Quantitative Aptitude PYQs

Given:

The numbers are: 1, 2, 4, 5, 6, 9, 11, and 18.

To find:

The average of these numbers.

Solution:

Step 1: Sum of the numbers

The sum of all the numbers is:

1 + 2 + 4 + 5 + 6 + 9 + 11 + 18 = 56

Step 2: Count the total number of numbers

There are 8 numbers in total.

Step 3: Calculate the average

The formula for the average is:

Average = (Sum of all numbers) / (Total number of numbers)

Substituting the values:

Average = 56 / 8 = 7

Answer:

The average of 1, 2, 4, 5, 6, 9, 11, and 18 is 7.

Shortcut Trick

Total discount from Dealer 1 = 15% of 5,00,000 = ₹75,000

Total discount from Dealer 2 = (12% of 3,00,000) + (6% of 2,00,000) + 6,000 (gift value)

Total discount from Dealer 2 = 36,000 + 12,000 + 6,000 = ₹54,000

Difference in Selling Prices = Difference in total discounts = 75,000 − 54,000 = ₹21,000

∴ The correct answer is ₹21,000.

Alternate Method

Given:

Marked Price (MP) = ₹5,00,000

Dealer 1 Discount = 15%

Dealer 2 Discount = 12% on first ₹3,00,000, 6% on next ₹2,00,000, and a gift of ₹6,000.

Formula:

Selling Price (SP) = MP − Discount

Marked Price: ₹5,00,000

Dealer 1 Discount (15%): ₹75,000

D2: 12% (3L)
6% (2L)
Gift

Difference = ₹21,000

Calculations:

For Dealer 1:

⇒ Discount₁ = 15/100 × 5,00,000 = ₹75,000

⇒ SP₁ = 5,00,000 − 75,000 = ₹4,25,000

For Dealer 2:

⇒ Discount on first ₹3,00,000 = 12/100 × 3,00,000 = ₹36,000

⇒ Discount on next ₹2,00,000 = 6/100 × 2,00,000 = ₹12,000

⇒ Gift value = ₹6,000

⇒ Total Discount₂ = 36,000 + 12,000 + 6,000 = ₹54,000

⇒ SP₂ = 5,00,000 − 54,000 = ₹4,46,000

Difference in Selling Prices:

⇒ Difference = SP₂ − SP₁ = 4,46,000 − 4,25,000 = ₹21,000

∴ The correct answer is ₹21,000.

Additional Information

Successive Discounts

A series of discounts a% and b% is equivalent to a single discount of [a + b − (ab/100)]%.

Effective Discount Rate

Total effective discount percentage = (Total Discount ÷ Marked Price) × 100.

Marked Price vs Selling Price

Marked Price is the listed price, while Selling Price is the actual transaction price after all discounts and value-additions are adjusted.

Given:

The product of the first four prime numbers is 6545, and the product of the last four prime numbers is 24871. We need to find the last prime number.

Let the five prime numbers be:

p₁, p₂, p₃, p₄, p₅ (in ascending order).

Step 1: The product of the first four prime numbers:

p₁ × p₂ × p₃ × p₄ = 6545

Factorize 6545:

6545 ÷ 5 = 1309

1309 ÷ 7 = 187

187 ÷ 11 = 17

Thus, the prime factors of 6545 are:

6545 = 5 × 7 × 11 × 17

So, the first four prime numbers are:

p₁ = 5, p₂ = 7, p₃ = 11, p₄ = 17

Step 2: The product of the last four prime numbers:

p₂ × p₃ × p₄ × p₅ = 24871

We already know that p₂ = 7, p₃ = 11, p₄ = 17, so substitute these values into the equation:

7 × 11 × 17 × p₅ = 24871

Calculate the product of the first three primes:

7 × 11 = 77

77 × 17 = 1309

Now, solve for p₅:

1309 × p₅ = 24871

p₅ = 24871 ÷ 1309

p₅ = 19

Therefore, the last prime number is:

p₅ = 19

Given:

The number of men = 1200, and the food lasts for 15 days.

