Quantitative Aptitude PYQs

Shortcut Trick

Net rate of filling = (1/6) − (1/7) = 1/42 part per minute.

Time to fill the full tank = 42 minutes.

Time to fill one-half (1/2) of the tank = 42 ÷ 2 = 21 minutes.

∴ The correct answer is 21 minutes.

Alternate Method

Given: Filling time = 6 min, Emptying time = 7 min.

Formula Used: Total Work = LCM of time; Efficiency = Work ÷ Time.

⇒ Combined efficiency of both pipes = 7 − 6 = 1 unit/min.

⇒ Total capacity to be filled = 1/2 × 42 = 21 units.

⇒ Time taken = 21 units ÷ 1 unit/min = 21 minutes.

∴ The correct answer is 21 minutes.

Additional Information

Inlet and Outlet Logic

Efficiency of an inlet pipe is always positive (+), while the efficiency of an outlet (leakage) pipe is always negative (−).

Combined Rate Formula

If pipe A fills in 'x' hours and pipe B empties in 'y' hours, net work done in 1 hour = (1/x) − (1/y).

Work and Wages

Wages are always distributed in the ratio of work done or the ratio of efficiencies of the pipes/workers.

Shortcut Trick

Identify the expression as the standard algebraic identity: (a + b)(a − b) = a² − b².

Comparing (x + 5)(x − 5) with (a + b)(a − b), we get a = x and b = 5.

Direct calculation: x² − 5² = x² − 25.

∴ The correct answer is x² − 25.

Alternate Method

Given: The expression to simplify is (x + 5)(x − 5).

Formula Used: Distributive Property (FOIL method) or (a + b)(a − b) = a² − b².

⇒ (x + 5)(x − 5)

⇒ x × (x − 5) + 5 × (x − 5)

⇒ (x × x) − (x × 5) + (5 × x) − (5 × 5)

⇒ x² − 5x + 5x − 25

⇒ x² + 0 − 25

⇒ x² − 25

∴ The correct answer is x² − 25.

Additional Information

Difference of Two Squares

The identity (a + b)(a − b) = a² − b² is known as the difference of two squares formula, used extensively in factoring and simplification.

Square of a Sum

The square of a binomial sum is given by (a + b)² = a² + 2ab + b².

Square of a Difference

The square of a binomial difference is given by (a − b)² = a² − 2ab + b².

Product of Binomials with Common Term

For (x + a)(x + b), the expansion is x² + (a + b)x + ab.

Shortcut Trick

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

Here, markup percentage a = +25% and discount percentage b = −68%.

Net change = 25 − 68 + (25 × (−68)) ÷ 100

Net change = −43 − 1700 ÷ 100 = −43 − 17 = −60%.

Negative sign indicates a loss.

∴ The correct answer is 60% loss.

Alternate Method

Given → Markup% = 25%, Discount% = 68%.
Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = (CP − SP) ÷ CP × 100.

⇒ Let Cost Price (CP) = 100.
⇒ Marked Price (MP) = 100 + 25% of 100 = 125.
⇒ Selling Price (SP) = 125 − 68% of 125.
⇒ SP = 125 × (32 ÷ 100) = 125 × 0.32 = 40.
⇒ Loss = CP − SP = 100 − 40 = 60.
⇒ Loss% = (60 ÷ 100) × 100 = 60%.

∴ The correct answer is 60% loss.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). The price reached after applying discount to MP is the Selling Price (SP).

Relationship Formula

MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%). If there is a loss, Profit% is replaced by −Loss%.

Shortcut Trick

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

Efficiency of A = 3, B = 2.

Work done by A in 3 days = 3 × 3 = 9 units.

Remaining Work = 24 − 9 = 15 units.

Combined Efficiency (A + B) = 3 + 2 = 5.

Extra Days = 15 ÷ 5 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given: Time taken by A = 8 days, Time taken by B = 12 days. A works alone for 3 days.

