Quantitative Aptitude PYQs

Given:

The expression: 0.125 × 4⁴ × 2⁻⁵ × 0.25² × 8²

Formula used:

a^m × aⁿ = a^m+n

(a^m)ⁿ = a^m×n

Calculation:

Convert each term into base 2:

0.125 = 1/8 = 1/2³ = 2⁻³

4⁴ = (2²)⁴ = 2⁸

2⁻⁵ = 2⁻⁵

0.25² = (1/4)² = (1/2²)² = 2⁻⁴

8² = (2³)² = 2⁶

Substitute these values back into the expression:

Expression =

Shortcut Trick

Assume total capacity of cistern = 12 units (LCM of 4 and 12).

Efficiency of Tap 1 = 3, Tap 2 = 1, and Tap 3 (emptying) = −1.

Combined efficiency when all opened = 3 + 1 − 1 = 3 units/hr.

Target work = 1/4 of total capacity = 12 ÷ 4 = 3 units.

Time required = Target units ÷ Combined efficiency = 3 ÷ 3 = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap A to fill = 4 hr

Time taken by Tap B to fill = 12 hr

Time taken by Tap C to empty = 12 hr

Formula Used:

Efficiency = Total Work ÷ Time

Total Efficiency = Efficiency_A + Efficiency_B − Efficiency_C

⇒ Net efficiency of all taps = 3 + 1 − 1 = 3 units/hr

⇒ Target to fill = 1/4 of cistern = 1/4 × 12 = 3 units

⇒ Time required = 3 units ÷ 3 units/hr = 1 hr

∴ The correct answer is 1 hr.

Additional Information

Pipes and Cisterns Basics

Filling pipes have positive efficiency, while emptying pipes (leakage) have negative efficiency.

Capacity Relation

Total capacity = Time taken to fill/empty × Efficiency of the pipe.

Time and Work Analogy

The concept is identical to men and work, where an emptying pipe acts like a person destroying the work.

Shortcut Trick

The product of two numbers is always equal to the product of their HCF and LCM.

Other number = (HCF × LCM) ÷ Given number

Other number = (36 × 7,560) ÷ 540

Other number = 7,560 ÷ 15 = 504

∴ The correct answer is 504.

Alternate Method

Given:

HCF = 36

LCM = 7,560

First Number (a) = 540

Formula Used:

a × b = HCF × LCM

Calculations:

⇒ 540 × b = 36 × 7,560

⇒ b = (36 × 7,560) ÷ 540

⇒ b = 272,160 ÷ 540

⇒ b = 504

∴ The correct answer is 504.

Additional Information

HCF and LCM Relation

For any two positive integers x and y, the product of the numbers is always equal to the product of their HCF and LCM: x × y = HCF(x, y) × LCM(x, y).

HCF Property

The Highest Common Factor (HCF) of a group of numbers is always a factor of their Least Common Multiple (LCM). Thus, LCM ÷ HCF is always an integer.

Co-prime Numbers

If the HCF of two numbers is 1, they are called co-prime or relatively prime. In this case, their LCM is simply the product of the two numbers.

Shortcut Trick

Total interest percentage = Rate × Time = 5% × 3 = 15%.

Simple Interest = 15% of ₹8,000.

SI = (15 ÷ 100) × 8,000 = 15 × 80 = 1,200.

∴ The correct answer is ₹1,200.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate of Interest (R) = 5% per annum

Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ SI = (8000 × 5 × 3) ÷ 100

⇒ SI = 80 × 15

⇒ SI = 1,200

∴ The correct answer is ₹1,200.

Additional Information

Amount in Simple Interest

The total amount (A) is the sum of the principal and the interest: A = P + SI = P(1 + (R × T) ÷ 100).

SI vs CI for Year 1

For the first year, if the interest is compounded annually, the Simple Interest and Compound Interest are always equal.

Finding Principal from SI

If SI, Rate, and Time are known, the Principal can be calculated as: P = (SI × 100) ÷ (R × T).

Shortcut Trick

Speed in m/s = 40 ÷ 8 = 5 m/s

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

Speed = 5 × (18/5) = 18 km/hr

∴ The correct answer is 18 km/hr.

