁿ = a^m−n

Calculations:

⇒ Term 1: x³/y² = x³y⁻²

⇒ Term 2: 1/x⁻¹ = x¹

⇒ Term 3: ÷ y⁻² = × (1/y⁻²) = × y²

⇒ Term 4: (1/x²)² = 1/x⁴ = x⁻⁴

⇒ Combining all: (x³y⁻²) × x¹ × y² × x⁻⁴

⇒ Expression = x^{(3 + 1 − 4)} × y^{(−2 + 2)}

⇒ Expression = x⁰ × y⁰ = 1 × 1 = 1

⇒ Square of Expression = (1)² = 1

∴ The correct answer is 1.

Additional Information

Laws of Exponents

When multiplying terms with the same base, add the powers: a^m × aⁿ = a^m+n. When dividing, subtract the powers: a^m ÷ aⁿ = a^m−n.

Zero Exponent Rule

Any non-zero base raised to the power of zero is always equal to 1 (a⁰ = 1).

Negative Exponents

A negative exponent denotes the reciprocal: a⁻ⁿ = 1/aⁿ. Conversely, 1/a⁻ⁿ = aⁿ.

Shortcut Trick

Find the deviation of the 8^th inning score from the old average: 217 − 59 = 158.

Distribute this excess score across the total number of innings (8).

Increase in average = 158 ÷ 8 = 19.75.

Rounding 19.75 to the nearest whole number gives 20.

∴ The correct answer is 20.

Alternate Method

Given: Average of 7 innings = 59, Score in 8^th inning = 217.

Formula Used: Average = Total Runs ÷ Number of Innings; Increase = New Average − Old Average.

⇒ Total runs in 7 innings = 7 × 59 = 413

⇒ Total runs in 8 innings = 413 + 217 = 630

⇒ New average score = 630 ÷ 8 = 78.75

⇒ Increase in average = 78.75 − 59 = 19.75

⇒ Rounded to the nearest whole number = 20

∴ The correct answer is 20.

Additional Information

Arithmetic Mean (Average)

The sum of all observations divided by the total number of observations. Formula: Avg = Σx ÷ n.

Deviation Method for Averages

When a new value is added, the change in average is (New Value − Old Average) ÷ New Total Count.

Rounding Off Rule

If the decimal part is 0.5 or greater, round up to the next integer; otherwise, round down. Here, 19.75 rounds to 20.

Shortcut Trick

List the first 13 prime numbers directly: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41.

Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 238.

∴ The correct answer is 238.

Alternate Method

Given: First 13 prime numbers.

Formula Used: Sum of numbers = sum of individual terms.

⇒ Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

⇒ First 13 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41.

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 10 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 17 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 28 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 41 + 17 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 58 + 19 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 77 + 23 + 29 + 31 + 37 + 41

⇒ Sum = 100 + 29 + 31 + 37 + 41

⇒ Sum = 129 + 31 + 37 + 41

⇒ Sum = 160 + 37 + 41

⇒ Sum = 197 + 41 = 238.

∴ The correct answer is 238.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The number 2 is the only even prime number.

Composite Numbers

Numbers greater than 1 that are not prime are called composite numbers. The number 1 is neither prime nor composite.

Co-prime Numbers

Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

Shortcut Trick

For four numbers a, b, c, d in proportion, the product of extremes equals the product of means: a × d = b × c.

Given a = 5, b = X, c = 3, and d = 63.

⇒ 5 × 63 = X × 3

⇒ X = (5 × 63) ÷ 3 = 5 × 21 = 105.

∴ The correct answer is 105.

Alternate Method

Given: First term (a) = 5, Second term (b) = X, Third term (c) = 3, Fourth proportion (d) = 63.

Formula: If a, b, c, d are in proportion, then a/b = c/d.

⇒ 5/X = 3/63

⇒ 5/X = 1/21

⇒ X = 5 × 21

⇒ X = 105

∴ The correct answer is 105.

Additional Information

Fourth Proportion

For three numbers a, b, and c, the fourth proportion d is defined such that a:b = c:d, which implies d = (b × c) ÷ a.

Third Proportion

For two numbers a and b, the third proportion c is defined such that a:b = b:c, which implies c = b² ÷ a.