⇒ Work done by A in 3 days = 3 × 3 = 9 units

⇒ Remaining Work = 24 − 9 = 15 units

⇒ Combined Efficiency of A and B = 3 + 2 = 5 units/day

⇒ Days required to finish remaining work = 15 ÷ 5

⇒ Time taken = 3 days

∴ The correct answer is 3 days.

Additional Information

Total Work and Efficiency

Total Work is usually taken as the LCM of individual times. Efficiency is work done per unit time (Efficiency = Total Work ÷ Time).

Combined Work Formula

When multiple people work together, their efficiencies are added to find the total rate of work. Time = Total Work ÷ Total Efficiency.

Pipes & Cisterns

This logic applies to pipes where an inlet pipe has positive efficiency and an outlet pipe has negative efficiency.

Shortcut Trick

Volume of rectangular tank = 8 × 6 × 4.62 = 221.76 m³

Rate of flow per minute = Area of pipe × Speed × 60 = (22/7) × (0.07)² × 5 × 60

Rate per minute = 0.0154 × 300 = 4.62 m³/min

Time required = Total Volume ÷ Rate per minute = 221.76 ÷ 4.62 = 48 minutes

∴ The correct answer is 48 minutes.

Alternate Method

Given:

Internal diameter of cylindrical pipe = 14 cm ⇒ Radius (r) = 7 cm = 0.07 m

Speed of water = 5 m/s

Dimensions of tank: Length (L) = 8 m, Width (W) = 6 m, Height (H) = 4.62 m

Formula Used: Volume = Area of cross-section × Speed × Time

Calculations:

⇒ Volume of the rectangular tank = L × W × H

⇒ Volume = 8 × 6 × 4.62 = 221.76 m³

⇒ Cross-sectional area of pipe = πr² = (22/7) × 0.07 × 0.07

⇒ Area = 22 × 0.01 × 0.07 = 0.0154 m²

⇒ Volume of water flowing per second = Area × Speed = 0.0154 × 5 = 0.077 m³/s

⇒ Total time taken (in seconds) = Total Volume ÷ Volume per second

⇒ Time = 221.76 ÷ 0.077 = 2880 seconds

⇒ Time in minutes = 2880 ÷ 60 = 48 minutes

∴ The correct answer is 48 minutes.

Additional Information

Volume of Cylindrical Flow

The volume of water discharged by a pipe in time 't' is given by V = πr²vt, where 'r' is the radius and 'v' is the velocity of flow.

Volume of a Cuboid

The volume of a rectangular tank (cuboid) is calculated as Length × Breadth × Height.

Unit Conversion

Always ensure all dimensions are in the same unit system (SI units like meters and seconds) before performing calculations.

Shortcut Trick

In a 30°−60°−90° triangle, the sides are in the ratio 1 : √3 : 2.

The side opposite to 30° (Base) = 8.5 m.

The side opposite to 60° (Height) = Base × √3.

Height = 8.5 × 1.7 = 14.45 m.

∴ The correct answer is 14.45 m.

Alternate Method

Given:

Angle of elevation (θ) = 60°

Distance from wall (Base) = 8.5 m

Value of √3 = 1.7

Formula Used:

tan(θ) = Perpendicular / Base

Calculations:

⇒ tan(60°) = Height / 8.5

⇒ √3 = Height / 8.5

⇒ Height = 8.5 × √3

⇒ Height = 8.5 × 1.7

⇒ Height = 14.45 m

∴ The correct answer is 14.45 m.

Additional Information

Trigonometric Ratios

Standard values for tan: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Right Angled Triangle Properties

In a right triangle, sin θ = P/H, cos θ = B/H, and Pythagoras theorem: P² + B² = H².

Angle of Elevation vs Depression

The angle of elevation from an observer to an object is always equal to the angle of depression from the object to the observer.

Shortcut Trick

Ratio of Curved Surface Area to Volume is 2πrh ÷ πr²h = 2/r.

Given 2/r = 1/7 ⇒ r = 14 units.

Diameter (d) = 2 × 14 = 28 units. Given d : h = 7 : 6.