Alternate Method

Given:

Distance traveled by the cyclist = 40 m

Time taken = 8 sec

Formula Used:

Speed = Distance ÷ Time

To convert speed from m/s to km/hr: Speed (m/s) × (18/5)

Calculations:

⇒ Speed = 40 ÷ 8

⇒ Speed = 5 m/s

⇒ Speed in km/hr = 5 × (18/5)

⇒ Speed = 18 km/hr

∴ The correct answer is 18 km/hr.

Additional Information

Unit Conversion (km/hr to m/s)

To convert speed from km/hr to m/s, multiply the value by 5/18.

Average Speed

Average Speed is defined as the Total Distance divided by the Total Time taken for the entire journey.

Relative Speed

When two objects move in the opposite direction, their Relative Speed is the sum of their individual speeds (u + v).

Shortcut Trick

In any right-angled triangle, if the angle of elevation is 45°, the Height (Perpendicular) is always equal to the Distance (Base).

Given Distance = 40 m.

∴ Height = 40 m.

∴ The correct answer is 40 m.

Alternate Method

Given:

Distance from the point to the building (Base) = 40 m

Angle of elevation (θ) = 45°

Formula Used:

tan θ = Perpendicular / Base

Let the height of the building be h.

⇒ tan 45° = h / 40

⇒ 1 = h / 40 (Since tan 45° = 1)

⇒ h = 40 × 1

⇒ h = 40 m

∴ The correct answer is 40 m.

Additional Information

Trigonometric Ratios Table

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

45-45-90 Triangle Property

In a triangle with angles 45°, 45°, and 90°, the sides are in the ratio 1 : 1 : √2.

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

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

Given values: a = (x + y)² and b = (x − y)².

Mean Proportional = √[(x + y)² × (x − y)²] = (x + y) × (x − y).

Using the algebraic identity (a + b)(a − b) = a² − b², we get x² − y².

∴ The correct answer is x² − y².

Alternate Method

Given:

First term (a) = (x + y)²

Third term (c) = (x − y)²

Formula Used:

If b is the mean proportional between a and c, then a : b = b : c or b² = ac ⇒ b = √ac.

Calculations:

⇒ Let the mean proportional be M.

⇒ M = √[(x + y)² × (x − y)²]

⇒ M = √[(x + y)(x − y)]²

⇒ M = (x + y)(x − y)

⇒ Applying identity: (a + b)(a − b) = a² − b²

⇒ M = x² − y²

∴ The correct answer is x² − y².

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is x such that a/x = x/b, giving x = √ab.

Third Proportional

If a, b, c are in continued proportion, then c is the third proportional and c = b²/a.

Fourth Proportional

For a, b, c, d in proportion (a:b :: c:d), d is the fourth proportional and d = (b × c)/a.

Algebraic Identity

Difference of squares: a² − b² = (a + b)(a − b).

Shortcut Trick

Given √x = 5, by squaring both sides we directly find x = 25.

The required value is √(x + 25).

Substituting x = 25: √(25 + 25) = √50.

Since 50 = 25 × 2, the value is 5√2.

∴ The correct answer is 5√2.

Alternate Method

Given:

The equation √x = 5.

Formula Used:

If √a = b, then a = b². Also, √(m × n) = √m × √n.

Calculations:

⇒ √x = 5

⇒ Squaring both sides: x = 5² = 25

⇒ Required expression = √(x + 25)

⇒ Substitute x = 25 into the expression: √(25 + 25)

⇒ Expression = √50

⇒ Expression = √(25 × 2)

⇒ 5√2

∴ The correct answer is 5√2.

Additional Information

Square Root Properties

For any non-negative numbers, the property √a × √b = √(ab) allows for the simplification of surds by factoring out perfect squares.

Solving Radical Equations

To solve for a variable under a radical, isolate the radical term and apply the inverse operation (squaring for square roots).

Perfect Squares

Memorizing perfect squares like 25, 36, 49, and 64 helps in quickly identifying factors that can be moved outside the √ symbol.

