Quantitative Aptitude PYQs

Shortcut Trick

When n identical cubes with edge a are joined in a row, the number of exposed square faces is (4n + 2).

Here, number of cubes n = 6 and edge a = 10 cm.

Total exposed faces = (4 × 6 + 2) = 26 faces.

Surface Area = Number of faces × a² ⇒ 26 × 10² = 2600 cm².

∴ The correct answer is 2600 cm².

Alternate Method

Given: Edge of each cube = 10 cm | Number of cubes = 6.

Formula Used: Total Surface Area (TSA) of Cuboid = 2(lb + bh + hl).

When cubes are joined end to end, the length of the resulting cuboid increases while breadth and height remain equal to the cube's edge.

⇒ Length (l) = 6 × 10 = 60 cm

⇒ Breadth (b) = 10 cm

⇒ Height (h) = 10 cm

⇒ TSA = 2(lb + bh + hl)

⇒ TSA = 2(60 × 10 + 10 × 10 + 10 × 60)

⇒ TSA = 2(600 + 100 + 600)

⇒ TSA = 2 × 1300

⇒ TSA = 2600 cm²

∴ The correct answer is 2600 cm².

Additional Information

Surface Area of a Cube

A cube has 6 identical square faces; hence the total surface area is 6a² and the lateral surface area is 4a².

Volume of the Resulting Solid

The volume remains the sum of individual volumes: Volume = n × a³. For 6 cubes, V = 6 × 10³ = 6000 cm³.

Diagonal of a Cuboid

The longest rod that can be placed in a cuboid is its diagonal, given by d = √(l² + b² + h²).

Shortcut Trick

Use the ratio relationship between Marked Price (MP) and Cost Price (CP): MP ÷ CP = (100 + Gain%) ÷ (100 − Discount%).

MP ÷ 651 = (100 + 68) ÷ (100 − 86) = 168 ÷ 14.

MP ÷ 651 = 12.

MP = 651 × 12 = 7,812.

∴ The correct answer is ₹7,812.

Alternate Method

Given: CP = ₹651, Gain% = 68%, Discount% = 86%. Formulas: SP = CP × (100 + Gain%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%).

⇒ Step 1: Find Selling Price (SP) = 651 × (168 ÷ 100) = 1,093.68.

⇒ Step 2: Since Discount is 86%, SP is 14% of Marked Price (MP).

⇒ Step 3: MP = 1,093.68 × 100 ÷ 14.

⇒ Step 4: MP = 109,368 ÷ 14 = 7,812.

∴ The correct answer is ₹7,812.

Additional Information

Relation between CP and MP

CP ÷ MP = (100 − D%) ÷ (100 + G%). This ratio is the fastest way to solve problems involving both discount and gain percentage.

Marked Price

The price listed on the label of an article is the Marked Price (List Price). Discounts are always calculated on this value.

Shortcut Trick

Use the Successive Discount formula: a + b − (a × b) ÷ 100.

Here, a = 16 and b = 22.

Equivalent Discount = 16 + 22 − (16 × 22) ÷ 100.

Equivalent Discount = 38 − (352 ÷ 100) = 38 − 3.52.

Equivalent Discount = 34.48%.

∴ The correct answer is 34.48%.

Alternate Method

Given: First discount (d₁) = 16%, Second discount (d₂) = 22%.

Formula: Net Discount = [1 − (1 − d₁/100) × (1 − d₂/100)] × 100

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 16% discount = 100 − 16 = 84.

⇒ Price after second discount (22% of 84) = 84 − (22/100 × 84) = 84 − 18.48.

⇒ Final Selling Price (SP) = 65.52.

⇒ Single equivalent discount = MP − SP = 100 − 65.52 = 34.48%.

∴ The correct answer is 34.48%.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The total effective discount is always less than the sum of individual discounts.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Selling Price (SP) = MP − Discount.

Shortcut Trick

Let Emma's age be 1 unit.

Sarah's age = 2 units + 5 years.

Total units = 1 + 2 + 5 = 29 ⇒ 3 units = 24.

1 unit = 8.

∴ The correct answer is 8 years.

Alternate Method

Given:

Let Emma's age = x years.