When the number of men increases by 1800, the total number of men becomes 1200 + 1800 = 3000.

Concept:

Food is inversely proportional to the number of men when the total amount of food remains constant. If the number of men increases, the number of days the food will last decreases.

Formula used:

Initial number of men × Initial number of days = Final number of men × Final number of days

Calculations:

1200 × 15 = 3000 × x

18000 = 3000 × x

x = 18000 / 3000

x = 6

Answer: The food will last for 6 days.

Given:

A shopkeeper sells 90 pens and makes a profit equal to the selling price of 30 pens.

Let:

Cost price of one pen = C

Selling price of one pen = S

Total selling price of 90 pens = 90 × S

Total cost price of 90 pens = 90 × C

Total profit = Selling price of 30 pens = 30 × S

Formula used:

Profit Percentage = (Profit / Cost Price) × 100

Step 1: Calculate the cost price of 90 pens:

Total profit = Total selling price - Total cost price

30 × S = (90 × S) - (90 × C)

Simplify:

30 × S = 90 × (S - C)

Simplify further:

S - C = (30 / 90) × S = (1 / 3) × S

So, Cost Price per pen (C) = S - (1 / 3) × S = (2 / 3) × S

Step 2: Calculate the profit percentage:

Profit per pen = S - C = S - (2 / 3) × S = (1 / 3) × S

Profit Percentage = (Profit per pen / Cost Price per pen) × 100

Profit Percentage = [(1 / 3) × S / (2 / 3) × S] × 100

Profit Percentage = (1 / 2) × 100 = 50%

Answer: The profit percentage is 50%.

Given:

Devendra leaves home at 8:25 am and reaches the office at 9:55 am every day. Therefore, the total time taken is:

9:55 am - 8:25 am = 1 hour 30 minutes (or 90 minutes).

On a particular day, he covers:

 3/10 of the total distance at 6/7 of his normal speed, and

 7/10 of the total distance at 7/6 of his normal speed.

To find: The time Devendra reaches the office on this particular day.

Solution:

Step 1: Normal time to cover the whole distance

Let the total distance be D and the normal speed be S.

The normal time taken to cover the entire distance is:

Time = D / S = 90 minutes

Step 2: Time for the first part of the journey (3/10 of the distance)

Distance for the first part = (3/10) × D

Speed for the first part = (6/7) × S

Time for the first part:

Time = (3/10 × D) / (6/7 × S)

Simplifying: Time = (3/10 × D) × (7/6 × 1/S) = (7/20) × (D / S)

Since we know D / S = 90 minutes from the normal time:

Time = (7/20) × 90 = 31.5 minutes

Step 3: Time for the second part of the journey (7/10 of the distance)

Distance for the second part = (7/10) × D

Speed for the second part = (7/6) × S

Time for the second part:

Time = (7/10 × D) / (7/6 × S)

Time = (7/10 × D) × (6/7 × 1/S) = (6/10) × (D / S)

Since we know D / S = 90 minutes from the normal time:

Time = (6/10) × 90 = 54 minutes

Step 4: Total time taken for the journey

Total time = Time for the first part + Time for the second part

Total time = 31.5 minutes + 54 minutes = 85.5 minutes

Step 5: Time Devendra reaches the office

Devendra leaves home at 8:25 am, and the time taken is 85.5 minutes (1 hour 25 minutes and 30 seconds).

Therefore, Devendra reaches the office at:

8:25 am + 1 hour 25 minutes 30 seconds = 9:50:30 am.

Answer: Devendra reaches the office at 9:50:30 am.

Given:

The LCM of two co-prime numbers is 667, and one of the numbers is 23. We need to find the other number.

Step 1: Understand the relationship between LCM and co-prime numbers:

When two numbers are co-prime, their LCM is the product of the two numbers. So, we can use the formula:

LCM(a, b) = a × b

Where a and b are the two co-prime numbers.

