Quantitative Aptitude PYQs

Shortcut Trick

Target value = 1/20 and Base value = 10/50 = 1/5

Required Percentage = (Target ÷ Base) × 100

Required Percentage = (1/20 ÷ 1/5) × 100

Required Percentage = (1/20 × 5) × 100

Required Percentage = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Target value = 1/20

Base value = 10/50

Formula Used:

Percentage = (Target ÷ Base) × 100

Base Value = 10/50 = 1/5 Target Value = 1/20 1/4 of Base = 25% Percentage = (1/20 ÷ 1/5) × 100 = 25%

Calculations:

⇒ Base value = 10/50 = 1/5

⇒ Percentage = (1/20 ÷ 1/5) × 100

⇒ Percentage = (1/20 × 5/1) × 100

⇒ Percentage = (5/20) × 100

⇒ Percentage = (1/4) × 100

⇒ Percentage = 25%

∴ The correct answer is 25%.

Additional Information

Percentage Formula

To find what percentage X is of Y, use the formula (X ÷ Y) × 100.

Fraction to Percentage Conversion

To convert any fraction to a percentage, multiply it by 100. For example, 1/4 = 1/4 × 100 = 25%.

Percentage of a Quantity

To find a given percentage P of a quantity Q, calculate (P ÷ 100) × Q.

Reciprocal Percentages

Knowing basic fraction to percentage conversions (like 1/2 = 50%, 1/4 = 25%, 1/5 = 20%) speeds up calculation significantly in competitive exams.

Shortcut Trick

Efficiency ratio A : B : C = 100 : (100 − 45) : 55 × 1.5 = 100 : 55 : 82.5 = 200 : 110 : 165 = 40 : 22 : 33.

Total Work = Efficiency of A × Time of A = 40 × 66 = 2640 units.

Work done by B + C in 24 days = (22 + 33) × 24 = 55 × 24 = 1320 units.

Remaining Work = 2640 − 1320 = 1320 units.

Time taken by A for remaining work = 1320 ÷ 40 = 33 days.

∴ The correct answer is 33 days.

Alternate Method

Given: Time taken by A = 66 days. B is 45% less efficient than A. C is 50% more efficient than B.

Formula Used: Total Work = Efficiency × Time; Time = Work ÷ Efficiency

⇒ Efficiency of A = 40 units/day (Assumed for calculation ease based on percentage).

⇒ Total Work = 40 × 66 = 2640 units.

⇒ Efficiency of B = 40 − 45% of 40 = 40 − 18 = 22 units/day.

⇒ Efficiency of C = 22 + 50% of 22 = 22 + 11 = 33 units/day.

⇒ Work completed by B and C in 24 days = (22 + 33) × 24 = 55 × 24 = 1320 units.

⇒ Remaining Work = 2640 − 1320 = 1320 units.

⇒ Time taken by A alone = 1320 ÷ 40 = 33 days.

∴ The correct answer is 33 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiencies is x : y, the ratio of time taken is y :

x. Work and Wages

Wages are generally distributed in the ratio of the work done by each person, which is equivalent to the ratio of their efficiencies if they work for the same duration.

Shortcut Trick

To check divisibility by 49, we can use the osculator method for 7² or simpler division.

A number is divisible by 49 if (Remaining digits + 5 × Last digit) is divisible by 49.

Checking 7132979: 713297 + 5(9) = 713342; 71334 + 5(2) = 71344; 7134 + 5(4) = 7154; 715 + 5(4) = 735.

Since 735 ÷ 49 = 15, the number is divisible.

∴ The correct answer is 7132979.

Alternate Method

Given: Numbers to check for divisibility by 49.

Method: Since 49 = 7 × 7, a number divisible by 49 must be divisible by 7 twice or simply perform long division for the target values.

⇒ Option 1: 7311476 ÷ 49 ≈ 149213.79 (Not divisible)

⇒ Option 2: 7132979 ÷ 49 = 145571 (Divisible)

⇒ Option 3: 7492470 ÷ 49 ≈ 152907.55 (Not divisible)

⇒ Option 4: 6697359 ÷ 49 ≈ 136680.79 (Not divisible)

∴ The correct answer is 7132979.

Additional Information

Divisibility by 7

Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the original number is also divisible by 7.