Sarah's age = 2x + 5 years.

Sum of their ages = 29 years.

Calculations:

⇒ x + (2x + 5) = 29

⇒ 3x + 5 = 29

⇒ 3x = 29 − 5

⇒ 3x = 24

⇒ x = 24 ÷ 3

⇒ x = 8

∴ The correct answer is 8 years.

Additional Information

Linear Equations in One Variable

An equation of the form ax + b = c, where x is the unknown variable and a, b, c are constants.

Age Relation Problems

Always define the base person's age as 'x' to simplify the relationship expressions for others.

Verification Method

Substitute the answer back: If Emma is 8, Sarah is (2 × 8) + 5 = 21. Sum: 8 + 21 = 29. Correct.

Shortcut Trick

Fourth Proportional of 10, 15, 18 = (15 × 18) ÷ 10 = 270 ÷ 10 = 27

Mean Proportional of 9 and 4 = √(9 × 4) = √36 = 6

Required Ratio = 27 : 6 = 9 : 2

∴ The correct answer is 9:2.

Alternate Method

Given: Numbers for Fourth Proportional (a=10, b=15, c=18) and Mean Proportional (x=9, y=4).

Formula Used: Fourth Proportional (d) = (b × c) ÷ a; Mean Proportional = √(x × y)

⇒ Fourth Proportional (d) = (15 × 18) ÷ 10

⇒ d = 270 ÷ 10 = 27

⇒ Mean Proportional (m) = √(9 × 4)

⇒ m = √36 = 6

⇒ Ratio = 27 : 6 = 9 : 2

∴ The correct answer is 9:2.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is such that a : b :: b : c, given by c = b² / a.

Mean Proportional Definition

The mean proportional between two numbers a and b is the geometric mean, calculated as √(a × b).

Continued Proportion

Three quantities a, b, c are in continued proportion if a/b = b/c, where b is the mean proportional.

Shortcut Trick

Let the number be N.

Translate the percentage and fraction: 80% = 4/5 and 10% = 1/10.

The equation becomes: N × (4/5) × (1/10) × (2/3) = 748.

Simplify: N × (8/150) = 748 ⇒ N = (748 × 150) ÷ 8.

N = 93.5 × 150 = 14025.

∴ The correct answer is 14025.

Alternate Method

Given:

Product result = 748

Percentage 1 = 80%

Percentage 2 = 10%

Fraction = 2/3

Calculations:

⇒ (80/100) × (10/100) × (2/3) × N = 748

⇒ (4/5) × (1/10) × (2/3) × N = 748

⇒ (8/150) × N = 748

⇒ N = (748 × 150) ÷ 8

⇒ N = 112200 ÷ 8

⇒ N = 14025

∴ The correct answer is 14025.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide the value by 100. For example, x% = x/100.

Successive Percentage Application

When multiple percentages or fractions are applied to a number, they are multiplied together as factors.

Reciprocal Multiplication

To find the original value from a final result, multiply the result by the reciprocal of each applied fraction.

Shortcut Trick

Total Cost Price (CP) of 9 articles = ₹16.

Total Selling Price (SP) of 9 articles = 9 × 2 = ₹18.

Profit = SP − CP = 18 − 16 = ₹2.

Profit % = (Profit ÷ CP) × 100 = (2 ÷ 16) × 100.

Profit % = (1 ÷ 8) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Alternate Method

Given → CP of 9 articles = ₹16, SP of 1 article = ₹2. Formulas: Profit = SP − CP | Profit % = (Profit ÷ CP) × 100.

⇒ Cost Price (CP) for 9 articles = ₹16. ⇒ Selling Price (SP) for 1 article = ₹2. ⇒ SP for 9 articles = 9 × 2 = ₹18. ⇒ Profit = 18 − 16 = ₹2. ⇒ Profit Percentage = (2 ÷ 16) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Additional Information

Profit and Loss Calculation

Profit occurs when SP > CP. Profit % is always calculated on the CP unless stated otherwise.

Unitary Method

You can also find CP of 1 article (16 ÷ 9) and compare it with SP of 1 article (2) to get the same result.

Shortcut Trick

Mode is defined as the value that appears most frequently in a data set.