Mean Proportion

The mean proportion between two numbers a and b is given by √(a × b).

Shortcut Trick
When distance is constant, Speed is inversely proportional to Time.
Speed Ratio (Usual : New) = 7 : 5
Time Ratio (Usual : New) = 5 : 7
Difference in time units = 7 − 5 = 2 units → 12 minutes
1 unit = 6 minutes
Usual Time = 5 units = 5 × 6 = 30 minutes
∴ The correct answer is 30 min.
Alternate Method
Given: New speed = 5/7 of usual speed, Time delay = 12 minutes.
Formula Used: Distance = Speed × Time

⇒ Let usual speed be s and usual time be t.
⇒ Distance D = s × t
⇒ New speed = 5s/7, New time = t + 12
⇒ D = (5s/7) × (t + 12)
⇒ s × t = (5s/7) × (t + 12)
⇒ 7t = 5t + 60
⇒ 2t = 60
⇒ t = 30 minutes
∴ The correct answer is 30 min.
Additional Information
Speed-Time Inverse Relation
If speed becomes a/b of the original speed, the time taken becomes b/a of the original time.
Late/Early Concept
Usual Time = (Late Time × a) / (b − a) where speed factor is a/b and b > a.
Average Speed
For constant distance, Average Speed = (2 × s1 × s2) / (s1 + s2).

Shortcut Trick

Area of a regular n-gon inscribed in a circle of radius R is given by: Area = (n ÷ 2) × R² × sin(360° ÷ n).

Ratio of areas = [6 × sin(60°)] : [5 × sin(72°)]

Ratio = [6 × (√3 ÷ 2)] : [5 × (√(10 + 2√5) ÷ 4)]

Ratio = 3√3 : [5√(10 + 2√5) ÷ 4]

Ratio = 12√3 : 5√(10 + 2√5)

∴ The correct answer is 12√3 : 5√(10 + 2√5).

Alternate Method

Given:

Radius of the circle = R

Number of sides of the first polygon (Hexagon), n₁ = 6

Number of sides of the second polygon (Pentagon), n₂ = 5

Formula Used:

Area of a regular polygon with n sides inscribed in a circle of radius R = (n ÷ 2) × R² × sin(2π ÷ n)

Calculations:

⇒ Area of Hexagon (A₁) = (6 ÷ 2) × R² × sin(360° ÷ 6)

⇒ A₁ = 3 × R² × sin(60°) = 3 × R² × (√3 ÷ 2) = (3√3 ÷ 2)R²

⇒ Area of Pentagon (A₂) = (5 ÷ 2) × R² × sin(360° ÷ 5)

⇒

Shortcut Trick

Side of the cube (a) = √³(Volume)

Side (a) = √³(1728) = 12 cm

Total Surface Area = 6 × a² = 6 × (12)²

Total Surface Area = 6 × 144 = 864 cm²

∴ The correct answer is 864 cm².

Alternate Method

Given:

Volume of the cube = 1728 cm³

Formula Used:

1. Volume of a cube (V) = a³

2. Total Surface Area (TSA) = 6a²

Where 'a' is the side of the cube.

Calculations:

⇒ a³ = 1728

⇒ a = √³(12 × 12 × 12)

⇒ a = 12 cm

⇒ Total Surface Area = 6 × (12)²

⇒ Total Surface Area = 6 × 144

⇒ Total Surface Area = 864 cm²

∴ The correct answer is 864 cm².

Additional Information

Lateral Surface Area (LSA)

The surface area of the four vertical sides of a cube is given by LSA = 4a².

Diagonal of a Cube

The longest distance between two opposite corners (space diagonal) is d = a√3.

Volume of a Cuboid

For a rectangular prism with length (l), breadth (b), and height (h), the volume is l × b × h.