Osculator for 49

For a number ending in 9 like 49, the positive osculator is (4 + 1) = 5. Multiply the last digit by 5 and add to the remaining part.

Divisibility by 7ⁿ

For higher powers of 7, it is often faster in exams to use long division or digital sum combined with prime factorization checks.

Shortcut Trick

The Selling Price (SP) is what remains after subtracting the Discount% from the Marked Price (MP).

MP × (100 − Discount)% = SP

MP × 40% = 49,834

MP = 49,834 ÷ 0.4 = 1,24,585.

∴ The correct answer is ₹1,24,585.

Alternate Method

Given: Discount = 60%, Selling Price (SP) = ₹49,834.

Formulas: SP = MP × (100 − Discount%) ÷ 100 | MP = (SP × 100) ÷ (100 − Discount%)

⇒ Let the Marked Price be MP.

⇒ SP = MP − 60% of MP = 40% of MP.

⇒ 40 ÷ 100 × MP = 49,834.

⇒ MP = (49,834 × 100) ÷ 40.

⇒ MP = 49,834 × 2.5 = 1,24,585.

∴ The correct answer is ₹1,24,585.

Additional Information

Marked Price & Discount

The Marked Price is the price printed on the label. Discount is always calculated on the Marked Price: SP = MP − Discount.

Ratio Method

60% discount means a ratio of 3:5. If MP is 5 units, Discount is 3 units, then SP is 2 units. Here, 2 units = 49,834, so 1 unit = 24,917. Therefore, 5 units = 1,24,585.

Shortcut Trick

Assume a base value for the Mean as 40.

Calculate deviations from 40: −19, −17, −2, +59, +24, −5, +8, +12, −24.

Net Deviation = (−19 − 17 − 2 − 5 − 24) + (59 + 24 + 8 + 12) = −67 + 103 = 36.

Average Deviation = 36 ÷ 9 = +4.

Correct Arithmetic Mean = 40 + 4 = 44.

∴ The correct answer is 44.

Alternate Method

Given:

Observations = 21, 23, 38, 99, 64, 35, 48, 52, 16

Total number of observations (n) = 9

Formula Used:

Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Sum = 21 + 23 + 38 + 99 + 64 + 35 + 48 + 52 + 16

⇒ Sum = 396

⇒ Arithmetic Mean = 396 ÷ 9

⇒ Arithmetic Mean = 44

∴ The correct answer is 44.

Additional Information

Combined Mean

If two groups have means x₁ and x₂ with sizes n₁ and n₂, the combined mean is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Effect of Change

If every observation is increased or decreased by a constant value 'k', the mean also increases or decreases by 'k'.

Sum of Deviations

The sum of the deviations of all observations from their arithmetic mean is always equal to zero (∑(x_i − Mean) = 0).

Shortcut Trick

Distance = 4 × Side = 4 × 20 = 80 m

Speed in m/s = 9 × (5/18) = 2.5 m/s

Time = Distance ÷ Speed = 80 ÷ 2.5 = 32 seconds

∴ The correct answer is 32 seconds.

Alternate Method

Given:

Side of square field (a) = 20 m

Speed of the boy (v) = 9 km/hr

Formula Used:

Distance (Perimeter) = 4 × Side

Time = Distance ÷ Speed

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

Calculations:

⇒ Total distance for one complete round = 4 × 20 m = 80 m

⇒ Speed in m/s = 9 × (5/18) = 5/2 = 2.5 m/s

⇒ Time taken = 80 ÷ 2.5

⇒ Time taken = 32 seconds

∴ The correct answer is 32 seconds.

Additional Information

Unit Conversion

1 km/hr = 5/18 m/s and 1 m/s = 18/5 km/hr. Always ensure distance and speed units are compatible.

Perimeter of Polygons

For a complete round, the distance covered is equal to the perimeter. For a rectangle, it is 2 × (length + breadth).

Average Speed

Average Speed = Total Distance ÷ Total Time. In uniform motion, speed remains constant throughout the journey.

Shortcut Trick

Apply BODMAS and solve orders (powers and roots) directly in one step.

Expression: √(5 + √25 + 5² − 3² − 10) − √4

⇒ √(5 + 5 + 25 − 9 − 10) − 2

⇒ √(35 − 19) − 2

⇒ √(16) − 2 = 4 − 2 = 2

∴ The correct answer is 2.