By observing the given data, the value 80 appears 5 times.

All other values such as 81 (3 times), 66 (3 times), and others appear less than 5 times.

Therefore, the maximum frequency corresponds to the number 80.

∴ The correct answer is 80.

Alternate Method

Given: Data set = {75, 70, 80, 70, 80, 71, 80, 66, 87, 82, 68, 73, 80, 75, 66, 64, 66, 87, 90, 68, 89, 82, 81, 81, 65, 81, 80}

To find the Mode, we construct a Frequency Table for the observed values:

⇒ Value 80: Observed at positions 3, 5, 7, 13, 27 → Frequency = 5

⇒ Value 81: Observed at positions 23, 24, 26 → Frequency = 3

⇒ Value 66: Observed at positions 8, 15, 17 → Frequency = 3

⇒ Value 75: Observed at positions 1, 14 → Frequency = 2

⇒ Value 70: Observed at positions 2, 4 → Frequency = 2

⇒ Value 87: Observed at positions 9, 18 → Frequency = 2

⇒ Value 82: Observed at positions 10, 22 → Frequency = 2

⇒ Value 68: Observed at positions 11, 20 → Frequency = 2

⇒ Other values (64, 65, 71, 73, 89, 90) each have a frequency of 1.

⇒ Comparing frequencies: 5 > 3 > 2 > 1.

⇒ Since 80 has the highest frequency (5), it is the Mode.

∴ The correct answer is 80.

Additional Information

Definition of Mode

The mode is the value that appears most often in a set of data. A data set can be unimodal (one mode), bimodal (two modes), or multimodal.

Empirical Relationship

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Median and Mean

Mean is the average of all observations, while Median is the middle value when data is arranged in ascending or descending order.

Shortcut Trick

To find the greatest number leaving the same remainder, calculate the HCF of the differences between the numbers.

Differences: (59 − 47) = 12, (74 − 59) = 15, and (74 − 47) = 27.

HCF of 12, 15, and 27 is 3.

∴ The correct answer is 3.

Alternate Method

Given: Numbers are 47, 59, and 74.

Formula: For the same remainder, the number is HCF(|x−y|, |y−z|, |z−x|).

⇒ Difference 1 = 59 − 47 = 12

⇒ Difference 2 = 74 − 59 = 15

⇒ Difference 3 = 74 − 47 = 27

⇒ Prime factors of 12 = 2² × 3

⇒ Prime factors of 15 = 3 × 5

⇒ Prime factors of 27 = 3³

⇒ HCF(12, 15, 27) = 3

∴ The correct answer is 3.

Additional Information

HCF with Same Remainder

The greatest number which divides x, y, and z leaving the same remainder in each case is HCF of (x−y), (y−z), and (z−x).

HCF with Different Remainders

To find the greatest number that divides x, y, z to leave remainders a, b, c, find HCF of (x−a), (y−b), and (z−c).

LCM Applications

LCM is used for finding the smallest number divisible by given values or finding when events (like bells ringing) occur together.

Shortcut Trick

The ratio of Total Surface Area (TSA) depends on the factor r(r + h) since 2π is constant.

For Original Cylinder: Factor₁ = 5 × (5 + 12) = 5 × 17 = 85

For Small Cylinder: Factor₂ = 2 × (2 + 3) = 2 × 5 = 10

Required Ratio (Small : Original) = 10 : 85 = 2 : 17

∴ The correct answer is 2 : 17.

Alternate Method

Given:

Original Cylinder: Radius (R) = 5 cm, Height (H) = 12 cm

Small Cylinder: Radius (r) = 2 cm, Height (h) = 3 cm

Formula Used:

Total Surface Area (TSA) of a Cylinder = 2πr(r + h)

Calculations:

⇒ TSA of Original Cylinder = 2 × π × R × (R + H)

⇒ TSA_Original = 2 × π × 5 × (5 + 12) = 10π × 17 = 170π cm²

⇒ TSA of one Small Cylinder = 2 × π × r × (r + h)

⇒ TSA_Small = 2 × π × 2 × (2 + 3) = 4π × 5 = 20π cm²

⇒ Required Ratio = TSA_Small : TSA_Original

⇒ Required Ratio = 20π : 170π = 20 : 170 = 2 : 17

∴ The correct answer is 2 : 17.