Shortcut Trick
Using the fundamental Trigonometric Identity: sin2t + cos2t = 1.
Given cos t = x, then sin2t = 1 − x2.
Since t is an acute angle (0 < t < 90°), sin t must be positive.
Taking the square root: sin t = √(1 − x2).
∴ The correct answer is √(1 − x2).
Alternate Method

Given: cos t = x, where t is an acute angle.
Formula Used: cos t = Base / Hypotenuse and sin t = Perpendicular / Hypotenuse.
⇒ Let Hypotenuse = 1 and Base = x.
⇒ Using Pythagoras Theorem: Perpendicular2 + Base2 = Hypotenuse2.
⇒ Perpendicular2 + x2 = 12.
⇒ Perpendicular2 = 1 − x2.
⇒ Perpendicular = √(1 − x2).
⇒ sin t = Perpendicular / Hypotenuse = √(1 − x2) / 1.
⇒ sin t = √(1 − x2).
∴ The correct answer is √(1 − x2).
Additional Information
Pythagorean Identities
Standard identities include sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, and 1 + cot2θ = cosec2θ.
ASTC Rule (Quadrants)
In the first quadrant (acute angles), all trigonometric ratios (sin, cos, tan, etc.) are positive.
Tangent in terms of x
If cos t = x, then tan t = sin t / cos t = √(1 − x2) / x.

Shortcut Trick

Effective interest rate for 2 years at 18% = 18 + 18 + (18 × 18) ÷ 100 = 36 + 3.24 = 39.24%.

Total Amount percentage = 100% (Principal) + 39.24% (Interest) = 139.24%.

Amount = 5000 × 139.24% = 50 × 139.24 = 6962.

∴ The correct answer is ₹6,962.

Alternate Method

Given:

Principal (P) = ₹5,000

Rate of Interest (R) = 18% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R ÷ 100)ⁿ

Calculations:

⇒ A = 5000 × (1 + 18 ÷ 100)²

⇒ A = 5000 × (1 + 0.18)²

⇒ A = 5000 × (1.18)²

⇒ A = 5000 × 1.3924

⇒ A = 6962

∴ The correct answer is ₹6,962.

Additional Information

Difference between CI and SI

The difference (D) between compound interest and simple interest for 2 years is given by: D = P(R ÷ 100)².

Effective Rate of Interest

For two years at rate R%, the effective rate is R + R + (R × R) ÷ 100. This follows the successive percentage change rule.

Compounded Half-Yearly

If interest is compounded half-yearly, the rate becomes R ÷ 2 and the time period becomes 2n.

Shortcut Trick

LCM of 2 and 3 is 6. Let Bibha buy 6 bananas of each type.

CP of first 6 = 6 ÷ 2 = 3. CP of next 6 = 6 ÷ 3 = 2. Total CP for 12 bananas = 3 + 2 = 5.

To make a 40% profit, use multiplier: SP = CP × (100 + P%) ÷ 100.

SP for a dozen (12 bananas) = 5 × 140 ÷ 100 = 7.

∴ The correct answer is 7.

Alternate Method

Given: CP1 = 1 rupee for 2 bananas, CP2 = 1 rupee for 3 bananas. Equal quantity bought. Desired Profit = 40%.

Formulas: Total CP = CP₁ + CP₂ | SP = CP × (100 + Profit%) ÷ 100

⇒ Let Bibha buy x bananas at each rate. Total bananas = 2x.

⇒ CP of x bananas at 2 for a rupee = x/2.

⇒ CP of x bananas at 3 for a rupee = x/3.

⇒ Total CP for 2x bananas = x/2 + x/3 = 5x/6.

⇒ To find CP of 12 bananas, let 2x = 12 ⇒ x = 6.

⇒ CP of 12 bananas = 5(6)/6 = 5.

⇒ SP = 5 × (140/100) = 5 × 1.4 = 7.

∴ The correct answer is 7.

Additional Information

Unitary Method in CP

Always calculate the cost price of a single unit (or a common multiple like LCM) to simplify calculations when items are bought at different rates.

Profit Calculation

Profit is always calculated on the total cost price incurred, which includes all batches of items purchased.

Shortcut Trick

Let the cost of a TV be 100.

Cost of Washing Machine (WM) = 100 − 25% of 100 = 75.

Initial total cost of 6 WM + 3 TV = (6 × 75) + (3 × 100) = 450 + 300 = 750.

New WM cost = 75 × 1.75 = 131.25; New TV cost = 100 × 0.51 = 51.

New total cost = (6 × 131.25) + (3 × 51) = 787.5 + 153 = 940.5.