Alternate Method

Given:

Expression = √(5 + √25 + 5² − 3² − 10) − √4

Concept Used:

Follow the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

Calculation:

First, resolve the squares and square roots (Orders) inside the main square root.

⇒ √25 = 5

⇒ 5² = 25

⇒ 3² = 9

⇒ √4 = 2

Substitute these values back into the expression:

⇒ √(5 + 5 + 25 − 9 − 10) − 2

Perform the addition inside the bracket:

⇒ √(10 + 25 − 9 − 10) − 2

⇒ √(35 − 9 − 10) − 2

Perform the subtraction inside the bracket:

⇒ √(26 − 10) − 2

⇒ √16 − 2

Resolve the final square root:

⇒ 4 − 2

⇒ 2

∴ The correct answer is 2.

Additional Information

BODMAS Rule

An acronym to remember the order of operations in arithmetic: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Perfect Squares and Roots

Squaring a number means multiplying it by itself (e.g., 5 × 5 = 25). The square root operation is its inverse, finding the value that was squared to yield the number (e.g., √25 = 5). For pure simplification, the principal (positive) square root is always considered.

Shortcut Trick

When speed becomes 3/4, the time becomes 4/3 of the usual time.

Difference in time ratio = 4 − 3 = 1 unit.

Given 1 unit = 8 minutes.

Usual time = 3 units = 3 × 8 = 24 minutes.

∴ The correct answer is 24 min.

Alternate Method

Given: New speed = 3/4 of Usual speed, Delay = 8 minutes.

Formula: Since Distance is constant, Speed ∝ 1/Time. (Ratio of speed = a : b → Ratio of time = b : a)

⇒ Ratio of Speed (Usual : New) = 1 : 3/4 = 4 : 3

⇒ Ratio of Time (Usual : New) = 3 : 4

⇒ Difference in time parts = 4 − 3 = 1 unit

⇒ 1 unit = 8 minutes

⇒ Usual time = 3 units = 3 × 8 = 24 minutes

∴ The correct answer is 24 min.

Additional Information

Inverse Proportion Rule

When distance is constant, speed and time are inversely proportional. If speed is multiplied by a/b, time is multiplied by b/a.

Direct Formula for Usual Time

If a person travels at a/b of his usual speed and is 't' minutes late, Usual Time = (a × t) / |b − a|.

Average Speed

For two equal distances covered at speeds x and y, Average Speed = 2xy / (x + y).

Shortcut Trick

Find the unit price of each item: Onions = 273 ÷ 13 = 21, Tomatoes = 462 ÷ 21 = 22, Potatoes = 594 ÷ 27 = 22.

Total cost = (21 × 21) + (25 × 22) + (32 × 22).

Total cost = 441 + 550 + 704 = 1,695.

∴ The correct answer is ₹1,695.

Alternate Method

Given → Cost of 13 kg onions = ₹273, 21 kg tomatoes = ₹462, 27 kg potatoes = ₹594.
Formulas → Unit Cost = Total Cost ÷ Quantity | Total Cost = Σ(Unit Cost × New Quantity)

⇒ Cost per kg of Onion = 273 ÷ 13 = ₹21.
⇒ Cost per kg of Tomato = 462 ÷ 21 = ₹22.
⇒ Cost per kg of Potato = 594 ÷ 27 = ₹22.
⇒ Cost of 21 kg Onions = 21 × 21 = ₹441.
⇒ Cost of 25 kg Tomatoes = 25 × 22 = ₹550.
⇒ Cost of 32 kg Potatoes = 32 × 22 = ₹704.
⇒ Total Cost = 441 + 550 + 704 = ₹1,695.

∴ The correct answer is ₹1,695.

Additional Information

Unitary Method

This method involves finding the value of a single unit first and then multiplying it by the required number of units to find the total value.

Calculation Tip

When multiplying by 22, you can multiply by 2 and then by 11. For example, 32 × 22 = 64 × 11 = 704.

Shortcut Trick

Ratio of students in the two groups = 14 : (38 − 14) = 14 : 24 = 7 : 12.

Assume the base average is 73. Deviation for the first group = 68 − 73 = −5.

Total deviation = 7 × (−5) = −35. This is distributed over the total ratio parts (7 + 12 = 19).

Net deviation per student = −35 ÷ 19 ≈ −1.84.

