Quantitative Aptitude PYQs

Shortcut Trick

Relative speed (opposite direction) = 99 + 81 = 180 km/h

Relative speed in m/s = 180 × 5/18 = 50 m/s

Total distance (sum of lengths) = Speed × Time = 50 × 9 = 450 m

Length of second train = Total distance − First train length

Length of second train = 450 − 220 = 230 m

∴ The correct answer is 230.

Alternate Method

Given:

Length of first train (L₁) = 220 m

Speed of first train (S₁) = 99 km/h

Speed of second train (S₂) = 81 km/h

Time taken to cross each other (T) = 9 seconds

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Total Distance = Length of Train 1 + Length of Train 2

Distance = Speed × Time

220 m
Train 1

99 km/h

L₂
Train 2

81 km/h

Opposite Direction (Relative Speed = S₁ + S₂)

Calculations:

⇒ Relative speed = 99 + 81 = 180 km/h

⇒ Speed in m/s = 180 × (5/18) = 50 m/s

⇒ Total distance = (L₁ + L₂)

⇒ L₁ + L₂ = Relative Speed × Time

⇒ 220 + L₂ = 50 × 9

⇒ 220 + L₂ = 450

⇒ L₂ = 450 − 220

⇒ L₂ = 230 m

∴ The correct answer is 230.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, relative speed is the difference of their speeds: |S₁ − S₂|.

Unit Conversion Shortcut

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

Train Crossing an Object with Length

When a train crosses another train or a platform, the total distance covered is the sum of their lengths: L_Total = L_Train + L_Object.

Train Crossing a Point Object

When a train crosses a pole or a stationary man, the distance covered is equal to the length of the train itself.

Given:

First month investment: Rs. 65,000

Rebate: 8%

Tax rate: 10%

Second month investment: Rs. 70,000

Rebate: 10%

Tax rate: 12%

Formula Used:

Net investment = (Initial investment - Rebate) + Tax

Percentage change = ((Second month investment - First month investment))/(First month investment) × 100

Calculation:

First month:

Rebate = 65,000 × 8% = 65,000 × 0.08 = 5,200

Net price after rebate = 65,000 - 5,200 = 59,800

Tax = 59,800 × 10% = 59,800 × 0.10 = 5,980

Net investment = 59,800 + 5,980 = 65,780

Second month:

Rebate = 70,000 × 10% = 70,000 × 0.10 = 7,000

Net price after rebate = 70,000 - 7,000 = 63,000

Tax = 63,000 × 12% = 63,000 × 0.12 = 7,560

Net investment = 63,000 + 7,560 = 70,560

Percentage change:

Change in investment = 70,560 - 65,780 = 4,780

Percentage change = (4,780)/(65,780) × 100

Percentage change ≈ 7.27%

∴ The percentage change in his investment from the first month to the second month is an increase between 7 and 8%.

Given:

Time (t) = 7 years

Final Amount (A) = 2 × Principal (P)

Formula used:

Simple Interest formula:

A = P + SI

SI = \(\dfrac{P × R × T}{100}\)

Calculation:

2P = P + \(\dfrac{P × R × 7}{100} \)

⇒ 2P - P = \(\dfrac{P × R × 7}{100}\)

⇒ P = \(\dfrac{P × R × 7}{100}\)

⇒ \(R = \dfrac{100}{7} \)

⇒ R = 14\(\frac{2}{7}\)%

∴ The required rate is 14\(\frac{2}{7}\)% per annum.

Given:

Ratio of numbers = 4 ∶ 5 ∶ 6

LCM = 180

Formula used:

Let the numbers be 4x, 5x, and 6x.

LCM of 4x, 5x, and 6x = LCM(4, 5, 6) × x

HCF = x

Calculation:

LCM(4,5,6) = 60

⇒ 60 × x = 180

⇒ x = 180 ÷ 60

⇒ x = 3

∴ HCF = 3

Given:

Numbers: 204, 507, 174

Formula Used:

To find the HCF of given numbers

Calculation:

Step 1: Find the prime factorization of each number.

204 = 2² × 3 × 17

507 = 3 × 13 × 13

174 = 2 × 3 × 29

Step 2: Identify the common factors.

The common factor is 3.