Shortcut Trick

The final price is calculated using successive multipliers.

Net Multiplier = (100 − 40)% × (100 − 22)% = 0.60 × 0.78 = 0.468

Amount paid = Marked Price × Net Multiplier = 1000 × 0.468 = 468

∴ The correct answer is ₹468.

Alternate Method

Given:

Marked Price (MP) of the oven = ₹1000

First Discount (d₁) = 40%

Additional Discount for credit card (d₂) = 22%

Formula Used:

Final Price = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ Price after the first 40% discount = 1000 − (40/100 × 1000)

⇒ Price after first discount = 1000 − 400 = ₹600

⇒ Additional 22% discount is applied on ₹600

⇒ Additional discount amount = 22/100 × 600 = ₹132

⇒ Final amount paid by Ritu = 600 − 132

⇒ Final amount paid = ₹468

∴ The correct answer is ₹468.

Additional Information

Single Equivalent Discount

For two successive discounts of x% and y%, the single equivalent discount is given by (x + y − xy/100)%.

Marked Price and Selling Price

The Selling Price (SP) is always Marked Price (MP) minus the Discount. Discount is always calculated on the Marked Price unless otherwise stated.

Successive Percentage Change

This concept is used when a value is changed multiple times consecutively. The order of discounts does not change the final selling price.

Shortcut Trick

Find the LCM of the divisors: LCM(12, 13, 7) = 1092.

Identify the largest multiple of 1092 below 9999: 1092 × 9 = 9828.

Add the common remainder to the multiple: 9828 + 3 = 9831.

∴ The correct answer is 9831.

Alternate Method

Given:

Divisors = 12, 13, 7

Remainder in each case = 3

Target = Largest 4-digit number

Calculations:

⇒ First, find the LCM of 12, 13, and 7.

⇒ 12 = 2² × 3

⇒ 13 = 13 (prime)

⇒ 7 = 7 (prime)

⇒ LCM = 12 × 13 × 7 = 1092

⇒ The largest 4-digit number is 9999.

⇒ Divide 9999 by the LCM: 9999 ÷ 1092

⇒ 9999 = (1092 × 9) + 171

⇒ The largest 4-digit multiple of 1092 = 9999 − 171 = 9828

⇒ Required number = Multiple + Remainder

⇒ Required number = 9828 + 3 = 9831

∴ The correct answer is 9831.

Additional Information

General Form for Constant Remainder

Any number that leaves a remainder 'r' when divided by x, y, and z can be expressed as: Number = [LCM(x, y, z) × k] + r, where k is an integer.

LCM of Prime Numbers

The LCM of a set of numbers that are co-prime (like 7, 13, and 12 in this case) is simply the product of those numbers.

Smallest n-digit Number

To find the smallest n-digit number satisfying a condition, divide the smallest n-digit base (like 1000 for 4-digits) by the LCM and find the next higher multiple.

Shortcut Trick

Order quantity 22 exceeds the 20 sweater limit, so the 25% discount is applicable.

Net Selling Price (SP) of one sweater = 75% of ₹600 = ₹450.

Total amount paid for 22 sweaters = 22 × 450 = ₹9,900.

∴ The correct answer is ₹9,900.

Alternate Method

Given:

Marked Price (MP) of 1 sweater = ₹600

Quantity ordered = 22 sweaters

Discount for > 20 sweaters = 25%

Formula Used:

Selling Price (SP) = MP − Discount

Total Amount = SP per unit × Quantity

Calculations:

⇒ Total MP of 22 sweaters = 22 × 600 = ₹13,200

⇒ Since 22 > 20, Discount rate = 25%

⇒ Total Discount amount = 25% of ₹13,200

⇒ Total Discount = (25 ÷ 100) × 13,200 = ₹3,300

⇒ Amount paid by retailer = Total MP − Total Discount

⇒ Amount paid = 13,200 − 3,300 = ₹9,900

∴ The correct answer is ₹9,900.

Additional Information

Marked Price (List Price)

The price mentioned on the product tag. Discount is always calculated on this value: SP = MP × (100 − D%)/100.