Since 7 units = 28, then 6 units (height) = 24 units.

Total Surface Area = 2 × (22/7) × 14 × (14 + 24) = 88 × 38 = 3344 unit².

∴ The correct answer is 3344 unit².

Alternate Method

Given:

Ratio of CSA to Volume = 1 : 7

Ratio of Diameter to Height = 7 : 6

Formula Used:

Curved Surface Area (CSA) = 2πrh

Volume (V) = πr²h

Total Surface Area (TSA) = 2πr(r + h)

Calculations:

⇒ CSA / Volume = 2πrh / πr²h = 2/r

⇒ 2/r = 1/7

⇒ r = 14 units

⇒ Diameter (d) = 2 × r = 2 × 14 = 28 units

⇒ d / h = 7 / 6

⇒ 28 / h = 7 / 6

⇒ h = (28 × 6) ÷ 7 = 4 × 6 = 24 units

⇒ TSA = 2 × (22/7) × 14 × (14 + 24)

⇒ TSA = 2 × 22 × 2 × 38

⇒ TSA = 88 × 38

⇒ TSA = 3344 unit²

∴ The correct answer is 3344 unit².

Additional Information

Cylinder Properties

For a cylinder with base radius r and height h, the volume is V = πr²h and the curved surface area is CSA = 2πrh.

Total Surface Area

TSA includes both the top and bottom circular faces plus the curved side: TSA = 2πrh + 2πr² = 2πr(r + h).

Hollow Cylinder

Volume of a hollow cylinder with outer radius R and inner radius r is V = π(R² − r²)h.

Shortcut Trick

Age ratio (8 years ago) = 13 : 1

Age ratio (after 12 years) = 3 : 1 (or 6 : 2 to match the age difference)

Equalizing the gap: 13 − 1 = 12 units and 3 − 1 = 2 units. Multiply second ratio by 6 → 18 : 6.

Ratio change: 13 → 18 (5 units) over 20 years (8 + 12). So, 1 unit = 4 years.

Age difference = (13 − 1) × 4 = 12 × 4 = 48 years.

∴ The correct answer is 48 years.

Alternate Method

Let the granddaughter's age 8 years ago be x and the grandmother's age be 13x.

Present ages: Granddaughter = x + 8, Grandmother = 13x + 8.

Ages after 12 years: Granddaughter = x + 20, Grandmother = 13x + 20.

According to the question:

⇒ 13x + 20 = 3 × (x + 20)

⇒ 13x + 20 = 3x + 60

⇒ 10x = 40

⇒ x = 4

Age difference (remains constant):

⇒ Difference = 13x − x = 12x

⇒ Difference = 12 × 4 = 48 years.

∴ The correct answer is 48 years.

Additional Information

Constant Age Difference

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

Ratio Balancing Method

To solve age problems quickly, make the difference between ratio parts equal by multiplying the ratios with the difference of the other ratio.

Linear Equation in Ages

For any "n" years in the future, add n to the present age; for "n" years in the past, subtract n from the present age.

Shortcut Trick

Check the unit digit: 6³ ends in 6 and 0³ ends in 0.

The unit digit of the result must be 6 − 0 = 6, which narrows it to 3096 or 3064.

Since 16³ is 4096 and 10³ is 1000, the difference is 3096.

∴ The correct answer is 3096.

Alternate Method

Given:

Expression = 16³ − 10³

Method Used: Direct expansion of cubes

Calculations:

⇒ 16³ = 16 × 16 × 16

⇒ 16³ = 256 × 16 = 4096

⇒ 10³ = 10 × 10 × 10 = 1000

⇒ Value = 4096 − 1000

⇒ Value = 3096

∴ The correct answer is 3096.

Additional Information

Difference of Cubes Identity

The expression can also be solved using the identity: a³ − b³ = (a − b)(a² + ab + b²).

Sum of Cubes Identity

Similarly, for addition, the formula used is: a³ + b³ = (a + b)(a² − ab + b²).