∴ HCF = 3

The HCF of 204, 507, and 174 is 3.

Shortcut Trick

When multiple spheres are melted to form a single sphere, the Conservation of Volume applies.

Sum of cubes of individual radii = Cube of the resulting radius (R³ = r₁³ + r₂³ + r₃³).

R³ = 6³ + 8³ + 10³

R³ = 216 + 512 + 1000 = 1728

R = 12 cm (Since 12 × 12 × 12 = 1728).

∴ The correct answer is 12 cm.

Alternate Method

Given:

Radius of 1^st sphere (r₁) = 6 cm

Radius of 2^nd sphere (r₂) = 8 cm

Radius of 3^rd sphere (r₃) = 10 cm

Formula Used:

Volume of a sphere = (4/3) × π × r³

Calculations:

⇒ Total Volume of 3 spheres = Volume of single large sphere

⇒ (4/3)πr₁³ + (4/3)πr₂³ + (4/3)πr₃³ = (4/3)πR³

⇒ (4/3)π × (6³ + 8³ + 10³) = (4/3)πR³

⇒ 6³ + 8³ + 10³ = R³

⇒ 216 + 512 + 1000 = R³

⇒ 1728 = R³

⇒ R = ³√1728

⇒ R = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Volume and Surface Area of a Sphere

The volume is given by V = (4/3)πr³ and the total surface area is S = 4πr².

Hemisphere Formulas

For a hemisphere, Volume = (2/3)πr³, Curved Surface Area = 2πr², and Total Surface Area = 3πr².

Melting and Recasting Rule

When one or more solids are melted to form another solid, the total volume remains constant, though the surface area usually changes.

Given:

Akash can finish a work in 18 days

Navin can finish the same work in 36 days

Formula used:

Work done by Akash in 1 day = 1/18

Work done by Navin in 1 day = 1/36

Total work done by Akash and Navin in 1 day = 1/18 + 1/36

Portion of work done by Navin = (Work done by Navin in 1 day) / (Total work done by Akash and Navin in 1 day)

Calculation:

Work done by Akash in 1 day = 1/18

Work done by Navin in 1 day = 1/36

Total work done by Akash and Navin in 1 day = 1/18 + 1/36

⇒ Total work done by Akash and Navin in 1 day = (2 + 1)/36

⇒ Total work done by Akash and Navin in 1 day = 3/36

⇒ Total work done by Akash and Navin in 1 day = 1/12

Portion of work done by Navin = (1/36) / (1/12)

⇒ Portion of work done by Navin = (1/36) × (12/1)

⇒ Portion of work done by Navin = 12/36

⇒ Portion of work done by Navin = 1/3

∴ The correct answer is option (1).

Shortcut Trick

Given:

A student misread the number 97 as 79 while doing a calculation.

Formula used:

Percentage Change = \(\dfrac{(New Value - Original Value)}{Original Value} \times 100\)

Calculation:

Original Value = 97

New Value = 79

Percentage Change = \(\dfrac{(79 - 97)}{97} \times 100\)

⇒ Percentage Change = \(\dfrac{-18}{97} \times 100\)

⇒ Percentage Change =   - 18.55%

∴ The correct answer is option (3).

Given:

Selling Price (SP) = ₹4,800

Loss = 20%

Profit = 12%

Formula used:

Cost Price (CP) = \(\dfrac{SP}{1 - \text{Loss %}}\)

Selling Price for profit = \(\text{CP} \times (1 + \text{Profit %})\)

Calculation:

CP = \(\dfrac{4800}{1 - \dfrac{20}{100}}\)

⇒ CP = \(\dfrac{4800}{0.8}\)

⇒ CP = ₹6,000

Selling Price for profit = \(\text{CP} \times (1 + \dfrac{12}{100})\)

⇒ Selling Price = \(6000 \times 1.12\)

⇒ Selling Price = ₹6,720

∴ The correct answer is option (3).

Shortcut Trick

Successive Percentage Increase for 3 years at 3% per year.

Effective rate for 3 years = 3 + 3 + 3 + (3×3 + 3×3 + 3×3)/100 + (3×3×3)/10000

Effective rate = 9 + 0.27 + 0.0027 = 9.2727%

Population after 3 years = 75556 × (1 + 9.2727/100) ≈ 75556 × 1.092727

Population ≈ 82562.11 → 82562 (neglecting decimal)

∴ The correct answer is 82562.