Successive Discount

If two discounts d₁% and d₂% are given, the effective discount is [d₁ + d₂ − (d₁ × d₂)/100]%.

Profit and Loss Relation

The relationship between MP and CP is given by: MP/CP = (100 + P%) / (100 − D%), where P is profit % and D is discount %.

Shortcut Trick

Net increase in daily riders = 4,860 − 4,500 = 360.

Percentage growth = (Change ÷ Initial Value) × 100.

Growth % = (360 ÷ 4,500) × 100 = 360 ÷ 45 = 8%.

∴ The correct answer is 8%.

Alternate Method

Given:

Average riders last month (Initial Value) = 4,500

Average riders this month (Final Value) = 4,860

Formula Used:

Percentage Increase = [(Final Value − Initial Value) ÷ Initial Value] × 100

Calculations:

⇒ Absolute increase = 4,860 − 4,500 = 360

⇒ Percentage growth = (360 ÷ 4,500) × 100

⇒ Percentage growth = (360 ÷ 45)

⇒ Percentage growth = 8%

∴ The correct answer is 8%.

Additional Information

Percentage Change Rule

Always use the initial or original value as the denominator (base) when calculating percentage growth or percentage reduction.

Multiplying Factor

An 8% increase is equivalent to multiplying the original value by (1 + 8/100) = 1.08. Thus, 4,500 × 1.08 = 4,860.

Fraction to Percentage

The ratio 360/4,500 simplifies to 36/450 = 4/50 = 2/25. Since 1/25 = 4%, then 2/25 = 8%.

Shortcut Trick

Total Surface Area (TSA) = 2 × 22/7 × 14 × (14 + 10) = 2112 cm².

Since 20% is blue, the remaining red portion is 80%.

Red Area = 80% of 2112 = 0.8 × 2112 = 1689.6 cm².

∴ The correct answer is 1689.6 cm².

Alternate Method

Given:

Radius (r) = 14 cm

Height (h) = 10 cm

Formula Used:

Total Surface Area (TSA) of a closed cylindrical reservoir = 2πr(r + h)

Calculations:

⇒ Total Surface Area = 2 × (22/7) × 14 × (14 + 10)

⇒ TSA = 2 × 22 × 2 × 24

⇒ TSA = 44 × 48 = 2112 cm²

⇒ Area coated in blue paint = 20% of 2112 = 422.4 cm²

⇒ Area coated in red paint = Total Area − Blue Area

⇒ Red Area = 2112 − 422.4 = 1689.6 cm²

∴ The correct answer is 1689.6 cm².

Additional Information

Curved Surface Area (CSA)

The lateral area of a cylinder excluding the top and bottom circles is 2πrh.

Volume of a Cylinder

The total capacity or space enclosed by a cylinder is given by πr²h.

Open Cylinder TSA

If a cylinder is open at one end, the surface area is πr² + 2πrh = πr(r + 2h).

Shortcut Trick

Effective Selling Price (SP) per 1 kg displayed = 60 × 1.20 × 0.90 = ₹64.80.

Actual Cost Price (CP) for 900 g given = (60 ÷ 1000) × 900 = ₹54.

Profit on each transaction = 64.80 − 54 = ₹10.80.

∴ The correct answer is ₹10.80.

Alternate Method

Given:

Cost Price (CP) of rice = ₹60 per kg

Mark-up percentage = 20%

Discount percentage = 10%

Faulty weight = 900 g instead of 1000 g

Step-by-step Calculation:

⇒ Cost Price (CP) of 1000 g = ₹60

⇒ Marked Price (MP) = 60 + (20% of 60) = 60 + 12 = ₹72

⇒ Selling Price (SP) = 72 − (10% of 72) = 72 − 7.20 = ₹64.80

⇒ Since the shopkeeper gives only 900 g when the buyer pays for 1 kg, we must find the cost of the actual quantity sold.

⇒ Actual CP for 900 g = (60 ÷ 1000) × 900 = ₹54

⇒ Profit = Selling Price − Actual Cost Price

⇒ Profit = 64.80 − 54 = ₹10.80

∴ The correct answer is ₹10.80.