Actual Average = 73 − 1.84 = 71.16 ≈ 71.2.

∴ The correct answer is 71.2.

Alternate Method

Given:

Total number of students = 38

Number of students in Group 1 (n₁) = 14 | Average score (A₁) = 68

Number of students in Group 2 (n₂) = 38 − 14 = 24 | Average score (A₂) = 73

Formula Used:

Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂)

Calculations:

⇒ Total score of Group 1 = 14 × 68 = 952

⇒ Total score of Group 2 = 24 × 73 = 1752

⇒ Total score of the whole class = 952 + 1752 = 2704

⇒ Average score of the whole class = 2704 ÷ 38

⇒ Average score = 1352 ÷ 19

⇒ Average score ≈ 71.1578...

⇒ Average score (rounded to one decimal place) = 71.2

∴ The correct answer is 71.2.

Additional Information

Weighted Average Concept

When groups of different sizes are combined, the weighted average is pulled closer to the average of the larger group.

Assumed Mean Method

New Average = Assumed Average + (Sum of Deviations ÷ Total Number of items). This simplifies large multiplications.

Rounding Rules

To round to one decimal place, look at the second decimal digit; if it is 5 or more, round up the first decimal digit.

Shortcut Trick

Convert the given decimals into fractions: 6.3 = 63/10 and 0.07 = 7/100.

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

LCM of 63 and 7 = 63.

HCF of 10 and 100 = 10.

LCM of given fractions = 63 ÷ 10 = 6.3.

∴ The correct answer is 6.3.

Alternate Method

Given:

The numbers are 6.3 and 0.07

Method Used:

Convert the given decimals to like decimals, treat them as integers to calculate the LCM, and then restore the decimal point.

Calculations:

⇒ Convert into like decimals (same number of decimal digits): 6.30 and 0.07

⇒ Remove the decimals to get integer values: 630 and 7

⇒ Now, find the LCM of 630 and 7

⇒ Since 630 is perfectly divisible by 7 (7 × 90 = 630), the LCM of 630 and 7 is 630.

⇒ Restore the decimal point after two places from the right (as per the like decimals): 6.30 = 6.3

∴ The correct answer is 6.3.

Additional Information

HCF of Fractions

HCF of fractions = HCF of numerators ÷ LCM of denominators.

LCM of Fractions

LCM of fractions = LCM of numerators ÷ HCF of denominators.

Relationship between LCM and HCF

For any two numbers A and B: LCM × HCF = A × B.

Shortcut Trick

When simple interests are equal for the same time period, the investments are in the inverse ratio of their rates.

Ratio of investments = P₁ : P₂ = R₂ : R₁

⇒ P₁ : P₂ = 9 : 10

Total ratio parts = 9 + 10 = 19

Sum invested at 10% (P₁) = (9 ÷ 19) × 20900 = 9900

∴ The correct answer is ₹9,900.

Alternate Method

Given:

Total investment = ₹20,900

Rate 1 (R₁) = 10% per annum

Rate 2 (R₂) = 9% per annum

Time (T) = 8 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Total: ₹20,900

Part 1 at 10%
Ratio: 9

Part 2 at 9%
Ratio: 10

Condition: Interest 1 = Interest 2

Calculation:

Let the sum invested at 10% be x and the sum invested at 9% be (20900 − x).

According to the question, the simple interest from both investments is equal after 8 years:

⇒ (x × 10 × 8) ÷ 100 = ((20900 − x) × 9 × 8) ÷ 100

⇒ 80x = 72(20900 − x)

⇒ 10x = 9(20900 − x)

⇒ 10x = 188100 − 9x

⇒ 10x + 9x = 188100

⇒ 19x = 188100

⇒ x = 188100 ÷ 19

⇒ x = 9900

∴ The correct answer is ₹9,900.

Additional Information

When Simple Interests are equal:

If parts P₁, P₂, P₃ are invested at rates R₁, R₂, R₃ for times T₁, T₂, T₃ such that their simple interests are equal, then P₁ : P₂ : P₃ = 1 ÷ (R₁ × T₁) : 1 ÷ (R₂ × T₂) : 1 ÷ (R₃ × T₃).