Alternate Method

Given:

Current Population (P) = 75,556

Rate of Increase (R) = 3% per annum

Time (n) = 3 years

Formula Used:

A = P × (1 + R/100)ⁿ

Calculations:

⇒ Population after 3 years = 75556 × (1 + 3/100)³

⇒ Population = 75556 × (1.03)³

⇒ Population = 75556 × (1.03 × 1.03 × 1.03)

⇒ Population = 75556 × 1.092727

⇒ Population = 82562.114212

⇒ Neglecting the decimal part, Population = 82562

∴ The correct answer is 82562.

Additional Information

Population Growth Formula

The formula for population growth is identical to the Compound Interest formula: A = P(1 + R/100)ⁿ.

Successive Percentage Change

For two successive changes of a% and b%, the net change is (a + b + ab/100)%.

Population Decrease

If the population decreases, the formula becomes A = P(1 − R/100)ⁿ, similar to Depreciation.

Shortcut Trick

Aprajita's initial salary = 150% of 6,00,000 = 9,00,000.

New salary after 12% hike = 112% of 9,00,000.

Calculation: 1.12 × 9,00,000 = 10,08,000.

∴ The correct answer is ₹10,08,000.

Alternate Method

Given:

• Sonia's initial salary = ₹6,00,000
• Aprajita's salary = 150% of Sonia's salary
• Hike in Aprajita's salary = 12%

Formula Used:

New Value = Initial Value × (1 + Increase%/100)

Calculations:

⇒ Aprajita's initial annual salary = 150% of 6,00,000

⇒ Initial salary = (150/100) × 6,00,000 = 1.5 × 6,00,000 = ₹9,00,000

⇒ Increment amount = 12% of 9,00,000

⇒ Increment = (12/100) × 9,00,000 = ₹1,08,000

⇒ Aprajita's salary now = 9,00,000 + 1,08,000

⇒ Aprajita's salary now = ₹10,08,000

∴ The correct answer is ₹10,08,000.

Additional Information

Percentage Multiplier

An increase of x% can be directly calculated by multiplying the base value with (1 + x/100). For example, a 12% increase means multiplying by 1.12.

Base Change Rule

Always identify the correct base for percentage calculations. In this problem, 150% was calculated on Sonia's salary, while 12% was calculated on Aprajita's own initial salary.

Successive Percentage Application

When one percentage is applied after another (like a salary being a % of another and then increasing), the net result is the product of their respective multipliers: Final = Base × m₁ × m₂.

Shortcut Trick

Let the two numbers be 33x and 33y, where x and y are coprime ratios.

The sum of their ratios is 594 ÷ 33 = 18.

We need pairs of coprime numbers (x, y) that add up to 18.

The possible coprime pairs summing to 18 are (1, 17), (5, 13), and (7, 11).

Thus, there are exactly 3 such valid pairs.

∴ The correct answer is 3 pairs.

Alternate Method

Given:

HCF of the two numbers = 33

Sum of the two numbers = 594

Formula Used:

If the HCF is H, the numbers can be written as Hx and Hy, where x and y are coprime.

Finding Coprime Pairs for Sum = 18

Pair 1
(1, 17)

Pair 2
(5, 13)

Pair 3
(7, 11)
Total 3 Valid Coprime Pairs

Calculations:

⇒ Let the two numbers be 33x and 33y.

⇒ According to the question, 33x + 33y = 594

⇒ 33(x + y) = 594

⇒ x + y = 594 ÷ 33

⇒ x + y = 18

⇒ Now, we need to find all possible pairs of (x, y) that sum to 18 such that their HCF is 1.

⇒ Checking combinations:

⇒ (1, 17) → Coprime (Valid)

⇒ (2, 16) → HCF is 2 (Invalid)

⇒ (3, 15) → HCF is 3 (Invalid)

⇒ (4, 14) → HCF is 2 (Invalid)

⇒ (5, 13) → Coprime (Valid)

⇒ (6, 12) → HCF is 6 (Invalid)

⇒ (7, 11) → Coprime (Valid)

⇒ (8, 10) → HCF is 2 (Invalid)

⇒ (9, 9) → HCF is 9 (Invalid)

⇒ The valid coprime pairs are (1, 17), (5, 13), and (7, 11).