Additional Information

Dishonest Dealer Concept

Profit occurs in two ways: by increasing the price (markup) and by reducing the weight given. Net Profit % = [ (True weight − False weight) / False weight ] × 100 (when sold at CP).

Successive Percentage Change

If a value is increased by x% and then decreased by y%, the effective change is [ x − y − (xy/100) ] %.

Relationship between CP, MP, and SP

MP = CP × (100 + Mark-up%)/100 and SP = MP × (100 − Discount%)/100.

Shortcut Trick

To find the larger value (Mary) when the sum and difference of two quantities are known:

Mary's age = (Combined Sum + Age Difference) ÷ 2

Mary's age = (27 + 3) ÷ 2 = 30 ÷ 2 = 15 years.

∴ The correct answer is 15 years.

Alternate Method

Given:

Combined age of Mary and Sister = 27 years

Difference in their ages = 3 years

Calculation:

Let the age of Mary's sister be x years.

⇒ Mary's age = x + 3

⇒ Total combined age = x + (x + 3) = 27

⇒ 2x + 3 = 27

⇒ 2x = 27 − 3

⇒ 2x = 24

⇒ x = 24 ÷ 2 = 12 years (Sister's age)

⇒ Mary's age = 12 + 3 = 15 years.

∴ The correct answer is 15 years.

Additional Information

Sum and Difference Formula

If the sum of two numbers is S and their difference is D, then the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Constant Age Difference

The age difference between two individuals remains constant throughout their lives; it does not change as they grow older.

Ratio Method for Ages

If ages are given in a ratio a:b, the difference between the ratio parts corresponds directly to the numerical age difference.

Shortcut Trick

Solve the nested radical from the innermost part: √(419 + 157) = √576 = 24.

Proceed to the next layer: √(3225 + 24) = √3249 = 57.

Solve the final outer layer: √(1624 + 57) = √1681 = 41.

Shortcut logic: The result of √(1624 + x) must be slightly more than √1624 ≈ 40.3. Among the options, 41 is the only possible integer.

∴ The correct answer is 41.

Alternate Method

Given Expression: √(1624 + √(3225 + √(419 + 157)))

Simplification Steps:

⇒ First, add the innermost numbers: 419 + 157 = 576

⇒ √(1624 + √(3225 + √576))

⇒ Since √576 = 24:

⇒ √(1624 + √(3225 + 24))

⇒ Add the values inside the next root: 3225 + 24 = 3249

⇒ √(1624 + √3249)

⇒ Since √3249 = 57 (as 50² = 2500 and 60² = 3600):

⇒ √(1624 + 57)

⇒ Add the remaining values: 1624 + 57 = 1681

⇒ √1681

⇒ 41

∴ The correct answer is 41.

Additional Information

Square Root Properties

For a number ending in 1, its square root ends in either 1 or 9. Since 41² = 1681 and 49² = 2401, 41 is the correct root for 1681.

Perfect Square Identification

A perfect square never ends with the digits 2, 3, 7, or 8. This helps in quickly eliminating incorrect results during multi-step simplification.

Nested Radicals

Always solve nested radicals from inside-out. Simplification of the innermost root provides a whole number that completes the next perfect square.

Shortcut Trick

Ratio of Officers to Clerks = 10 : 30 = 1 : 3

Assume base average = ₹25,000 (lower value)

Extra from 1 Officer = 45,000 − 25,000 = ₹20,000

Distribute this extra among total parts (1 + 3 = 4): 20,000 ÷ 4 = ₹5,000

Net Average = 25,000 + 5,000 = ₹30,000

∴ The correct answer is ₹30,000.

Alternate Method

Given:

Number of officers = 10 | Average salary of officers = ₹45,000

Number of clerks = 30 | Average salary of clerks = ₹25,000

Formula: Average = (Total Salary of all staff) ÷ (Total number of staff)

⇒ Total salary of 10 officers = 10 × 45,000 = ₹4,50,000

⇒ Total salary of 30 clerks = 30 × 25,000 = ₹7,50,000

⇒ Total salary of entire staff = 4,50,000 + 7,50,000 = ₹12,00,000

⇒ Total number of staff members = 10 + 30 = 40

⇒ Average salary of staff = 12,00,000 ÷ 40

⇒ Average salary = ₹30,000

∴ The correct answer is ₹30,000.