When Amounts are equal:

If the final amounts received from the investments are equal, then P₁ : P₂ : P₃ = 1 ÷ (100 + R₁ × T₁) : 1 ÷ (100 + R₂ × T₂) : 1 ÷ (100 + R₃ × T₃).

Simple Interest vs Compound Interest:

For the first year, simple interest and compound interest on a given sum at the same rate are always equal when compounded annually. From the second year onwards, compound interest generates interest on previously earned interest.

Shortcut Trick

65% of 65 = 65² ÷ 100 = 4225 ÷ 100 = 42.25

2/5 of 55 = 2 × 11 = 22

Difference = 42.25 − 22 = 20.25

∴ The correct answer is 20.25.

Alternate Method

Given:

First value = 65% of 65

Second value = 2/5 of 55

Calculation:

⇒ First value = (65 ÷ 100) × 65 = 4225 ÷ 100 = 42.25

⇒ Second value = (2 ÷ 5) × 55 = 2 × 11 = 22

⇒ Required difference = First value − Second value

⇒ Required difference = 42.25 − 22 = 20.25

∴ The correct answer is 20.25.

Additional Information

Squaring numbers ending in 5

To find the square of a number ending in 5 (like N5), multiply N by (N+1) and append 25 at the end. For example, 65² → 6 × 7 = 42, so 4225.

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide the value by 100 and simplify. For example, 65% = 65 ÷ 100 = 13/20.

BODMAS Rule

When simplifying arithmetic expressions, follow the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

To check divisibility by 3, simply calculate the sum of all digits of the given number.

Sum of digits = 6 + 4 + 8 + 6 + 7 + 9 + 0 + 5 + 9 = 54

Since 54 is a multiple of 3 (54 ÷ 3 = 18), the entire number is perfectly divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 648679059

Concept Used:

Check standard divisibility rules for each given option (10, 11, 3, 8) systematically to find the correct match.

Calculations:

⇒ Divisibility by 10: The last digit must be 0. Here, the last digit is 9, so it is not divisible by 10.

⇒ Divisibility by 11: The difference between the sum of digits at odd places and even places must be 0 or a multiple of 11.

⇒ Sum of digits at odd places = 6 + 8 + 7 + 0 + 9 = 30

⇒ Sum of digits at even places = 4 + 6 + 9 + 5 = 24

⇒ Difference = 30 − 24 = 6. Since 6 is not a multiple of 11, it is not divisible by 11.

⇒ Divisibility by 8: The number formed by the last three digits must be divisible by 8.

⇒ The last three digits form 059, which is not divisible by 8.

⇒ Divisibility by 3: The sum of all digits must be divisible by 3.

⇒ Sum = 6 + 4 + 8 + 6 + 7 + 9 + 0 + 5 + 9 = 54

⇒ 54 is completely divisible by 3, confirming the whole number is divisible by 3.

∴ The correct answer is 3.

Additional Information

Divisibility by 4 & 8

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. It is divisible by 8 if its last three digits are divisible by 8.

Divisibility by 9

Similar to the rule for 3, a number is completely divisible by 9 if the sum of all its digits is a multiple of 9.

Divisibility by 6

A number must be divisible by both 2 (ending in an even digit) and 3 (sum of digits is a multiple of 3) to be perfectly divisible by 6.

Shortcut Trick

For operations on the same base, combine the exponents directly using the laws of indices from left to right.

Exponents are subtracted for division and added for multiplication.

Given expression: 57⁻⁹ ÷ 57⁴ × 57⁻¹⁰

⇒ 57^{−9 − 4 + (−10)}

⇒ 57⁻²³

∴ The correct answer is 57⁻²³.

Alternate Method

Given:

57⁻⁹ ÷ 57⁴ × 57⁻¹⁰

Formula Used:

Law of Division: a^m ÷ aⁿ = a^{m − n}

Law of Multiplication: a^m × aⁿ = a^{m + n}

Calculation:

⇒ According to the order of operations, evaluate division and multiplication from left to right.

⇒ Apply the division rule to the first two terms:

⇒ (57⁻⁹ ÷ 57⁴) × 57⁻¹⁰

⇒ 57^{−9 − 4} × 57⁻¹⁰

⇒ 57⁻¹³ × 57⁻¹⁰

⇒ Now, apply the multiplication rule to the remaining expression:

⇒ 57^{−13 + (−10)}

⇒ 57^{−13 − 10}

⇒ 57⁻²³

∴ The correct answer is 57⁻²³.

