Quantitative Aptitude PYQs

Given:

Curved surface area (CSA) of a cone = 1518 cm²

Radius (r) = 21 cm

Formula used:

CSA = πrl

Where, l = slant height

Calculation:

1518 = \(\frac{22}{7}\) × 21 × l

⇒ 1518 = 22 × 3 × l

⇒ 1518 = 66l

⇒ l = \(\frac{1518}{66}\)

⇒ l = 23 cm

∴ The correct answer is option (3).

Given:

Total Amount = ₹2,500

Interest Rate for first part = 5% p.a.

Interest Rate for second part = 6% p.a.

Total Annual Interest = ₹140

Formula used:

Simple Interest = Principal x Rate x Time

Calculations:

Let the amount lent at 5% p.a. be ₹x.

Then, the amount lent at 6% p.a. = ₹2500 - x.

Interest from first part = 0.05 x x

Interest from second part = 0.06 x (2500 - x)

Total Interest = 0.05x + 0.06(2500 - x) = 140

⇒ 0.05x + 150 - 0.06x = 140

⇒ -0.01x + 150 = 140

⇒ -0.01x = -10

⇒ x = 1000

Answer:

₹1,000 is lent at 5% p.a.

Given:

Principal (P) = ₹5,000

Amount after 1 year (A₁) = ₹5,450

Time (t) = 3 years

Formula used:

A = P(1 + r/100)^t

Where, r = rate of interest

Calculation:

First, find the rate of interest (r):

5450 = 5000(1 + r/100)

⇒ 5450/5000 = 1 + r/100

⇒ 1.09 = 1 + r/100

⇒ r/100 = 0.09

⇒ r = 9%

Now, calculate the amount after 3 years:

A = 5000(1 + 9/100)³

⇒ A = 5000(1.09)³

⇒ A = 5000(1.295029)

⇒ A = 6475.145

∴ The amount due at the end of 3 years is ₹6,475 (to the nearest rupee).

∴ The correct answer is option (1).

Shortcut Trick

Total shirts in transaction = 2 (Paid) + 1 (Free) = 3 shirts.

Total Cost Price (CP) = 3 × ₹300 = ₹900.

Total Selling Price (SP) needed for 30% profit = 900 × 130% = ₹1170.

Since the customer pays for only 2 shirts, the SP of one shirt = ₹1170 ÷ 2 = ₹585.

∴ The correct answer is ₹585.

Alternate Method

Given:

CP of 1 shirt = ₹300

Scheme = Buy 2, Get 1 Free

Desired Profit = 30%

Shirt 1
CP: ₹300

Shirt 2
CP: ₹300

Free Shirt
CP: ₹300

Total Shirts Handed Over = 3
Total CP = ₹900

Calculations:

⇒ Total Cost Price for 3 shirts = 3 × 300 = ₹900

⇒ Profit desired = 30% of ₹900 = (30 ÷ 100) × 900 = ₹270

⇒ Total Selling Price required = Total CP + Profit = 900 + 270 = ₹1170

⇒ Number of shirts paid for by the customer = 2

⇒ Selling price of one shirt = Total SP ÷ Number of shirts paid = 1170 ÷ 2

⇒ Selling price of one shirt = ₹585

∴ The correct answer is ₹585.

Additional Information

Discount Percentage in Free Schemes

Discount % = (Number of free items ÷ Total number of items) × 100. For 'Buy 2 Get 1', Discount = (1 ÷ 3) × 100 = 33.33%.

Effective Profit Relation

Total SP = Total CP × (1 + Profit/100). The total SP is then distributed only over the items actually sold (paid for).

Marked Price and Discount

If a discount is offered on Marked Price (MP), then SP = MP × (1 − Discount/100). In 'Buy X Get Y' schemes, MP is often the price of one item.

Given:

The average of the marks of Ram, Ramesh, Somesh, and Soham in mathematics is 75.

The average of the marks of Ramesh, Somesh, Soham, and Mohan in mathematics is 80.

The marks scored by Mohan alone in mathematics is 70.

Formula used:

Average = (Sum of marks) / (Number of students)

Calculation:

Let the marks of Ram be R, Ramesh be Ra, Somesh be S, and Soham be So.

From the first average:

(R + Ra + S + So) / 4 = 75

⇒ R + Ra + S + So = 75 × 4

⇒ R + Ra + S + So = 300

From the second average:

(Ra + S + So + Mohan) / 4 = 80

⇒ Ra + S + So + 70 = 80 × 4

⇒ Ra + S + So + 70 = 320

⇒ Ra + S + So = 320 - 70

⇒ Ra + S + So = 250

Now, subtract the sum of Ra, S, and So from the sum of R, Ra, S, and So:

R + Ra + S + So - (Ra + S + So) = 300 - 250

⇒ R = 50

∴ The correct answer is option (1).