Step 2: Apply the given information:

Let the other number be x. Since the LCM of the two co-prime numbers is 667, we have:

23 × x = 667

Step 3: Solve for x:

x = 667 / 23

x = 29

Therefore, the other number is 29.

Given:

The employee's salary is first increased by 45% and then decreased by 45%.

To find:

The effective change in the employee's salary.

Solution:

Step 1: Let the initial salary be S

Initial salary = S

Step 2: Increase by 45%

The salary after a 45% increase is:

New salary = S + 45% of S = S + 0.45S = 1.45S

Step 3: Decrease by 45%

Now, the salary is decreased by 45%. The new salary after a 45% decrease on the increased salary (1.45S) is:

New salary = 1.45S - 45% of 1.45S = 1.45S × (1 - 0.45) = 1.45S × 0.55 = 0.7975S

Step 4: Calculate the effective change

The effective change is the difference between the new salary (0.7975S) and the initial salary (S):

Effective change = 0.7975S - S = -0.2025S

Step 5: Convert the change to percentage

The effective change as a percentage of the initial salary is:

Effective change percentage = (-0.2025S / S) × 100 = -20.25%

Answer:

The effective change in the employee's salary is a decrease of 20.25%.

Given:

The edges of a triangular board are 12 m, 35 m, and 37 m. The cost of painting is ₹ 8 per square meter. We need to find the total cost of painting the triangular board.

Step 1: Use Heron's formula to find the area of the triangle:

The sides of the triangle are 12 m, 35 m, and 37 m.

Let the semi-perimeter (s) be:

s = (12 + 35 + 37) / 2 = 84 / 2 = 42 m

Now, apply Heron's formula to find the area (A):

A = √[s(s - a)(s - b)(s - c)]

Where, a = 12 m, b = 35 m, and c = 37 m.

A = √[42(42 - 12)(42 - 35)(42 - 37)]

A = √[42 × 30 × 7 × 5]

A = √[44100]

A = 210 m²

Step 2: Calculate the cost of painting:

Cost of painting = Area × Rate

Cost of painting = 210 m² × ₹ 8/m²

Cost of painting = ₹ 1680

Therefore, the cost of painting the triangular board is ₹ 1680.

Given:

A certain sum doubles in 5 years at simple interest. We need to find the time it will take for the same sum to become six times under the same interest rate.

Step 1: Use the formula for simple interest:

The formula for simple interest is:

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

Where: P = Principal, R = Rate of interest, T = Time in years.

Let's assume the principal is P and the rate of interest is R. According to the problem, the sum doubles in 5 years, which means the total amount becomes 2P after 5 years.

The total amount A is given by:

A = P + SI

So, after 5 years, we have:

2P = P + (P × R × 5) / 100

Step 2: Solve for the rate of interest:

2P - P = (P × R × 5) / 100

P = (P × R × 5) / 100

100 = R × 5

R = 20%

Step 3: Calculate the time to become six times:

Now, we need to find the time when the sum becomes 6P. Using the formula for the total amount:

A = P + (P × R × T) / 100

6P = P + (P × 20 × T) / 100

6P - P = (P × 20 × T) / 100

5P = (P × 20 × T) / 100

5 × 100 = 20 × T

500 = 20 × T

T = 500 / 20

T = 25 years

Therefore, the same sum will become six times in 25 years.

Given:

The value of (0.2× 0.2× 0.2+0.5× 0.5× 0.5)/(0.02× 0.02× 0.02+0.05× 0.05× 0.05)

Formula used:

Basic arithmetic operations

Calculations:

(0.2× 0.2× 0.2 + 0.5× 0.5× 0.5)/(0.02× 0.02× 0.02 + 0.05× 0.05× 0.05)

⇒ (0.008 + 0.125)/(0.000008 + 0.000125)

⇒ 0.133/0.000133

⇒ 1000

∴ The correct answer is option (1).