Additional Information

Volume of a Cylinder

The amount of space occupied by a cylinder is given by the formula V = πr²h.

Curved Surface Area (CSA)

The area of the curved part of the cylinder is 2πrh. Total surface area includes the two circular bases: TSA = CSA + 2πr².

Melting and Recasting Rule

When a solid is melted and recast into other shapes, its total Volume remains constant. Volume_Original = n × Volume_Small.

Shortcut Trick

If sinA = 1/2, then the angle A = 30°.

Substituting A in sin3A → sin(3 × 30°) = sin90°.

The value of sin90° is 1.

∴ The correct answer is 1.

Alternate Method

Given: sinA = 1/2

Formula Used: sin3A = 3sinA − 4sin³A

⇒ sin3A = 3 × (1/2) − 4 × (1/2)³

⇒ sin3A = 3/2 − 4 × (1/8)

⇒ sin3A = 3/2 − 1/2

⇒ sin3A = 2/2 = 1

∴ The correct answer is 1.

Additional Information

Trigonometric Triple Angle Formulas

The formula for sin3A is 3sinA − 4sin³A and for cos3A is 4cos³A − 3cosA.

Standard Values of Sine

sin0° = 0, sin30° = 1/2, sin45° = 1/√2, sin60° = √3/2, sin90° = 1.

Range of Sine Function

The value of sinθ for any real θ always lies in the interval [−1, 1].

Shortcut Trick

In ΔBDC, ∠C = 90° − ∠DBC = 90° − 76° = 14°.

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

By the Exterior Angle Theorem: ∠AEB = ∠CAE + ∠C.

∠AEB = 53° + 14° = 67°.

∴ The correct answer is 67°.

Alternate Method

Given: In ΔABC, BD ⊥ AC at D, ∠DBC = 76°, and ∠CAE = 53°.

Formula Used: Angle Sum Property of a triangle and Exterior Angle Theorem.

In ΔBDC:

⇒ ∠BDC = 90° (Given BD ⊥ AC)

⇒ ∠DBC = 76° (Given)

⇒ ∠C = 180° − (∠BDC + ∠DBC)

⇒ ∠C = 180° − (90° + 76°) = 14°.

In ΔAEC:

⇒ ∠CAE = 53° (Given)

⇒ ∠ACE = 14° (Calculated above)

⇒ ∠AEB is the exterior angle to ΔAEC at vertex E.

⇒ ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 53° + 14° = 67°.

∴ The correct answer is 67°.

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 of any triangle is always 180°.

Complementary Angles

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

Shortcut Trick

Let TV cost = 100, then Washing Machine (WM) cost = 75.

Ratio of cost WM : TV = 3 : 4. For 8 WM and 4 TV, cost ratio = (8 × 3) : (4 × 4) = 24 : 16 = 3 : 2.

Using Weighted Average: % Change = [(3 × 30%) + (2 × −19%)] ÷ (3 + 2).

% Change = (90 − 38) ÷ 5 = 52 ÷ 5 = 10.4%.

Since the result is positive, it is an increase.

∴ The correct answer is Increase by 10.4%.

Alternate Method

Given → TV cost = 100, WM cost = 75 (25% less). Formulas: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂) | % Change = (Change ÷ Original) × 100.

⇒ Initial Total Cost = (8 × 75) + (4 × 100) = 600 + 400 = 1000. ⇒ New WM Cost = 75 + 30% of 75 = 75 + 22.5 = 97.5. ⇒ New TV Cost = 100 − 19% of 100 = 81. ⇒ New Total Cost = (8 × 97.5) + (4 × 81) = 780 + 324 = 1104. ⇒ Percentage Change = [(1104 − 1000) ÷ 1000] × 100 = (104 ÷ 1000) × 100 = 10.4%.

∴ The correct answer is Increase by 10.4%.

Additional Information

Weighted Average Percentage

When different components of a total change by different percentages, the net change is the sum of (weight × individual change) divided by the total weight.