Percentage change = (940.5 − 750) ÷ 750 × 100 = 190.5 ÷ 750 × 100 = 25.4%.

∴ The correct answer is Increase by 25.4%.

Alternate Method

Given: Cost of WM = 75% of TV cost. Number of units: WM = 6, TV = 3.
Formulas: Total Cost = (Price₁ × Quantity₁) + (Price₂ × Quantity₂) | % Change = (New − Old) ÷ Old × 100.

⇒ Let TV cost = 100x and WM cost = 75x.
⇒ Initial Total Cost = 6(75x) + 3(100x) = 450x + 300x = 750x.
⇒ Increase in WM cost = 75% of 75x = 56.25x. Decrease in TV cost = 49% of 100x = 49x.
⇒ Total Net Change = 6(56.25x) − 3(49x) = 337.5x − 147x = 190.5x.
⇒ Percentage change = (190.5x ÷ 750x) × 100 = 25.4%.

∴ The correct answer is Increase by 25.4%.

Additional Information

Percentage Change in Aggregates

When multiple items are involved, the total percentage change is the weighted average of the individual percentage changes.

Effective Change Calculation

Net % Change = (Total Final Value − Total Initial Value) ÷ Total Initial Value × 100.

Shortcut Trick

Convert the fraction directly into a decimal.

2/5 = 0.4

Now, add it to the given decimal value:

Expression = 0.4 + 1.4 = 1.8

∴ The correct answer is 1.8.

Alternate Method

Given:

Expression = 2/5 + 1.4

Calculation:

⇒ Convert the decimal 1.4 into a fraction.

⇒ 1.4 = 14/10

⇒ Simplify the fraction: 14/10 = 7/5

⇒ Substitute the fraction back into the expression:

⇒ Expression = 2/5 + 7/5

⇒ Since the denominators are the same, add the numerators directly:

⇒ Expression = (2 + 7) / 5 = 9/5

⇒ Convert the resulting fraction back to a decimal:

⇒ 9/5 = 1.8

∴ The correct answer is 1.8.

Additional Information

BODMAS Rule

Always simplify expressions using the sequence: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

To convert a fraction to a decimal, divide the numerator by the denominator. If the denominator is a factor of 10, 100, etc., multiply both the numerator and denominator by the necessary multiplier to easily convert it into a decimal (e.g., 2/5 = (2 × 2) / (5 × 2) = 4/10 = 0.4).

Shortcut Trick

Ratio of A : B is 3 : 1. Let their ages be 3x and x.

New condition: (3x + 4) = 5 × (x − 6)

⇒ 3x + 4 = 5x − 30 ⇒ 2x = 34 ⇒ x = 17.

Current age of A = 3 × 17 = 51 years.

Age of A two years ago = 51 − 2 = 49 years.

∴ The correct answer is 49 years.

Alternate Method

Given: A = 3B

Formula Used: Linear equation in one variable.

⇒ Let B's present age be x years.

⇒ A's present age = 3x years.

⇒ According to question: (3x + 4) = 5(x − 6)

⇒ 3x + 4 = 5x − 30

⇒ 5x − 3x = 30 + 4

⇒ 2x = 34 ⇒ x = 17.

⇒ A's present age = 3 × 17 = 51 years.

⇒ A's age two years ago = 51 − 2 = 49 years.

∴ The correct answer is 49 years.

Additional Information

Ratio Method for Ages

If the age ratio changes over time, equate the parts by finding the difference between ratio terms and multiplying cross-ratio values.

Age Difference Constant

The difference between the ages of two people remains constant regardless of how many years pass or are subtracted.

Base Change in Ages

Always identify the present age as the base (x) to easily calculate past (x − n) or future (x + n) ages.

Shortcut Trick

Total deduction = 79 + 25 + 19 = ₹123

Remaining amount = 5,403 − 123 = ₹5,280

Ratio of shares = 14 : 5 : 11 (Total parts = 30)

1 part = 5,280 ÷ 30 = ₹176

B's remaining = 176 × 5 = ₹880; C's remaining = 176 × 11 = ₹1,936

Difference = (1,936 + 19) − (880 + 25) = 1,955 − 905 = ₹1,050

∴ The correct answer is ₹1,050.