Given:

Numbers: 65, 75, 90

Formula used:

To find the greatest common divisor (GCD), use the prime factorization method or the Euclidean algorithm.

Calculation:

Prime factorization:

65 = 5 × 13

75 = 3 × 5 × 5

90 = 2 × 3 × 3 × 5

Common factors: 5

⇒ GCD = 5

∴ The correct answer is option 1.

Given:

Marked Price (MP) = ₹3,675

Discount = 12%

Loss = 2%

Formula used:

Selling Price (SP) = MP - (Discount% × MP)

Cost Price (CP) = \(\dfrac{SP}{(1 - \frac{Loss%}{100})}\)

Calculation:

SP = 3675 - (12% × 3675)

⇒ SP = 3675 - (0.12 × 3675)

⇒ SP = 3675 - 441

⇒ SP = 3234

CP = \(\dfrac{3234}{(1 - \frac{2}{100})}\)

⇒ CP = \(\dfrac{3234}{0.98}\)

⇒ CP = 3300

∴ The correct answer is option 2.

Shortcut Trick

For a conical tent, the ratio between Volume and Base Area is given by (1/3) × Height.

Direct formula: Vertical Height (h) = (3 × Volume) ÷ Base Area.

Since the number of persons is the same for both area and volume, we can use the individual requirements: h = 3 × (300 ÷ 45).

h = 3 × (20/3) = 20 m.

∴ The correct answer is 20 m.

Alternate Method

Given:

Number of persons = 3

Ground space required per person = 45 m²

Air volume required per person = 300 m³

Value of π = 22/7

Calculations:

⇒ Total Base Area (A) = 3 × 45 = 135 m²

⇒ Total Volume (V) = 3 × 300 = 900 m³

⇒ Formula for Volume of a Cone: V = (1/3) × A × h

⇒ 900 = (1/3) × 135 × h

⇒ 900 = 45 × h

⇒ h = 900 ÷ 45

⇒ h = 20 m

∴ The correct answer is 20 m.

Additional Information

Volume of a Cone

The volume of a cone is exactly one-third the volume of a cylinder with the same base radius and height: V = (1/3)πr²h.

Slant Height and Surface Area

Slant height (l) = √(r² + h²). The Curved Surface Area (CSA) is πrl and the Total Surface Area (TSA) is πr(l + r).

Relationship with Cylinder

If a cone and a cylinder have the same height and radius, the ratio of their volumes is 1 : 3.

Given:

Basket A has 9 apples and 12 bananas.

Basket B has 10 apples and 15 bananas.

Basket C has 8 apples and 16 bananas.

Formula used:

Ratio = Number of Apples / Number of Bananas

Calculations:

Ratio of apples to bananas in Basket A = 9 / 12

⇒ Ratio in Basket A = 3 / 4

Ratio of apples to bananas in Basket B = 10 / 15

⇒ Ratio in Basket B = 2 / 3

Ratio of apples to bananas in Basket C = 8 / 16

⇒ Ratio in Basket C = 1 / 2

Comparing the ratios:

Basket A = 3 / 4

Basket B = 2 / 3

Basket C = 1 / 2

⇒ 3 / 4 (Basket A) > 2 / 3 (Basket B)

∴ The correct answer is option (4).

Given:

Principal (P) = ₹3,125

Amount (A) = ₹3,515.20

Rate of interest (r) = 8% per annum

Compounded half-yearly

Formula Used:

A = P(1 + r/200)^2t

Where:

A = Amount, P = Principal, r = Rate of interest per annum, t = Time in years

Calculations:

Substitute the values in the formula:

3515.20 = 3125(1 + 8/200)^2t

3515.20 / 3125 = (1 + 8/200)^2t

⇒ 1.125 = (1 + 0.04)^2t

⇒ 1.125 = (1.04)^2t

Take logarithm on both sides:

log(1.125) = 2t × log(1.04)

⇒ log(1.125) / log(1.04) = 2t

Using values log(1.125) ≈ 0.0504 and log(1.04) ≈ 0.0170:

⇒ 0.0504 / 0.0170 = 2t

⇒ 2t ≈ 2.9647

⇒ t ≈ 2.9647 / 2

⇒ t ≈ 1.4823

Answer:

The time is  \(1\frac{1}{2}\) years or 1.5 years.