BODMAS Rule in Powers

According to the BODMAS rule, exponents or orders (cubes, squares) are calculated before performing addition or subtraction.

Shortcut Trick

Total decimal places in the numerator = 1 + 4 + 1 = 6.

Total decimal places in the denominator = 3 + 4 + 1 = 8.

Adjust the decimal difference by multiplying the numerator by 10^{8 − 6} = 100.

Simplify the integer fraction: (95 × 85 × 189) ÷ (21 × 17 × 19) = 5 × 5 × 9 = 225.

Evaluate the full expression: √16 + √900 + √(225 × 100) = 4 + 30 + 150 = 184.

∴ The correct answer is 184.

Alternate Method

Given:

Expression: \( \sqrt{16} + \sqrt{900} + \sqrt{\frac{9.5 × 0.0085 × 18.9}{0.021 × 0.0017 × 1.9}} \)

Calculation:

⇒ \( \sqrt{16} + \sqrt{900} + \sqrt{\frac{9.5 × 0.0085 × 18.9}{0.021 × 0.0017 × 1.9}} \)

⇒ \(4 + 30 + \sqrt{\frac{95 × 85 × 189}{0.21 × 17 × 19}} \)

⇒ \(34 + \sqrt{\frac{5 × 5 × 189 × 100}{21 }} \)

⇒ \(34 + \sqrt{{5 × 5 × 9 × 100}} \)

⇒ 34 + 5 × 3 × 10

⇒ 4 + 30 + 150

⇒ 184

∴ The correct answer is 184.

Additional Information

BODMAS Rule

When solving simplifications, follow the order of operations: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Properties of Square Roots

Multiplication and division inside roots can be separated: √(a × b) = √a × √b and √(a ÷ b) = √a ÷ √b. This allows for simplifying complex terms easily before evaluating the root.

Exponent Division Rule

When dividing algebraic terms or numbers with the same base, subtract the exponent of the denominator from the exponent of the numerator: x^a ÷ x^b = x^{a − b}. In this problem, 10⁻⁶ ÷ 10⁻⁸ = 10².

Shortcut Trick

In decimal multiplication, the total number of decimal places in the product is the sum of the decimal places in the multipliers.

Here, the number 4.3 has 1 decimal place.

So, 4.3 × 4.3 will have 1 + 1 = 2 decimal places in total.

We are given that 43 × 43 = 1849.

Placing the decimal point 2 places from the right gives 18.49.

∴ The correct answer is 18.49.

Alternate Method

Given:

43 × 43 = 1849

Calculation:

⇒ We need to find the exact value of 4.3 × 4.3

⇒ First, express the decimal numbers as fractions:

⇒ 4.3 = 43 ÷ 10

⇒ 4.3 × 4.3 = (43 ÷ 10) × (43 ÷ 10)

⇒ 4.3 × 4.3 = (43 × 43) ÷ (10 × 10)

⇒ Substitute the given value of 43 × 43 = 1849

⇒ 4.3 × 4.3 = 1849 ÷ 100

⇒ 4.3 × 4.3 = 18.49

∴ The correct answer is 18.49.

Additional Information

Multiplying Decimals Rule

Multiply the numbers as if they were whole numbers ignoring the decimals. Then, count the total number of decimal digits in the question and place the decimal point so the product has the exact same total number of decimal places.

Dividing by Powers of 10

To divide any number by 10, 100, 1000, etc., shift the decimal point to the left by the exact number of zeros present in the denominator.

Squares Ending in 3

The square of any number that ends with the digit 3 will always have 9 as its unit digit. For example, 3² = 9, 13² = 169, 23² = 529, and 43² = 1849.

Shortcut Trick

When the distance traveled is the same for both parts of the journey, the Average Speed is the harmonic mean of the speeds.

Average Speed = (2 × s₁ × s₂) ÷ (s₁ + s₂)

Average Speed = (2 × 52 × 118) ÷ (52 + 118) = 12272 ÷ 170

Average Speed = 72.188... ≈ 72.19 km/hr

∴ The correct answer is 72.19 km/hr.