Percentage Increase/Decrease

Percentage change = [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive value indicates an increase, while a negative value indicates a decrease.

Shortcut Trick

Let Total Work = LCM(30, 45, 24) = 360 units.

Efficiencies: A = 12, B = 8, C = −15.

Work done by A + B in first 6 minutes = (12 + 8) × 6 = 120 units.

Remaining work = 360 − 120 = 240 units. Combined efficiency (A+B+C) = 12 + 8 − 15 = 5.

Time for remaining part = 240 ÷ 5 = 48 minutes. Total time = 6 + 48 = 54 minutes.

∴ The correct answer is 54 min.

Alternate Method

Given: Pipe A fills in 30 min, Pipe B fills in 45 min, Pipe C empties in 24 min.

⇒ Work done in first 6 min = (Eff_A + Eff_B) × 6

⇒ Work done = (12 + 8) × 6 = 120 units

⇒ Remaining work = 360 − 120 = 240 units

⇒ Combined efficiency of A, B, and C = 12 + 8 − 15 = 5 units/min

⇒ Time taken to complete remaining work = 240 ÷ 5 = 48 min

⇒ Total time required = Initial time + Remaining time = 6 + 48 = 54 min

∴ The correct answer is 54 min.

Additional Information

Pipes and Cisterns Efficiency

Filling pipes have positive efficiency, while emptying (outlet) pipes have negative efficiency in calculations.

Total Work Calculation

Total capacity of the tank is usually taken as the LCM of the individual times to simplify division.

Combined Rate

When pipes operate together, the net rate is the algebraic sum of their individual rates: Net Rate = R₁ + R₂ − R_outlet.

Shortcut Trick

Given a + b = 4 and ab = 3.

Use the direct identity: a³ + b³ = (a + b)³ − 3ab(a + b).

Substitute the values: 4³ − 3 × 3 × 4

Value = 64 − 36 = 28.

∴ The correct answer is 28.

Alternate Method

Given:

Sum of terms (a + b) = 4

Product of terms (ab) = 3

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

⇒ (4)³ = a³ + b³ + 3 × 3 × 4

⇒ 64 = a³ + b³ + 36

⇒ a³ + b³ = 64 − 36

⇒ a³ + b³ = 28

∴ The correct answer is 28.

Additional Information

Sum of Cubes Formula

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

Difference of Cubes Formula

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

Standard Binomial Expansion

The cube of a sum is expanded as (a + b)³ = a³ + 3a²b + 3ab² + b³, which simplifies to a³ + b³ + 3ab(a + b).

Shortcut Trick

Ratio of Income : Expenditure = 8 : 5.

Saving = Income − Expenditure = 8 − 5 = 3 units.

Saving % = (Saving ÷ Income) × 100 = (3 ÷ 8) × 100.

Since 1/8 = 12.5%, then 3/8 = 3 × 12.5% = 37.5%.

∴ The correct answer is 37.5%.

Alternate Method

Given: Ratio of Income (I) to Expenditure (E) is 8 : 5.

Formula Used: Saving (S) = Income − Expenditure; Saving % = (S/I) × 100

⇒ Let Income = 8x and Expenditure = 5x

⇒ Saving = 8x − 5x = 3x

⇒ Saving Percentage = (3x ÷ 8x) × 100

⇒ Saving Percentage = 0.375 × 100

⇒ Saving Percentage = 37.5%

∴ The correct answer is 37.5%.

Additional Information

Income-Expenditure Identity

Total Income is always the sum of Expenditure and Savings (I = E + S).

Percentage Conversion

To find the percentage of Savings, always use Income as the base (denominator) unless specified otherwise.

Common Fraction to Percentage

1/8 = 12.5%, 2/8 (1/4) = 25%, 3/8 = 37.5%, 4/8 (1/2) = 50%.

Shortcut Trick

Combined efficiency of A, B, and C = (1/3 + 1/3 + 1/12) = 9/12 = 3/4 units/day.

Days taken for same work = 1 ÷ (3/4) = 4/3 days.

Days taken for thrice the work = 3 × (4/3) = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Days taken by A = 3, B = 3, C = 12.

