Quantitative Aptitude PYQs

Given:

Final Selling Price (SP) = ₹680

Successive Discounts = 15% and 20%

Formula used:

SP = MP × (1 - Discount1) × (1 - Discount2)

Where, MP = Marked Price

Calculation:

680 = MP × (1 - 0.15) × (1 - 0.20)

⇒ 680 = MP × 0.85 × 0.80

⇒ 680 = MP × 0.68

⇒ MP = 680 / 0.68

⇒ MP = 1000

∴ The correct answer is option 2.

Given:

Provisions for 60 pupils for 18 days.

Strength increased to 72 pupils.

Formula used:

Total Provisions = Number of Pupils × Number of Days

Calculations:

Total Provisions = 60 × 18 = 1080 pupil-days

Let the number of days the provisions will last for 72 pupils be d.

Total Provisions = 72 × d

⇒ 72 × d = 1080

⇒ d = 1080 ÷ 72

⇒ d = 15

Answer:

The provisions will finish off in 15 days if the strength is increased to 72 pupils.

Given:

Compound interest (CI) = ₹4,058

Rate (R) = 8% p.a.

Time (t) = 3 years

Simple Interest (SI) rate = 10% p.a.

Simple Interest (SI) time = 4 years

Formula used:

CI = P\((1+\frac{R}{100})^t - P\)

SI = \(\frac{P \times R \times T}{100}\)

Calculations:

4058 = P \((1+\frac{8}{100})^3 - P\)

⇒ \(\frac{4058}{P} = (1+\frac{8}{100})^3 - 1\)

⇒ \(\frac{4058}{P} = (1.08)^3 - 1\)

⇒ \(\frac{4058}{P} = 1.259712 - 1\)

⇒ \(\frac{4058}{P} = 0.259712\)

⇒ P = \(\frac{4058}{0.259712}\)

⇒ P = ₹15,627

Now, calculating Simple Interest:

SI = \(\frac{P \times R \times T}{100}\)

⇒ SI = \(\frac{15627 \times 10 \times 4}{100}\)

⇒ SI = ₹6,250

∴ The correct answer is option 3.

Given:

The selling price of an item is 15% more than the cost price and shopkeeper is able to earn ₹25.

Formula used:

Selling Price (SP) = Cost Price (CP) + Profit

Profit = CP × Profit%

Calculation:

Let CP be x.

Profit = x × 15% = 0.15x

Given Profit = ₹25

⇒ 0.15x = 25

⇒ x = 25 / 0.15

⇒ x = ₹166.67

Selling Price (SP) = CP + Profit

⇒ SP = 166.67 + 25

⇒ SP = ₹191.67

∴ The correct answer is option 3.

Given:

Fractions are 1.75 ÷ 9, 5.6 ÷ 12, and 7 ÷ 18.

Formula used:

HCF of fractions = HCF of numerators ÷ LCM of denominators

Calculations:

Convert decimal numerators to fractions:

1.75 = 7 ÷ 4

5.6 = 28 ÷ 5

7 = 7 ÷ 1

 Express the given fractions with integer numerators:

1.75 ÷ 9 = (7 ÷ 4) ÷ 9 = 7 ÷ 36

5.6 ÷ 12 = (28 ÷ 5) ÷ 12 = 28 ÷ 60 = 7 ÷ 15

7 ÷ 18 = 7 ÷ 18

 Determine HCF of numerators:

HCF(7, 7, 7) = 7

 Determine LCM of denominators:

Denominators are 36, 15, and 18.

Factorization:

36 = 2² × 3²

15 = 3 × 5

18 = 2 × 3²

LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

Calculate HCF of fractions:

HCF = 7 ÷ 180

Answer:

The HCF of 1.75 ÷ 9, 5.6 ÷ 12, and 7 ÷ 18 is 7 ÷ 180.

Shortcut Trick

When a solid is recast, its volume remains constant.

2/3 × π × (r_h)³ = 1/3 × π × (r_c)² × h_c

2 × 8³ = (r_c)² × 64

1024 = 64 × (r_c)²

(r_c)² = 16 ⇒ r_c = 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

Given:

Radius of the lead hemisphere (r) = 8 cm

Height of the right circular cone (H) = 64 cm

Value of pi (π) = 22/7

Formula Used:

Volume of a Hemisphere = 2/3 × π × r³

Volume of a Cone = 1/3 × π × R² × H

Calculations:

⇒ Volume of Hemisphere = Volume of Cone

⇒ 2/3 × π × (8)³ = 1/3 × π × R² × 64

⇒ 2 × 512 = R² × 64

⇒ 1024 = 64 × R²

⇒ R² = 1024 ÷ 64

⇒ R² = 16

⇒ R = √16

⇒ R = 4 cm

∴ The correct answer is 4 cm.