Alternate Method

Given: Distance (D) = 966 km, Forward Speed (s₁) = 52 km/hr, Return Speed (s₂) = 118 km/hr.

Formula Used: Average Speed = Total Distance ÷ Total Time

⇒ Total Distance = 966 + 966 = 1932 km

⇒ Time to go (t₁) = 966 ÷ 52 = 18.5769 hours

⇒ Time to return (t₂) = 966 ÷ 118 = 8.1864 hours

⇒ Total Time = 18.5769 + 8.1864 = 26.7633 hours

⇒ Average Speed = 1932 ÷ 26.7633 = 72.188...

∴ The correct answer is 72.19 km/hr.

Additional Information

Average Speed (Equal Distances)

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

Average Speed (Equal Time)

If a person travels for equal time intervals at speeds x and y, the average speed is (x + y) ÷ 2.

General Average Speed Formula

Average Speed = (Total Distance) ÷ (Total Time taken for the journey).

Shortcut Trick

Total Capacity (LCM of 12, 18) = 36 units

Efficiency (P + Q) = 36 ÷ 12 = 3 units/hr

Efficiency (P) = 36 ÷ 18 = 2 units/hr

Efficiency (Q) = 3 − 2 = 1 unit/hr

Time for Q to fill 1/4^th = (36 × 1/4) ÷ 1 = 9 hours

∴ The correct answer is 9 hours.

Alternate Method

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

⇒ Efficiency of Q = 3 − 2 = 1 unit/hr

⇒ Total capacity of cistern = 36 units

⇒ One-fourth of the cistern = 36 ÷ 4 = 9 units

⇒ Time taken by Q = 9 units ÷ 1 unit/hr = 9 hours

∴ The correct answer is 9 hours.

Additional Information

Pipes & Cisterns Efficiency

The rate of work done by a pipe is its Efficiency. Total Work = Time taken × Efficiency.

Inlet and Outlet Pipes

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

Time Calculation

If a pipe fills a tank in 'x' hours, its 1-hour work is 1/x of the total capacity.

Shortcut Trick
The best scheme for the shopkeeper is the one offering the lowest effective discount.
Scheme A: 19 + 25 − (19 × 25) ÷ 100 = 44 − 4.75 = 39.25%
Scheme B: [6 ÷ (7 + 6)] × 100 = 6 ÷ 13 × 100 ≈ 46.15%
Scheme C: 50% (Fixed single discount)
Scheme D: 3 + 39 − (3 × 39) ÷ 100 = 42 − 1.17 = 40.83%
Comparing discounts: 39.25% < 40.83% < 46.15% < 50%.
∴ The correct answer is A.

Alternate Method
Given → Four different discount schemes A, B, C, and D.
Formulas: Successive Discount = a + b − (ab ÷ 100) | Buy X Get Y Free Discount% = [Y ÷ (X + Y)] × 100.

⇒ Calculate effective discount for Scheme A: 19 + 25 − 4.75 = 39.25%.
⇒ Calculate effective discount for Scheme B (Buy 7 Get 6 Free): (6/13) × 100 = 46.15%.
⇒ Calculate effective discount for Scheme D: 3 + 39 − 1.17 = 40.83%.
⇒ Scheme C is given as a flat 50% discount.
⇒ For a shopkeeper, the minimum discount results in the maximum profit.
⇒ Minimum value among {39.25, 46.15, 50, 40.83} is 39.25%.
∴ The correct answer is A.

Additional Information
Perspective of Best Scheme
The Best Scheme for Customer is the one with the highest discount, whereas the Best Scheme for Shopkeeper is the one with the lowest discount.
Buy X Get Y Free
Discount% = [Number of free items ÷ Total number of items (Paid + Free)] × 100.

Shortcut Trick

Initial votes share = 63%

Decrease in support = 24%

Remaining support = 100% − 24% = 76%

Final percentage = 63 × 0.76 = 47.88%

∴ The correct answer is 47.88%.