Additional Information

Law of Multiplication

When multiplying terms with the same base, add their exponents: a^m × aⁿ = a^m+n.

Law of Division

When dividing terms with the same base, subtract the exponent of the divisor from the dividend's exponent: a^m ÷ aⁿ = a^m−n.

Power of a Power Rule

When raising a power to another power, multiply the exponents: (a^m)ⁿ = a^m×n.

Negative Exponent Rule

A negative exponent indicates the reciprocal of the base raised to the positive power: a^−m = 1 ÷ a^m.

Shortcut Trick

For any 3D spherical object, the surface area is directly proportional to the square of its radius (Area ∝ r²).

Given the balloon's radius becomes 5 times its original size.

Ratio of surface areas = (New Radius)² : (Old Radius)²

⇒ Ratio = 5² : 1² = 25 : 1

∴ The correct answer is 25:1.

Alternate Method

Given:

⇒ Initial radius of the balloon = r

⇒ Final radius of the balloon after pumping air = 5r

Formula Used:

⇒ Surface Area of a sphere = 4πr²

r
Before Pumping
Area = S₁

5r
After Pumping
Area = S₂

Calculations:

⇒ Initial surface area (S₁) = 4πr²

⇒ Final surface area (S₂) = 4π(5r)²

⇒ S₂ = 4π(25r²) = 100πr²

Ratio of final surface area to initial surface area:

⇒ Ratio = S₂ : S₁

⇒ Ratio = 100πr² : 4πr²

⇒ Ratio = 25 : 1

∴ The correct answer is 25:1.

Additional Information

Volume of a Sphere

The volume of a sphere is given by V = (4/3)πr³. If the radius is increased by 5 times, the new volume will become 5³ = 125 times the original volume.

Surface Area of Hemisphere

For a hemisphere of radius r, the curved surface area (CSA) is 2πr² and the total surface area (TSA) is 3πr².

Concept of Similarity in 3D Shapes

For any two similar 3D objects with a scale factor of k (ratio of corresponding sides), the ratio of their surface areas is k², and the ratio of their volumes is k³.

Shortcut Trick

The third proportional 'c' to two numbers 'a' and 'b' is calculated using the relation c = b² ÷ a.

Substitute the given values a = 60 and b = 30.

x = (30 × 30) ÷ 60

x = 900 ÷ 60 = 15

∴ The correct answer is 15.

Alternate Method

Ratio and Proportion Scale

60

30

x = 15

× 0.5

× 0.5

Continued Proportion: 60 : 30 :: 30 : x

Given:

⇒ First term (a) = 60

⇒ Second term (b) = 30

⇒ Third proportional (c) = x

Formula Used:

⇒ If three quantities a, b, and c are in continued proportion, then a : b :: b : c

⇒ Product of Extremes = Product of Means ⇒ a × c = b²

Calculations:

⇒ 60 : 30 = 30 : x

⇒ 60 × x = 30 × 30

⇒ 60x = 900

⇒ x = 900 ÷ 60

⇒ x = 15

∴ The correct answer is 15.

Additional Information

Mean Proportional

The mean proportional between two quantities a and b is given by √(a × b).

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b = c : d), then d is known as the fourth proportional, evaluated as d = (b × c) ÷ a.

Compound Ratio

For any two ratios a : b and c : d, their compound ratio is obtained by multiplying the antecedents and consequents respectively, given by (a × c) : (b × d).

Continued Proportion

Three non-zero numbers a, b, and c are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third (a : b = b : c).

Shortcut Trick

The average of the first n even numbers is directly given by the formula (n + 1).

Given the number of terms, n = 156.

⇒ Average = 156 + 1 = 157

∴ The correct answer is 157.

Alternate Method

2

4

6

...

312

Average = 157

Arithmetic Progression of First 156 Even Numbers

Given:

First even number (a) = 2

Total numbers (n) = 156

Common difference (d) = 2

Formula Used:

Last term (L) = a + (n − 1) × d

Average of an arithmetic progression = (First term + Last term) ÷ 2

Calculations:

⇒ Last term = 2 + (156 − 1) × 2

⇒ Last term = 2 + 155 × 2

⇒ Last term = 2 + 310 = 312

⇒ Average = (2 + 312) ÷ 2

⇒ Average = 314 ÷ 2

⇒ Average = 157

∴ The correct answer is 157.