Alternate Method

Given: Total amount = ₹5,403. Deductions = ₹79, ₹25, ₹19. Ratio = 14 : 5 : 11.

⇒ Net Amount = 5,403 − (79 + 25 + 19) = ₹5,280

⇒ Let the shares after deduction be 14x, 5x, and 11x.

⇒ 14x + 5x + 11x = 5,280

⇒ 30x = 5,280 ⇒ x = 176

⇒ B's original share = 5x + 25 = (5 × 176) + 25 = 880 + 25 = ₹905

⇒ C's original share = 11x + 19 = (11 × 176) + 19 = 1,936 + 19 = ₹1,955

⇒ Difference = 1,955 − 905 = ₹1,050

∴ The correct answer is ₹1,050.

Additional Information

Ratio & Proportion Basics

If a sum S is divided into a ratio a:b:c, the share of the first part is [a / (a+b+c)] × S.

Division with Adjustments

When fixed amounts are deducted from or added to shares, first adjust the total sum accordingly before applying the ratio division.

Difference between shares

Difference between shares can be calculated by finding individual values or by (Ratio Difference × Value of 1 part) ± Adjustment difference.

Shortcut Trick

Total Expenditure % = 25% + 15% + 10% = 50%

Savings % = 100% − 50% = 50%

Monthly Savings = 50% of ₹60,000 = ₹30,000

∴ The correct answer is ₹30,000.

Alternate Method

Given:

Monthly Income = ₹60,000

Rent = 25%, Groceries = 15%, Transportation = 10%

Calculations:

⇒ Total Expenditure = (25 + 15 + 10) % of Income

⇒ Total Expenditure = 50% of ₹60,000 = ₹30,000

⇒ Savings = Total Income − Total Expenditure

⇒ Savings = ₹60,000 − ₹30,000

⇒ Savings = ₹30,000

∴ The correct answer is ₹30,000.

Additional Information

Percentage Expenditure

When percentages are given on the original income, they are additive: Total % = P₁ + P₂ + ... + P_n.

Remaining Income Concept

If the question states spending of the remaining income, use the successive formula: Remaining = Income × (1 − r₁/100) × (1 − r₂/100).

Savings Ratio

Savings = Income − Expenditure. The ratio of Expenditure to Savings in this case is 50:50 or 1:1.

Shortcut Trick

Convert percentages and fractions into a single multiplier: 0.9 × 0.8 × (2/7) × Number = 576.

Number = 576 ÷ (0.72 × 2/7) = 576 ÷ (1.44/7).

Number = (576 × 7) ÷ 1.44 = 400 × 7 = 2800.

∴ The correct answer is 2800.

Alternate Method

Given: 90% of 80% of 2/7 of x = 576

⇒ (90/100) × (80/100) × (2/7) × x = 576

⇒ (9/10) × (4/5) × (2/7) × x = 576

⇒ (72/350) × x = 576

⇒ x = (576 × 350) ÷ 72

⇒ x = 8 × 350

⇒ x = 2800

∴ The correct answer is 2800.

Additional Information

Percentage to Fraction Conversion

Always simplify percentages into fractions (e.g., 80% = 4/5, 90% = 9/10) to make large multiplications easier to cancel out.

Successive Percentage Rule

When applying multiple percentages to a number, the order does not matter; a% of b% of x is the same as (a × b / 100)% of x.

Base Value Identification

If "x" is the original number, any percentage of it is x × (rate/100). For "of" operations, always use multiplication.

Shortcut Trick

Relative Speed in opposite direction = 58 + 68 = 126 km/h

Conversion to m/s = 126 × (5/18) = 7 × 5 = 35 m/s

Total Distance = 316 + 349 = 665 m

Time taken = 665 ÷ 35 = 19 seconds

∴ The correct answer is 19 sec.

Alternate Method

Given:

Length of first train (L₁) = 316 m

Length of second train (L₂) = 349 m

Speed of first train (S₁) = 58 km/h

Speed of second train (S₂) = 68 km/h

Formula Used:

Time = Total Distance ÷ Relative Speed

Relative Speed (Opposite Direction) = S₁ + S₂

Calculations:

⇒ Total distance = 316 + 349 = 665 m

⇒ Relative speed = 58 + 68 = 126 km/h

⇒ Speed in m/s = 126 × (5/18) = 35 m/s

⇒ Time taken = 665 ÷ 35

⇒ Time taken = 19 seconds

∴ The correct answer is 19 sec.