Alternate Method

Given:

Votes in first round = 63% of total

Decrease in second round = 24% of his current share

Calculations:

⇒ Amount of decrease = 24% of 63%

⇒ Decrease = (24 ÷ 100) × 63 = 0.24 × 63 = 15.12%

⇒ Final votes = 63% − 15.12%

⇒ Final votes = 47.88%

∴ The correct answer is 47.88%.

Additional Information

Percentage Decrease

To calculate a percentage decrease on a base value, multiply the base by (100 − decrease %)/100.

Successive Percentage

When a value changes successively, the final value is A × (1 ± x/100) × (1 ± y/100).

Base Value Concept

Always identify if the percentage change is applied to the original total or the intermediate value.

Shortcut Trick

For Mean Proportion b between a and c: b² = a × c

Given b = 42 and c = 294, find x (where x is a).

x = (42 × 42) ÷ 294

x = 1764 ÷ 294 = 6

∴ The correct answer is 6.

Alternate Method

Given:

Mean Proportion (b) = 42

First term (a) = x

Third term (c) = 294

Formula Used:

If b is the mean proportion between a and c, then a : b = b : c ⇒ b² = ac

Calculations:

⇒ 42² = x × 294

⇒ 1764 = x × 294

⇒ x = 1764 ÷ 294

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Mean Proportion

The mean proportional between two numbers a and b is √(a × b).

Third Proportion

If a : b = b : c, then c is called the third proportion to a and b, calculated as c = b² ÷ a.

Fourth Proportion

If a : b = c : d, then d is the fourth proportion, calculated as d = (b × c) ÷ a.

Shortcut Trick

Check options for numerator = denominator − 6.

(1) 5 − 4 = 1 | (2) 11 − 6 = 5 | (3) 8 − 5 = 3 | (4) 13 − 7 = 6.

Only 7/13 has a difference of 6. Let us verify the condition: (7+2)/(13+2) = 9/15 = 3/5.

Wait, if the condition becomes 2/3, let us check (x − 6 + 2) / (x + 2) = 2/3 ⇒ (x − 4) / (x + 2) = 2/3 ⇒ 3x − 12 = 2x + 4 ⇒ x = 16. Fraction = 10/16 = 5/8.

∴ The correct answer is 5/8.

Alternate Method

Let the denominator be x.

Then, the numerator = x − 6.

Original Fraction = (x − 6) / x

According to the question, if 2 is added to both:

⇒ ((x − 6) + 2) / (x + 2) = 2/3

⇒ (x − 4) / (x + 2) = 2/3

⇒ 3 × (x − 4) = 2 × (x + 2)

⇒ 3x − 12 = 2x + 4

⇒ 3x − 2x = 4 + 12

⇒ x = 16

⇒ Numerator = 16 − 6 = 10

⇒ Fraction = 10 / 16

⇒ Simplest form = 5 / 8

∴ The correct answer is 5/8.

Additional Information

Simplest Form of a Fraction

A fraction is in its simplest form when the Highest Common Factor (HCF) of the numerator and denominator is 1.

Proper vs Improper Fractions

A proper fraction has a numerator less than the denominator (e.g., 5/8), while an improper fraction has a numerator greater than or equal to the denominator.

Ratio and Proportion

Fractional equality a/b = c/d can be solved using cross-multiplication: a × d = b × c.

Shortcut Trick

Total Profit = (Profit in Transaction 1) + (Profit or Loss in Transaction 2)

Transaction 1: CP₁ = 24,000 + 3,000 = 27,000; SP₁ = 30,000 ⇒ Profit₁ = 3,000

Transaction 2: Loss₂ = 1,200

Net Profit = 3,000 − 1,200 = 1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given: Initial purchase price = ₹24,000, Repair cost = ₹3,000, First selling price = ₹30,000, Buy back price = ₹28,000, Loss on second sale = ₹1,200.