∴ The correct answer is 3 pairs.

Additional Information

Highest Common Factor (HCF)

The largest positive integer that divides two or more numbers without leaving a remainder. Any two numbers can be written as H × a and H × b, where H is the HCF and a, b are coprime.

Coprime Numbers

Two integers are coprime (or mutually prime) if their only common divisor is 1. They do not necessarily have to be prime numbers themselves.

LCM and HCF Property

For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF × LCM.

Shortcut Trick

Identify the largest 3-digit number which is 999 and check downward for primality.

999: Divisible by 3 (Sum of digits 9+9+9 = 27).

998: Divisible by 2 (Even number).

997: Not divisible by any prime numbers up to √997 ≈ 31.

Testing 997 with 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 shows no factors.

∴ The correct answer is 997.

Alternate Method

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

To find the largest 3-digit prime number, we evaluate numbers starting from 999 downwards:

⇒ Number 999: Sum of digits = 9 + 9 + 9 = 27. Since 27 is divisible by 3, 999 is not prime.

⇒ Number 998: It is an even number ending in 8, so it is divisible by 2. Not prime.

⇒ Number 997: We test for primality by checking divisibility by primes less than √997 ≈ 31.57.

⇒ Primes to check: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.

⇒ 997 ÷ 7 = 142.42 | 997 ÷ 11 = 90.63 | 997 ÷ 13 = 76.69 | 997 ÷ 17 = 58.64 | 997 ÷ 19 = 52.47.

⇒ Since 997 is not divisible by any of these, 997 is the largest 3-digit prime number.

∴ The correct answer is 997.

Additional Information

Smallest 3-Digit Prime

The smallest 3-digit prime number is 101.

Prime Numbers Count

There are exactly 168 prime numbers between 1 and 1000.

Twin Primes

Pairs of prime numbers that have a difference of 2 are called twin primes (e.g., 3 and 5, 11 and 13).

Even Prime

The number 2 is the only even prime number; all other prime numbers are odd.

Given:

Marked price (MP) = ₹9,000

Discount = 25%

Gain with discount = 8%

Formula used:

Selling price (SP) = MP × (1 - Discount%)

Cost price (CP) = SP / (1 + Gain%)

New selling price without discount = MP

New gain% = (New SP - CP) / CP × 100

Calculations:

SP = 9000 × (1 - 0.25)

⇒ SP = 9000 × 0.75 = 6750

CP = 6750 / (1 + 0.08)

⇒ CP = 6750 / 1.08

⇒ CP = 6250

New SP without discount = 9000

New gain% = (9000 - 6250) / 6250 × 100

⇒ New gain% = 2750 / 6250 × 100

⇒ New gain% = 0.44 × 100

⇒ New gain% = 44%

∴ The correct answer is option (1).

Shortcut Trick

Volume of a cone depends on r²h.

Successive increase of 25% (ratio 4 → 5) for three dimensions (r, r, h).

Initial Volume : Final Volume = (4 × 4 × 4) : (5 × 5 × 5) = 64 : 125.

Percentage Increase = (125 − 64) ÷ 64 × 100 = 61 ÷ 64 × 100 = 95.3125%.

∴ The correct answer is 95.31%.

Alternate Method

Given: Increase in radius (r) = 25%, Increase in height (h) = 25%.

Formula Used: Volume of Cone (V) = 1/3 × π × r² × h

⇒ Let initial radius = r and initial height = h.

⇒ Initial Volume (V₁) = 1/3 × π × r² × h

⇒ New radius (r') = r + 0.25r = 1.25r

⇒ New height (h') = h + 0.25h = 1.25h

⇒ New Volume (V₂) = 1/3 × π × (1.25r)² × (1.25h)

⇒ V₂ = 1/3 × π × 1.5625r² × 1.25h = 1.953125 × (1/3 × π × r² × h)

⇒ V₂ = 1.953125 × V₁

⇒ % Increase = (V₂ − V₁) ÷ V₁ × 100 = (1.953125 − 1) × 100 = 95.3125%

∴ The correct answer is 95.31%.