Additional Information

Volume of a Sphere

The volume of a complete sphere is 4/3 × π × r³, which is exactly double that of a hemisphere.

Surface Area of a Hemisphere

Curved Surface Area = 2πr² and Total Surface Area (including base) = 3πr².

Properties of Recasting

During the conversion of one solid shape into another, the total volume remains constant, though the surface area usually changes.

Slant Height of a Cone

The slant height (l) is given by l = √(R² + H²).

Given:

(2 × 2 × 3 × 3 × 5 × 7 × 7 × 11), (2 × 3 × 3 × 5 × 11), (2 × 3 × 3 × 5 × 7 × 11 × 11), (2 × 3 × 3 × 5 × 5 × 11), (2 × 3 × 5 × 7 × 7)

Formula used:

HCF = Product of lowest powers of common prime factors

Calculations:

Common primes: 2, 3, 5

Lowest powers: 2¹, 3¹, 5¹

HCF = 2 × 3 × 5 ⇒ 30

Answer:

The HCF is 30.

Given:

Total distance = 600 km

Time for 120 km by train and remaining by car = 8 hours

Time for 200 km by train and remaining by car = 8 hours 20 minutes

Formula used:

Speed = Distance / Time

Calculations:

Let speed of train = T km/hr

Let speed of car = C km/hr

⇒ 120/T + 480/C = 8 ...(i)

⇒ 200/T + 400/C = 8 + 1/3

⇒ 200/T + 400/C = 25/3 ...(ii)

Multiplying (i) by 3:

⇒ 3(120/T + 480/C) = 3*8

⇒ 360/T + 1440/C = 24 ...(iii)

Subtracting (ii) from (iii):

⇒ (360/T + 1440/C) - (200/T + 400/C) = 24 - 25/3

⇒ 160/T + 1040/C = 72/3 - 25/3

⇒ 160/T + 1040/C = 47/3

⇒ 160/T + 1040/C = 47/3

Dividing by 8:

⇒ 20/T + 130/C = 47/24

Using (i):

⇒ 20/T = 47/24 - 130/C

From (i):

120/T = 8 - 480/C

⇒ 8 - 480/C = 47/24 - 130/C

⇒ 8*24/24 - 480/C = 47/24 - 130/C

⇒ 192/24 - 480/C = 47/24 - 130/C

⇒ 192 - 47/24 = 480/C - 130/C

⇒ 145/24 = 350/C

⇒ C = 350 × 24/145

⇒ C = 56 km/hr

Using (i):

⇒ 120/T + 480/56 = 8

⇒ 120/T + 8.571 = 8

⇒ 120/T = -0.571

⇒ T = 120/8

⇒ T = 15 km/hr

Therefore, the ratio of the speed of the train to that of the car is:

⇒ T/C = 15/20

⇒ 3/4

∴ The correct answer is option (3).

Given:

Kiran sells an object to Varun at a profit of 10%. Varun sells that object to Arun for ₹1,155 and makes a profit of 5%.

Formula used:

Selling Price (SP) = Cost Price (CP) × (1 + Profit Percentage / 100)

Calculations:

Varun's Selling Price (SP_Varun) = ₹1,155

Varun's Profit Percentage = 5%

⇒ Varun's Cost Price (CP_Varun) = SP_Varun / (1 + Profit Percentage / 100)

⇒ CP_Varun = 1155 / (1 + 5 / 100)

⇒ CP_Varun = 1155 / 1.05

⇒ CP_Varun = ₹1,100

Kiran's Selling Price (SP_Kiran) = CP_Varun = ₹1,100

Kiran's Profit Percentage = 10%

⇒ Kiran's Cost Price (CP_Kiran) = SP_Kiran / (1 + Profit Percentage / 100)

⇒ CP_Kiran = 1100 / (1 + 10 / 100)

⇒ CP_Kiran = 1100 / 1.10

⇒ CP_Kiran = ₹1,000

∴ The correct answer is option 1.

Given:

The price of a commodity increased by 22%.

Formula used:

Let the original consumption be 100 units.

New price = 100 + 22 = 122

Total expenditure should remain the same.