Additional Information

Average of First n Odd Numbers

The sum of the first n odd numbers is always n², which means their average is simply equal to n.

Average of First n Natural Numbers

The sum of the first n consecutive natural numbers is calculated as n(n + 1) ÷ 2. Therefore, their average is (n + 1) ÷ 2.

Average of an Arithmetic Progression

For any sequence of numbers with a constant difference (like consecutive integers, evens, or odds), the average strictly falls in the exact center and is equal to (First term + Last term) ÷ 2.

Shortcut Trick

Let Cost Price (CP) = 100 units

Marked Price (MP) = 100 + 25% = 125 units

Selling Price (SP) = 125 − (16% of 125) = 105 units

Profit = SP − CP = 105 − 100 = 5 units

Given 5 units = ₹64 ⇒ 125 units = 25 × 64 = ₹1600

∴ The correct answer is ₹1600.

Alternate Method

Given:

Markup = 25%

Discount = 16%

Profit = ₹64

Formula Used:

MP = CP × (1 + Markup% ÷ 100)

SP = MP × (1 − Discount% ÷ 100)

Profit = SP − CP

CP = 100x

MP = 125x

SP = 105x

+25% Markup

-16% Discount

Profit = 5x (₹64)
5x = ₹64 → 125x = 25 × 64 = ₹1600

Calculations:

⇒ Let Cost Price (CP) = 100x

⇒ Marked Price (MP) = 100x + (25% of 100x) = 125x

⇒ Discount = 16% of 125x = 20x

⇒ Selling Price (SP) = 125x − 20x = 105x

⇒ Profit = SP − CP = 105x − 100x = 5x

⇒ Given Profit = ₹64

⇒ 5x = ₹64

⇒ x = 64 ÷ 5 = 12.8

⇒ List Price (MP) = 125 × 12.8 = ₹1600

∴ The correct answer is ₹1600.

Additional Information

Effective Discount Formula

For two successive discounts of a% and b%, the effective single discount is (a + b − ab ÷ 100)%.

Direct Relation between CP and MP

The ratio of Cost Price to Marked Price is directly given by: CP/MP = (100 − Discount%) ÷ (100 + Profit%).

Markup Percentage

Markup is always calculated on the Cost Price: Markup% = (MP − CP) ÷ CP × 100.

Shortcut Trick

Let the sum of the two consecutive even numbers be S.

The numbers can be symmetrically represented as (S/2 − 1) and (S/2 + 1).

Sum of their squares = (S/2 − 1)² + (S/2 + 1)² = S²/2 + 2

Given, S²/2 + 2 = 2452 ⇒ S²/2 = 2450

⇒ S² = 4900 ⇒ S = 70

∴ The correct answer is 70.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 2452.

Formula Used:

(a + b)² = a² + 2ab + b²

Calculation:

Let the two consecutive even natural numbers be x and (x + 2).

⇒ x² + (x + 2)² = 2452

⇒ x² + (x² + 4x + 4) = 2452

⇒ 2x² + 4x − 2448 = 0

Dividing the entire equation by 2:

⇒ x² + 2x − 1224 = 0

Splitting the middle term to factorise:

⇒ x² + 36x − 34x − 1224 = 0

⇒ x(x + 36) − 34(x + 36) = 0

⇒ (x − 34)(x + 36) = 0

⇒ x = 34 or x = −36

Since the numbers are natural numbers, we discard the negative value. So, x = 34.

The two consecutive even numbers are 34 and 36.

Sum of the numbers = 34 + 36 = 70

∴ The correct answer is 70.

Additional Information

Consecutive Even Numbers

Even numbers that follow each other in order have a constant difference of 2. They can be generally represented as 2n, 2n+2, 2n+4, etc., where n is an integer.

Key Algebraic Identity

The sum of squares of two variables can be related using the identity: (a − b)² + (a + b)² = 2(a² + b²). This is highly useful for simplifying symmetric expressions quickly without forming large quadratics.

Quadratic Equation Roots

For a standard quadratic equation ax² + bx + c = 0, the nature of roots is determined by the discriminant (D = b² − 4ac). Sum of the roots is given by −b/a and the product of roots is given by c/a.