Given:

Initial Side Length (s) = 12 फुट

Cut Side Length = 4 फुट

Formula used:

Area of a square = side²

Percentage Decrease = \(\left(\frac{\text{Initial Area} - \text{New Area}}{\text{Initial Area}}\right) × 100\)

Calculation:

Initial Area = 12²

⇒ Initial Area = 144 फुट²

New Side Length = 12 - 4 = 8 फुट

New Area = 8 × 12 = 96

⇒ New Area = 96 फुट²

Percentage Decrease = \(\left(\frac{144 - 96}{144}\right) × 100\)

⇒ Percentage Decrease = \(\left(\frac{48}{144}\right) × 100\)

⇒ Percentage Decrease = 33.33%

∴ The correct answer is option (4).

Given:

The number of prime numbers which are greater than 100 and less than 150 is:

Formula used:

List all prime numbers between 100 and 150.

Calculation:

Prime numbers between 100 and 150 are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149

⇒ Total prime numbers = 10

∴ The correct answer is option (3).

Given:

Loss Percentage (L) = 13%

Selling Price (SP) = ₹4,524

New Selling Price (SP') = ₹6,400

Formula used:

Cost Price (CP) = \(\dfrac{SP \times 100}{100 - L}\)

Profit Percentage (P) = \(\dfrac{SP' - CP}{CP} \times 100\)

Calculations:

CP = \(\dfrac{4524 \times 100}{100 - 13}\)

⇒ CP = \(\dfrac{4524 \times 100}{87}\)

⇒ CP = 5200

P = \(\dfrac{6400 - 5200}{5200} \times 100\)

⇒ P = \(\dfrac{1200}{5200} \times 100\)

⇒ P = \(0.2308 \times 100\)

⇒ P = 23.08%

∴ The correct answer is option (4).

Given:

If the diameter of a circle is increased by 24%, then the percentage increase in the area is:

Formula used:

Area of a circle = πr²

New diameter = original diameter × (1 + 24/100)

New radius = new diameter / 2

Percentage increase in area = ((New Area - Original Area) / Original Area) × 100%

Calculation:

Let the original diameter be D, then the original radius r = D/2

New diameter = D × (1 + 24/100) = D × 1.24

New radius = (D × 1.24) / 2 = 0.62D

Original Area = πr² = π(D/2)² = πD²/4

New Area = π(0.62D)² = π(0.3844D²) = 0.3844πD²

Percentage increase in area = ((New Area - Original Area) / Original Area) × 100%

⇒ Percentage increase in area = ((0.3844πD² - πD²/4) / (πD²/4)) × 100%

⇒ Percentage increase in area = ((0.3844 - 0.25) / 0.25) × 100%

⇒ Percentage increase in area = (0.1344 / 0.25) × 100%

⇒ Percentage increase in area = 53.76%

∴ The correct answer is option (2).

Given:

Diameter (d) = 28 cm

Height (h) = 20 cm

π = 22/7

Formula used:

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

Where, r = radius

Calculation:

Radius (r) = Diameter/2 = 28/2 = 14 cm

TSA = 2 × (22/7) × 14 × (20 + 14)

⇒ TSA = 2 × (22/7) × 14 × 34

⇒ TSA = 2 × (22/7) × 476

⇒ TSA = 2 × 22 × 68

⇒ TSA = 2992 cm²

∴ The correct answer is option (1).

Given:

Length of the circular track = 2000 m = 2 km

Speed of X = 15 km/h

Speed of Y = 33 km/h

Formula used:

Time to meet = Distance / Relative speed

Calculation:

Relative speed = 15 + 33 = 48 km/h (since they are moving in opposite directions)

Time to meet = 2 / 48 = 1/24 hours

Convert to minutes: (1/24) × 60 = 2.5 minutes × 60 = 150 seconds

X and Y will meet next after 150 sec.

Given:

The sum of three prime numbers is 72.

One of them exceeds another by 24.

Formula used:

P₁ + P₂ + P₃ = 72

P₃ = P₂ + 24

Calculations:

Since the sum is even and the sum of three odd numbers is odd, one prime must be 2.

2 + P₂ + (P₂ + 24) = 72

⇒ 2 + 2P₂ + 24 = 72

⇒ 2P₂ + 26 = 72

⇒ 2P₂ = 46

⇒ P₂ = 23

P₃ = 23 + 24 = 47

Answer:

One of the prime numbers is 47.

Shortcut Trick

Use the identity a² − b² = (a − b)(a + b) for the numerator and factor out 0.3 from the denominator.

Numerator = (11.12 − 8.12) × (11.12 + 8.12) = 3 × 19.24

Denominator = 0.3 × (0.3 + 18.94) = 0.3 × 19.24

Final Value = (3 × 19.24) ÷ (0.3 × 19.24) = 3 ÷ 0.3 = 10

∴ The correct answer is 10.

Alternate Method

Given:

Expression = ((11.12)² − (8.12)²) ÷ ((0.3)² + (0.3)(18.94))

Formula Used:

1. Difference of squares: a² − b² = (a − b)(a + b)
2. Common factoring: xy + xz = x(y + z)

Calculations:

⇒ Numerator = (11.12 − 8.12)(11.12 + 8.12)

⇒ Numerator = 3 × 19.24

⇒ Denominator = 0.3(0.3 + 18.94)

⇒ Denominator = 0.3 × 19.24

⇒ Total Expression = (3 × 19.24) ÷ (0.3 × 19.24)

⇒ Expression = 3 ÷ 0.3

⇒ Expression = 30 ÷ 3 = 10

∴ The correct answer is 10.