Calculations:

Original expenditure = 100 × 100

New expenditure = 122 × (new consumption)

⇒ 10000 = 122 × (new consumption)

⇒ new consumption = 10000/122

⇒ new consumption ≈ 81.97 units

Reduction in consumption = 100 - 81.97

⇒ Reduction in consumption ≈ 18.03 units

Percentage reduction = (18.03/100) × 100

⇒ Percentage reduction ≈ 18.03%

∴ The correct answer is option (4).

Given:

Marked Price (MP) = ₹4,000

Discount = 10%

Profit = 20%

Formula used:

Selling Price (SP) = MP - Discount

SP = Cost Price (CP) + Profit

Calculation:

Discount = 10% of ₹4,000

⇒ Discount = 0.10 × 4,000 = ₹400

Selling Price (SP) = MP - Discount

⇒ SP = 4,000 - 400 = ₹3,600

Profit = 20% of CP

⇒ SP = CP + Profit

⇒ 3,600 = CP + 0.20 × CP

⇒ 3,600 = 1.20 × CP

⇒ CP = 3,600 / 1.20

⇒ CP = ₹3,000

∴ The correct answer is option (4).

Given:

Diameter of base (d) = 8 cm

Height (h) = 35 cm

Formula used:

Volume of cylinder (V) = πr²h

Where, r = radius of base

Calculation:

Radius (r) = d/2 = 8/2 = 4 cm

Volume (V) = πr²h

⇒ V = (22/7) × 4² × 35

⇒ V = (22/7) × 16 × 35

⇒ V = (22 × 16 × 35) / 7

⇒ V = (12320) / 7

⇒ V = 1760 cm³

∴ The correct answer is option 1.

Given:

The average weight of the first 14 persons among 15 persons is 80 kg.

The weight of the 15^th person is 42 kg more than the average weight of all 15 persons.

Formula used:

Average weight of 15 persons = Total weight of 15 persons / 15

Total weight of 14 persons = Average weight of 14 persons × 14

Calculation:

Total weight of 14 persons = 80 × 14

⇒ Total weight of 14 persons = 1120 kg

Let the average weight of all 15 persons be x kg.

Weight of the 15^th person = x + 42 kg

Total weight of 15 persons = 1120 + (x + 42)

⇒ 15x = 1120 + x + 42

⇒ 15x - x = 1120 + 42

⇒ 14x = 1162

⇒ x = 83 kg

Weight of the 15^th person = 83 + 42

⇒ Weight of the 15^th person = 125 kg

∴ The correct answer is option (3).

Given:

Altitude = Base - 7 cm

Hypotenuse = 17 cm

Formula used:

In a right-angled triangle, (Hypotenuse)² = (Base)² + (Altitude)²

Calculations:

Let the base be x cm. Then, the altitude is (x - 7) cm.

Using the Pythagorean theorem:

(17)² = x² + (x - 7)²

⇒ 289 = x² + (x² - 14x + 49)

⇒ 289 = 2x² - 14x + 49

⇒ 2x² - 14x + 49 = 289

⇒ 2x² - 14x - 240 = 0

Solving the quadratic equation:

x² - 7x - 120 = 0

⇒ (x - 15)(x + 8) = 0

⇒ x = 15 or x = -8

Since base length cannot be negative, x = 15 cm.

Altitude = 15 - 7 = 8 cm

Area of the triangle = (1/2) × Base × Altitude

⇒ Area = (1/2) × 15 × 8

⇒ Area = 60 square cm

∴ The correct answer is option (4).

Given:

Amount (A) = ₹1,46,410

Rate (r) = 10% per annum

Time (t) = 4 years

Formula used:

A = P(1 + \(\frac{r}{100}\))^t

Where, P = Principal

Calculation:

1,46,410 = P(1 + \(\frac{10}{100}\))⁴

⇒ 1,46,410 = P(1.1)⁴

⇒ 1,46,410 = P × 1.4641

⇒ P = \(\frac{1,46,410}{1.4641}\)

⇒ P = ₹1,00,000

∴ The correct answer is option 1.

Shortcut Trick

Total Simple Interest (SI) = 12,400 − 10,000 = Rs. 2,400

SI for 1 year = 2,400 ÷ 4 = Rs. 600

Rate of Interest (R) = (SI for 1 year ÷ Principal) × 100

⇒ R = (600 ÷ 10000) × 100 = 6%

∴ The correct answer is 6% p.a.