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

⇒ Total Efficiency of A + B + C = 4 + 4 + 1 = 9 units/day.

⇒ Work to be completed = 3 × Total Work = 3 × 12 = 36 units.

⇒ Time taken = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Work-Efficiency Relation

Efficiency is inversely proportional to the time taken to complete the work. More efficiency means less time.

Work and Wages

Wages are distributed in the ratio of work done or the ratio of efficiencies when all persons work for the same duration.

Pipes and Cisterns

Similar logic applies where filling pipes have positive efficiency and leakage/outlet pipes have negative efficiency.

Shortcut Trick

Let the price of a short notebook be x. Then the long notebook price is x + 20.

According to the question: 4(x + 20) + 5x = 305

9x + 80 = 305 ⇒ 9x = 225 ⇒ x = 25.

Price of long notebook = 25 + 20 = 45.

∴ The correct answer is 45, 25.

Alternate Method

Given:

Total cost of 4 long and 5 short notebooks = ₹305.

Difference in price = ₹20.

Formula Used:

Formation of Linear Equations: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂).

Calculations:

Let L be the price of a long notebook and S be the price of a short notebook.

⇒ 4L + 5S = 305 --- (Eq. 1)

⇒ L − S = 20 ⇒ L = S + 20 --- (Eq. 2)

Substituting the value of L from (Eq. 2) into (Eq. 1):

⇒ 4(S + 20) + 5S = 305

⇒ 4S + 80 + 5S = 305

⇒ 9S = 305 − 80

⇒ 9S = 225

⇒ S = 25

Now, finding the value of L:

⇒ L = 25 + 20 = 45

∴ The correct answer is 45, 25.

Additional Information

Substitution Method

A process of solving a system of equations by expressing one variable in terms of the other and substituting it into the remaining equation.

Elimination Method

A technique where equations are added or subtracted to eliminate one variable, making it easier to solve for the other.

Linear Equation in Two Variables

An algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line, typically written as ax + by = c.

Shortcut Trick

Convert all to base 2:

32 = 2⁵, 64 = 2⁶, 16 = 2⁴, 4 = 2²

Expression inside root:

= 2⁵ × (2⁶)⁻² × 2 × (2⁴)⁻³ × (2²)^m+3

⇒ = 2^{5 − 12 + 1 − 12 + 2m + 6}

⇒ = 2^{2m − 12}

Square root ⇒ power becomes half:

⇒ = 2^{(2m − 12)/2} = 2^{m − 6}

Given value = 2 ⇒ 2^m−6 = 2¹

⇒ m − 6 = 1 ⇒ m = 7

∴ Answer = 7

Alternate Method

Given:

√(32 × 64⁻² × 2 / 16³ × 4^m+3) = 2

Formula used:

a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn

Calculation:

⇒ Convert to base 2 and combine powers

⇒ Inside root = 2^{2m − 12}

⇒ After square root = 2^{m − 6}

⇒ 2^{m − 6} = 2

⇒ m − 6 = 1

⇒ m = 7

∴ The correct answer is 7.

Additional Information

Exponent Rule

Convert all numbers to same base for easy simplification.

Square Root Rule

√(aⁿ) = a^n/2

Power Matching

If bases are same, equate exponents.

Shortcut Trick

Speed is constant → Distance is directly proportional to Time.

Ratio of Time = 6 : 4.5 = 4 : 3

4 units = 240 km

1 unit = 60 km

3 units = 3 × 60 = 180 km

∴ The correct answer is 180 km.

Alternate Method

Given:

Distance₁ = 240 km

Time₁ = 6 hours

Time₂ = 4.5 hours

Formula Used:

Speed = Distance ÷ Time

Distance = Speed × Time

Calculations:

⇒ Speed of the bus = 240 ÷ 6

⇒ Speed = 40 kmph

⇒ New Distance = Speed × New Time

⇒ New Distance = 40 × 4.5

⇒ New Distance = 180 km

∴ The correct answer is 180 km.

Additional Information

Direct Proportionality

When speed is constant, distance (D) is directly proportional to time (T), expressed as D₁/T₁ = D₂/T₂.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds u and v, Average Speed = 2uv/(u+v).

Unit Conversion

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