Given:

3 men and 4 women can do a work in 7 days

2 men and 1 woman can do it in 14 days

Formula used:

Let 1 man's 1 day work = M

Let 1 woman's 1 day work = W

Then,

3M + 4W = 1/7

2M + 1W = 1/14

Calculations:

3M + 4W = 1/7

⇒ 42M + 56W = 1

2M + 1W = 1/14

⇒ 28M + 14W = 1

Now, multiply the second equation by 2:

⇒ 56M + 28W = 2

Subtract the first equation from this:

⇒ (56M + 28W) - (42M + 56W) = 2 - 1

⇒ 14M - 28W = 1

⇒ 14M = 1 + 28W

⇒ M = (1 + 28W)/14

Substitute M in 2M + 1W = 1/14:

⇒ 2((1 + 28W)/14) + W = 1/14

⇒ (2 + 56W)/14 + W = 1/14

⇒ (2 + 70W)/14 = 1/14

⇒ 2 + 70W = 1

⇒ 70W = -1

⇒ W = -1/70

Now, substitute W in M:

⇒ M = (1 + 28(-1/70))/14

⇒ M = (1 - 28/70)/14

⇒ M = (1 - 2/5)/14

⇒ M = 3/5 * 1/14

⇒ M = 3/70

Now, 7 women will complete the work in:

7W = 7 * (-1/70)

⇒ 7W = -1/10

⇒ Time taken = 10 days

∴ The correct answer is option (2).

Given:

Neelam's initial speed = 5 km/h

Time difference = 10 min late

Increased speed = 6 km/h

Time difference = 15 min early

Formula used:

Distance = Speed × Time

Calculation:

Let the distance be D km

Time taken at 5 km/h = D/5 hours

Time taken at 6 km/h = D/6 hours

According to the problem:

D/5 = Normal time + 10/60 (late by 10 min)

D/6 = Normal time - 15/60 (early by 15 min)

Equating the normal times:

D/5 - 10/60 = D/6 + 15/60

⇒ D/5 - D/6 = 25/60

⇒ (6D - 5D)/30 = 25/60

⇒ D/30 = 25/60

⇒ D = (25/60) × 30

⇒ D = 12.5 km

∴ The correct answer is option (2).

Given:

The cost price (CP) and selling price (SP) of an article are in the ratio of 13:17.

Formula used:

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

Calculation:

Let CP = 13x and SP = 17x

⇒ Percentage Profit = \(\dfrac{17x - 13x}{13x} \times 100\)

⇒ Percentage Profit = \(\dfrac{4x}{13x} \times 100\)

⇒ Percentage Profit = \(\dfrac{400}{13}\) %

⇒ Percentage Profit = 30¹⁰/₁₃ %

∴ The correct answer is option (4).

Given:

Total number of voters = 85800

Percentage of votes polled = 80%

Percentage of polled votes cast in favor of Soham = 65%

Formula used:

Total votes polled = Total voters × Percentage of votes polled

Votes received by Soham = Polled votes × Percentage of votes cast in favor of Soham

Votes received by Ram = Polled votes - Votes received by Soham

Calculation:

Total votes polled = 85800 × 80%

⇒ Total votes polled = 85800 × 0.80

⇒ Total votes polled = 68640

Votes received by Soham = 68640 × 65%

⇒ Votes received by Soham = 68640 × 0.65

⇒ Votes received by Soham = 44616

Votes received by Ram = 68640 - 44616

⇒ Votes received by Ram = 24024

∴ The correct answer is option (4).