Alternate Method

Given:

Principal (P) = Rs. 10,000

Amount (A) = Rs. 12,400

Time (T) = 4 years

0 Years
P = ₹10,000

4 Years
A = ₹12,400

• SI = ₹2,400

SI per year = ₹2400 ÷ 4 = ₹600

Formula Used:

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

SI = (P × R × T) ÷ 100

Calculations:

⇒ SI = 12400 − 10000 = Rs. 2400

⇒ 2400 = (10000 × R × 4) ÷ 100

⇒ 2400 = 100 × R × 4

⇒ 2400 = 400 × R

⇒ R = 2400 ÷ 400

⇒ R = 6

∴ The correct answer is 6% p.a.

Additional Information

Difference between SI and CI

Simple Interest is calculated only on the principal amount, while Compound Interest is calculated on the principal as well as the accumulated interest.

SI vs CI for 2 Years

The difference between Compound Interest and Simple Interest on the same principal for 2 years is given by: Difference = P(R ÷ 100)²

Rule of 72

A quick way to estimate the number of years required to double your money at a given annual fixed interest rate (R) is approximately 72 ÷ R.

Given:

The largest 4-digit prime number is:

Formula used:

Prime Number: A number that is greater than 1 and has no positive divisors other than 1 and itself.

Calculation:

Checking the options:

9991: Not a prime number (divisible by 97)

9967: Prime number

9983: Not a prime number (divisible by 67)

9973: Prime number

Since, 9973 > 9967

Hence, largest 4-digit prime number is 9973.

∴ The correct answer is option (4).

Given:

A man takes 40 minutes to cover a certain distance at a speed of 9 km/h.

Formula used:

Time = Distance / Speed

Calculation:

First, find the distance covered:

Distance = Speed × Time

Distance = 9 km/h × (40/60) h

⇒ Distance = 9 × (2/3)

⇒ Distance = 6 km

Now, calculate the time taken at a speed of 12 km/h:

Time = Distance / Speed

Time = 6 km / 12 km/h

⇒ Time = 0.5 hours

Convert hours into minutes:

Time = 0.5 × 60 minutes

⇒ Time = 30 minutes

∴ The correct answer is option 4.

Given:

Ram and Naresh together can complete the work in 20 days.

Naresh and Shyam together can complete the work in 15 days.

All three (Ram, Naresh, and Shyam) together can complete the work in 12 days.

Formula used:

The rate of work is the reciprocal of the time taken to complete the work.

If Ram, Naresh, and Shyam together complete the work in 12 days, then their combined rate of work is:

Rate of work = 1 / Time taken = 1 / 12

Let:

Ram's rate of work = R_r

Naresh's rate of work = R_n

Shyam's rate of work = R_s

From the given data, we can form the following equations based on the rates of work:

1. R_r + R_n = 1 / 20 (Ram and Naresh together complete the work in 20 days)
2. R_n + R_s = 1 / 15 (Naresh and Shyam together complete the work in 15 days)
3. R_r + R_n + R_s = 1 / 12 (All three together complete the work in 12 days)

Calculation:

From equation (1): R_r + R_n = 1 / 20

From equation (2): R_n + R_s = 1 / 15

From equation (3): R_r + R_n + R_s = 1 / 12

Now, subtract equation (1) from equation (3):

(R_r + R_n + R_s) - (R_r + R_n) = 1 / 12 - 1 / 20

⇒ R_s = (1 / 12 - 1 / 20)

Find a common denominator for the fractions:

⇒ R_s = (5 - 3) / 60

⇒ R_s = 2 / 60 = 1 / 30

So, Shyam's rate of work is 1 / 30.

Now, substitute R_s = 1 / 30 into equation (2):

R_n + 1 / 30 = 1 / 15

⇒ R_n = 1 / 15 - 1 / 30

⇒ R_n = (2 - 1) / 30

⇒ R_n = 1 / 30

Now, substitute R_n = 1 / 30 into equation (1):

R_r + 1 / 30 = 1 / 20

⇒ R_r = 1 / 20 - 1 / 30

Find a common denominator for the fractions:

⇒ R_r = (3 - 2) / 60

⇒ R_r = 1 / 60

Finally, calculate the rate of work of Ram and Shyam together:

R_r + R_s = 1 / 60 + 1 / 30

⇒ R_r + R_s = (1 + 2) / 60

⇒ R_r + R_s = 3 / 60 = 1 / 20

Therefore, Ram and Shyam together can complete the work in 20 days.

∴ The time taken by Ram and Shyam together to complete the work is 20 days.

Given:

3y = A

6A = 3B

Formula used:

We need to find the value of (y)/(B).

Calculation:

3y = A

⇒ y = A/3

6A = 3B

⇒ 2A = B

Therefore, (y)/(B) = (A/3)/(2A)

⇒ (y)/(B) = 1/(6)

∴ The correct answer is option (4).