Additional Information

Algebraic Identities

Using identities like a² − b² simplifies calculations involving large decimals significantly compared to direct squaring.

Factorization Method

Taking out a common factor (like 0.3 in the denominator) helps in canceling terms and reducing the complexity of division.

BODMAS Rule

Always solve terms inside the brackets first before performing the final division between the numerator and denominator.

Given:

 Dileep can complete the work in 20 days.

 Rajesh can complete the work in 30 days.

 They work on alternate days, starting with Rajesh.

Formula used:

Work done = Work rate × Time

Total work = 1

Calculations:

Determine the work rates.

Rajesh's work rate = 1/30 per day.

Dileep's work rate = 1/20 per day.

 Calculate work done in a 2-day cycle.

Day 1: Rajesh works, completes 1/30 of the work.

Day 2: Dileep works, completes 1/20 of the work.

Total work in 2 days = 1/30 + 1/20 = (2 + 3)/60 = 5/60 = 1/12.

 Determine the number of cycles needed to complete the work.

Total work = 1.

Work per cycle = 1/12.

Number of cycles = 1 ÷ (1/12) = 12.

Calculate the total number of days.

Each cycle = 2 days.

Total days = 12 × 2 = 24 days.

Answer:

The entire work will be completed in 24 days.

Given:

\(\frac{1}{a}\) : \(\frac{1}{b}\) : \(\frac{1}{c}\) = 2 : 3 : 5

Formula used:

Invert the ratios to find a : b : c

Calculations:

a : b : c = \(\frac{1}{2}\) : \(\frac{1}{3}\) : \(\frac{1}{5}\)

⇒ Multiply each part by 30 to eliminate fractions

⇒ a : b : c = 15 : 10 : 6

∴ The ratio a : b : c is 15 : 10 : 6.

Given:

The ratio of the age of a father to that of his son is 7:4.

The product of their ages (in years) is 1008.

Formula used:

Let the age of the father be 7x and the age of the son be 4x.

Product of their ages = (7x) × (4x) = 1008

Calculations:

28x² = 1008

⇒ x² = 1008 ÷ 28

⇒ x² = 36

⇒ x = 6

Father's age = 7x = 7 × 6 = 42

Son's age = 4x = 4 × 6 = 24

After 10 years:

Father's age = 42 + 10 = 52

Son's age = 24 + 10 = 34

The ratio of the age of the father to that of the son after 10 years = 52:34 = 26:17

∴ The correct answer is option (1).

Given:

Find the smallest fraction among:

1721172117211721

, 911911911911

, 1619161916191619

, 1315131513151315

Formula used:

Compare the fractions by converting them to decimal form.

Calculations:

1721172117211721

≈ 0.8095

911911911911

≈ 0.8182

1619161916191619

≈ 0.8421

1315131513151315

≈ 0.8667

⇒ The smallest fraction is 1721172117211721

∴ The correct answer is option 1.

Given:

Simple Interest for 8 years = 40% of Principal (P)

Formula used:

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

Calculation:

Let R be the rate of interest.

SI for 8 years = 40% of P

⇒ P × R × 8 / 100 = 40% of P

⇒ P × R × 8 / 100 = 0.4 × P

⇒ R × 8 = 40

⇒ R = 5%

We need to find the time (T) when SI equals the Principal (P).

⇒ P × 5 × T / 100 = P

⇒ 5T / 100 = 1

⇒ T = 100 / 5

⇒ T = 20 years

∴ The correct answer is option 1.

Given:

The average of four numbers is 20, and one of these four numbers is 26.

Formula used:

The average of the remaining three numbers = (Total sum of four numbers - Given number) / 3

Calculation:

Total sum of four numbers = Average × 4

⇒ Total sum of four numbers = 20 × 4

⇒ Total sum of four numbers = 80

Total sum of remaining three numbers = Total sum of four numbers - Given number

⇒ Total sum of remaining three numbers = 80 - 26

⇒ Total sum of remaining three numbers = 54

Average of remaining three numbers = Total sum of remaining three numbers / 3

⇒ Average of remaining three numbers = 54 / 3

⇒ Average of remaining three numbers = 18

∴ The correct answer is option 1.

Given:

A temple top is a hollow hemispherical dome with an inner radius of 2 units and an outer radius of 3 units.

Formula used:

Volume of a hollow hemisphere = \(\dfrac{2}{3} \pi (R^3 - r^3)\)

Where, R = outer radius and r = inner radius

Calculation:

R = 3 units, r = 2 units

Volume of concrete used = \(\dfrac{2}{3} \pi (3^3 - 2^3)\)

⇒ Volume of concrete used = \(\dfrac{2}{3} \pi (27 - 8)\)

⇒ Volume of concrete used = \(\dfrac{2}{3} \pi \times 19\)

⇒ Volume of concrete used = \(\dfrac{38\pi}{3}\)

∴ The correct answer is option 2.