Additional Information

Successive Percentage Formula

For two successive changes a% and b%, net change = (a + b + ab/100)%. For three changes, apply the formula twice.

Volume Proportionality

For a cone, V ∝ r²h. If only radius changes by x%, volume changes by (x + x + x²/100)%. If both change, it is a triple successive change.

Curved Surface Area (CSA)

CSA of a cone = πrl, where l = √(r² + h²). If r and h increase by k%, l also increases by k%, and CSA increases by (k + k + k²/100)%.

Given:

P ∶ (Q + R) = 2 ∶ 5

R ∶ (P + Q) = 6 ∶ 7

Formula used:

Let P = 2k and (Q + R) = 5k

Let R = 6m and (P + Q) = 7m

Calculation:

P = 2k

⇒ Q + R = 5k

⇒ Q = 5k - R

R = 6m

⇒ P + Q = 7m

⇒ P = 7m - Q

Given P = 2k, so 2k = 7m - Q

Also, Q = 5k - 6m

⇒ 2k = 7m - (5k - 6m)

⇒ 2k = 7m - 5k + 6m

⇒ 2k + 5k = 13m

⇒ 7k = 13m

⇒ m = (7/13)k

Now, Q = 5k - 6m

⇒ Q = 5k - 6(7/13)k

⇒ Q = 5k - (42/13)k

⇒ Q = (65/13)k - (42/13)k

⇒ Q = (23/13)k

P + R = 2k + 6m

⇒ P + R = 2k + 6(7/13)k

⇒ P + R = 2k + (42/13)k

⇒ P + R = (26/13)k + (42/13)k

⇒ P + R = (68/13)k

⇒ Q ∶ (P + R) = (23/13)k ∶ (68/13)k

⇒ Q ∶ (P + R) = 23 ∶ 68

∴ The correct answer is option (3).

Given:

\(\dfrac{79(7)}{40}\) is equivalent to:

Formula used:

Simple Division

Calculations:

\(79\frac{7}{40}\)

**⇒ (**79 × 40) + 7 / 40 

⇒ (3160 + 7) /40 

⇒ 3167 / 40 

⇒  79.175

Given:

Principal (P) = ₹85,000

Rate (R) = 7.5% per annum

Time (T) = 4 years

Formula used:

Simple Interest (SI) = \(\dfrac{P \times R \times T}{100}\)

Amount (A) = Principal (P) + Simple Interest (SI)

Calculations:

SI = \(\dfrac{85,000 \times 7.5 \times 4}{100}\)

⇒ SI = \(\dfrac{85,000 \times 30}{100}\)

⇒ SI = ₹25,500

Amount (A) = ₹85,000 + ₹25,500

⇒ A = ₹1,10,500

∴ The correct answer is option 1.

Given:

Speed of boat in still water (b) = 30 km/h

Downstream distance = 60 km

Upstream distance = 60 km

Total time = 4 hours 30 minutes = 9/2 hours

Formula used:

Downstream speed: d = b + w

Upstream speed: u = b - w

Where w = speed of stream

Calculations:

According to the question:

60/(30 + w) + 60/(30 - w) = 9/2

⇒ 3600 = (9/2) × (900 - w²)

⇒ 900 - w² = 800

⇒ w² = 100

⇒ w = 10

∴ Speed of the stream = 10 km/h.

Given:

Average marks in 5 subjects = 62

Average marks in the first four subjects are 5 less than the marks in the fifth subject

Formula used:

Sum of marks in 5 subjects = Average marks × Number of subjects

Calculations:

Sum of marks in 5 subjects = 62 × 5

⇒ Sum of marks in 5 subjects = 310

Let the marks in the fifth subject be x.

Then, average marks in the first four subjects = x - 5

Sum of marks in the first four subjects = (x - 5) × 4

⇒ Sum of marks in the first four subjects = 4x - 20

Sum of marks in the first four subjects + Marks in the fifth subject = 310

⇒ 4x - 20 + x = 310

⇒ 5x - 20 = 310

⇒ 5x = 310 + 20

⇒ 5x = 330

⇒ x = 330 / 5

⇒ x = 66

∴ The correct answer is 66.