Additional Information

Weighted Average Formula

When two groups are combined, the combined average A is given by (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Alligation Rule

The ratio of the number of items in two groups is (A₂ − A_mix) : (A_mix − A₁), where A_mix is the average of the combined group.

Average Property

The combined average always lies between the lowest and the highest individual averages of the constituent groups.

Shortcut Trick

Ratio of E : F : G : H = 9 : 7 : 5 : 3

Difference between F and G units = 7 − 5 = 2 units

Given, 2 units = ₹4000 ⇒ 1 unit = ₹2000

E's share = 9 units = 9 × 2000 = ₹18000

∴ The correct answer is ₹18000.

Alternate Method

Given:

Ratio of distribution among E, F, G, H = 9 : 7 : 5 : 3

Difference between share of G and F = ₹4000

Formula Used:

Value of one unit = Difference in values ÷ Difference in ratio units

Individual share = Ratio unit × Value of one unit

Calculations:

⇒ Let the amounts received by E, F, G, and H be 9x, 7x, 5x, and 3x.

⇒ According to the question, F − G = 4000

⇒ 7x − 5x = 4000

⇒ 2x = 4000

⇒ x = 4000 ÷ 2 = 2000

⇒ Share of E = 9x = 9 × 2000

⇒ Share of E = ₹18000

∴ The correct answer is ₹18000.

Additional Information

Ratio Equality

A ratio remains unchanged if both terms are multiplied or divided by the same non-zero number (e.g., a:b = ak:bk).

Proportional Distribution

To divide a total amount T in the ratio a:b:c, the first part is [a / (a+b+c)] × T.

Compound Ratio

For two ratios a:b and c:d, the compounded ratio is (ac) : (bd).

Given:

Expression = 3.46 + 3 1/4 − 5 1/8 + 4.25 + 0.165

Formula used:

Conversion of mixed fractions to decimal form and BODMAS rule for basic arithmetic.

Calculations:

First, convert the mixed fractions into decimals:

⇒ 3 1/4 = 3 + 0.25 = 3.25

⇒ 5 1/8 = 5 + 0.125 = 5.125

Now, substitute these values into the given expression:

⇒ 3.46 + 3.25 − 5.125 + 4.25 + 0.165

Group the positive terms together to simplify the addition:

⇒ (3.46 + 3.25 + 4.25 + 0.165) − 5.125

⇒ 11.125 − 5.125

⇒ 6

∴ The correct answer is option (2).

Shortcut Trick

Ratio of Amount (A) to Principal (P) = 21218 : 20000

Divide both sides by 2 to get perfect squares: 10609 : 10000

Since the time is 2 years, take the square root: √10609 : √10000 = 103 : 100

Rate of Interest = (103 − 100) / 100 × 100 = 3%

∴ The correct answer is 3%.

Alternate Method

Given:

Principal (P) = ₹20,000

Amount (A) = ₹21,218

Time (n) = 2 years

Formula Used:

A = P(1 + r/100)ⁿ

Calculations:

⇒ 21218 = 20000(1 + r/100)²

⇒ 21218 ÷ 20000 = (1 + r/100)²

⇒ 10609 ÷ 10000 = (1 + r/100)²

⇒ (103/100)² = (1 + r/100)²

⇒ 103/100 = 1 + r/100

⇒ 1.03 = 1 + r/100

⇒ r/100 = 1.03 − 1 = 0.03

⇒ r = 0.03 × 100 = 3

∴ The correct answer is 3%.

Additional Information

Effective Rate of Interest

For 2 years at r% compound interest, the effective rate is given by R_eff = r + r + (r × r)/100.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is P × (r/100)².

Rule of 72

A quick way to estimate how long it takes for an investment to double is dividing 72 by the annual rate of interest (t ≈ 72/r).