Formulas: Total CP = Purchase Price + Overheads | Profit = SP − CP | Loss = CP − SP | Net Profit = Total SP − Total CP

⇒ Total Cost Price for Transaction 1 = 24,000 + 3,000 = 27,000

⇒ Profit from Transaction 1 = 30,000 − 27,000 = 3,000

⇒ Cost Price for Transaction 2 = 28,000

⇒ Selling Price for Transaction 2 = CP − Loss = 28,000 − 1,200 = 26,800

⇒ Total CP = 27,000 + 28,000 = 55,000

⇒ Total SP = 30,000 + 26,800 = 56,800

⇒ Overall Profit = 56,800 − 55,000 = 1,800

∴ The correct answer is ₹1,800.

Additional Information

Treatment of Overhead Expenses

Any money spent on repairs, transportation, or labor after buying an item is added to the purchase price to calculate the Effective Cost Price.

Net Profit/Loss in Multiple Transactions

The Overall Profit or Loss is the algebraic sum of results from each individual transaction. Profit is taken as positive (+), and Loss is taken as negative (−).

Shortcut Trick

Number 648XYZ is divisible by 10 ⇒ Z = 0.

Divisible by 4 ⇒ YZ must be divisible by 4. Since Z = 0, smallest Y = 0 (00 is divisible by 4).

Divisible by 9 ⇒ 6 + 4 + 8 + X + 0 + 0 = 18 + X must be divisible by 9 ⇒ X = 0 or 9.

Using X = 0, Y = 0, Z = 0: X + Z − Y = 0 + 0 − 0 = 0.

∴ The correct answer is 0.

Alternate Method

Given: Number 648XYZ is divisible by 4, 9, and 10. Y is the smallest possible digit.

Calculations:

⇒ Divisibility by 10: The last digit must be 0. ∴ Z = 0.

⇒ Divisibility by 4: The last two digits (YZ) must be divisible by 4. Since Z = 0, Y can be 0, 2, 4, 6, 8.

⇒ Smallest Y: The question states Y is the smallest possible number. ∴ Y = 0.

⇒ Divisibility by 9: Sum of digits (6 + 4 + 8 + X + Y + Z) must be divisible by 9.

⇒ 18 + X + 0 + 0 = 18 + X.

⇒ For 18 + X to be divisible by 9, X can be 0 or 9.

⇒ If X = 0, then X + Z − Y = 0 + 0 − 0 = 0.

⇒ If X = 9, then X + Z − Y = 9 + 0 − 0 = 9 (Not in options).

∴ The correct answer is 0.

Additional Information

Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4 or are 00.

Divisibility by 10

A number is divisible by 10 if its unit digit is strictly 0.

Shortcut Trick

Arrange the Basketball scores in ascending order: 2, 5, 7, 8, 10, 10, 10, 11, 12, 14, 17, 18, 24, 25, 27.

Total number of match scores (n) = 15 (Odd).

For odd n, Median = ((n + 1) ÷ 2)^th term.

Median = ((15 + 1) ÷ 2)^th term = 8^th term.

The 8^th value in the sorted list is 11.

∴ The correct answer is 11 points.

Alternate Method

Given Data: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 10, 8, 10, 12, 11.

Formula Used: Median = ((n + 1) ÷ 2)^th term (where n is odd).

⇒ Total observations (n) = 15

⇒ Sorted Data in ascending order: 2, 5, 7, 8, 10, 10, 10, 11, 12, 14, 17, 18, 24, 25, 27

⇒ Median index = (15 + 1) ÷ 2 = 8^th term

⇒ Median value = Value of 8^th term

⇒ Median = 11

∴ The correct answer is 11 points.

Additional Information

Median for Even n

If the number of observations (n) is even, the median is the average of the (n ÷ 2)^th and ((n ÷ 2) + 1)^th terms.

Mode of Data

The mode is the most frequently occurring value. In this set, 10 occurs three times, so Mode = 10.

Arithmetic Mean

Mean = (Sum of all values) ÷ (Total number of values). It represents the central balance point of the data.