Additional Information

Relative Speed Concept

When moving in opposite directions, speed = S₁ + S₂. When moving in the same direction, speed = |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.

Crossing Objects

When a train crosses another train/platform, Distance = Length of Train 1 + Length of Train 2/Platform.

Shortcut Trick

Mode is the observation that appears most frequently in a data set.

List of numbers: 115, 190, 110, 185, 85, 110, 115, 102, 90, 95, 85, 68, 85

Frequency of 85 = 3 times (Highest among all numbers).

Frequency of 115 = 2 times | Frequency of 110 = 2 times | Others = 1 time.

∴ The correct answer is 85.

Alternate Method

Given: Observations = 115, 190, 110, 185, 85, 110, 115, 102, 90, 95, 85, 68, 85

Formula Used: Mode = Value with maximum frequency.

To find the Mode, we construct a frequency table for the given data:

⇒ Number 85 appears: 3 times

⇒ Number 115 appears: 2 times

⇒ Number 110 appears: 2 times

⇒ Number 190, 185, 102, 90, 95, 68 appear: 1 time each

⇒ The maximum frequency is 3, which corresponds to the number 85.

∴ Mode = 85

∴ The correct answer is 85.

Additional Information

Definition of Mode

The mode is the value that occurs most frequently in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal.

Empirical Relationship

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

Median

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

Mean (Arithmetic Average)

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

Shortcut Trick

Let the Marked Price (MP) be 100.

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

SP = 100 × 0.90 × 0.73 × 0.92

SP = 90 × 0.6716 = 60.444

Effective price percentage = (60.444 ÷ 100) × 100 ≈ 60.44%

∴ The correct answer is 60.44%.

Alternate Method

Given → Discounts = 10%, 27%, 8%. Formulas → Net Selling Price = MP × (100−d₁)% × (100−d₂)% × (100−d₃)%.

⇒ First discount: 100 − 10% of 100 = 90.
⇒ Second discount: 90 − 27% of 90 = 90 − 24.3 = 65.7.
⇒ Third discount: 65.7 − 8% of 65.7 = 65.7 − 5.256 = 60.444.
⇒ Effective Price % = (60.444 ÷ 100) × 100 = 60.444%.

∴ The correct answer is 60.44%.

Additional Information

Successive Discount Formula

For two discounts a% and b%, the single equivalent discount is (a + b − ab/100)%.

Effective Price vs Discount

The effective price percentage is (100 − Equivalent Discount)%. In this problem, the total discount is approx 39.56%.

Shortcut Trick

Assume all 14 shots were 2-pointers: Total points = 14 × 2 = 28 points.

The difference from actual score is 36 − 28 = 8 points.

Each 3-point shot adds 1 point (3 − 2 = 1) over a 2-point shot.

Number of 3-point shots = 8 ÷ 1 = 8.

Number of 2-point shots = 14 − 8 = 6.

∴ The correct answer is Number of 2-point shots = 6; Number of 3-point shots = 8.

Alternate Method

Given: Total shots = 14 | Total score = 36 points.

Formula: Let x be the number of 2-point shots and y be the number of 3-point shots.

⇒ Equation 1 (Total Shots): x + y = 14

⇒ Equation 2 (Total Score): 2x + 3y = 36

⇒ From (1), x = 14 − y. Substitute this into (2):

⇒ 2(14 − y) + 3y = 36

⇒ 28 − 2y + 3y = 36

⇒ 28 + y = 36 ⇒ y = 8 (3-point shots)

⇒ x = 14 − 8 = 6 (2-point shots)

∴ The correct answer is Number of 2-point shots = 6; Number of 3-point shots = 8.

Additional Information

Linear Equations in Two Variables

Equations like ax + by = c represent a relationship where two different quantities contribute to a fixed total.

Substitution Method

Solve one equation for one variable and substitute it into the other to reduce the problem to a single variable.

The Alligation Rule

Ratio of quantities = (Higher value − Average) : (Average − Lower value). This is highly effective for mixed value problems.