Quantitative Aptitude PYQs

Given:

Radius of the hemisphere (r) = 9 cm

Height of the cone (h) = 72 cm

Formula Used:

Volume of the hemisphere = \(\frac{2}{3} \pi r^3\)

Volume of the cone = \(\frac{1}{3} \pi R^2 h\)

Calculation:

Volume of the hemisphere = \(\frac{2}{3} \pi (9)^3\)

Volume of the hemisphere = \(\frac{2}{3} \pi × 729\)

Volume of the hemisphere = \(\frac{1458}{3} \pi\)

Volume of the hemisphere = 486π

Volume of the cone = Volume of the hemisphere

\(\frac{1}{3} \pi R^2 × 72 = 486 \pi\)

⇒ R^2 × 24 = 486

⇒ R^2 = \(\frac{486}{24}\)

⇒ R^2 = 20.25

⇒ R = \(\sqrt{20.25}\)

⇒ R = 4.5 cm

The radius of the base of the cone is 4.5 cm.

Given:

Principal (P) = Rs. 8,000

Amount (A) = Rs. 9,200

Time (T) = 3 years

Rate of interest (r) = ?

Formula Used:

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

Amount (A) = P + SI

Calculation:

Simple Interest (SI) = A - P

⇒ SI = 9,200 - 8,000

⇒ SI = 1,200

Using SI formula:

⇒ 1,200 = (8,000 × r × 3) / 100

⇒ 1,200 = 24,000r / 100

⇒ 1,200 = 240r

⇒ r = 1,200 / 240

⇒ r = 5%

New rate of interest = r + 2% = 5% + 2% = 7%

New Simple Interest (SI) = (P × new rate × T) / 100

⇒ New SI = (8,000 × 7 × 3) / 100

⇒ New SI = 168,000 / 100

⇒ New SI = 1,680

New Amount = P + New SI

⇒ New Amount = 8,000 + 1,680

⇒ New Amount = 9,680

The amount Bhumika would have received is Rs. 9,680.

Given:

The monthly incomes of A and B are in the ratio of 3 ∶ 4.

Their monthly expenditures are in the ratio 2 ∶ 3.

Each saves Rs. 6,000 per month.

Formula Used:

Income - Expenditure = Savings

Calculation:

Let the monthly income of A be 3x and that of B be 4x.

Let the monthly expenditure of A be 2y and that of B be 3y.

Savings of A: 3x - 2y = 6000

Savings of B: 4x - 3y = 6000

We have the following system of equations:

1. 3x - 2y = 6000

2. 4x - 3y = 6000

To solve these equations, multiply the first equation by 3 and the second equation by 2:

3. 9x - 6y = 18000

4. 8x - 6y = 12000

Subtract the second equation from the first equation:

⇒ 9x - 6y - (8x - 6y) = 18000 - 12000

⇒ x = 6000

Substitute the value of x in the first equation:

⇒ 3(6000) - 2y = 6000

⇒ 18000 - 2y = 6000

⇒ 2y = 12000

⇒ y = 6000

Now, the monthly income of B is:

⇒ 4x = 4 × 6000 = 24000

The monthly income of B is Rs. 24,000.

Given:

Total payment for the work = Rs. 36,000.

Radha can complete the work in 20 days.

Shama can complete the work in 60 days.

With the help of Rama, they finished the work in 10 days.

Formula Used:

Work done by a person in 1 day = 1 / Number of days the person can complete the work.

Share of payment = (Work done by the person in 1 day × Total work days) × Total payment.

Calculation:

Work done by Radha in 1 day = 1 / 20

Work done by Shama in 1 day = 1 / 60

Let the work done by Rama in 1 day = 1 / x

Total work done in 1 day by all three = (1 / 20) + (1 / 60) + (1 / x)

Since they finished the work in 10 days, the total work done in 1 day by all three = 1 / 10

⇒ (1 / 20) + (1 / 60) + (1 / x) = 1 / 10

⇒ (\(\frac{3}{60}\) + \(\frac{1}{60}\) + \(\frac{1}{x}\)) = 1 / 10

⇒ \(\frac{4}{60}\) + \(\frac{1}{x}\) = 1 / 10

⇒ \(\frac{4 + 1/x}{60}\) = 1 / 10

⇒ 4 + 1 / x = 6

⇒ 1 / x = 2

⇒ x = 30

So, Rama can complete the work in 30 days.

Work done by Rama in 10 days = 10 / 30 = 1 / 3

Rama's share of the payment = (1 / 3) × Rs. 36,000

Rama's share of the payment = Rs. 12,000

The correct answer is option 3.

Shortcut Trick

Fraction value of 44.44% = 4/9.

If consumption increases by a/b, price must decrease by a/(a+b) to keep expenditure constant.

Here, a = 4, b = 9, so decrease in price = 4 / (4 + 9) = 4/13.

Percentage decrease (P) = (4/13) × 100% = 400/13% = 30\(10\over13\) %.

∴ The correct answer is 30\(10\over13\) %.

Alternate Method

Given:

Increase in consumption = 44.44%

Expenditure remains constant.

Formula Used:

Expenditure = Price × Consumption

Calculations:

⇒ 44.44% = 4 × 11.11% = 4/9

⇒ Let the initial consumption be 9 units.

⇒ Final consumption = 9 + 4 = 13 units.

⇒ Since Expenditure is constant, Price is inversely proportional to Consumption.

⇒ Ratio of initial to final consumption = 9 : 13

⇒ Ratio of initial to final price = 13 : 9

⇒ Reduction in price = 13 − 9 = 4 ratio units.

⇒ Percentage reduction (P) = (4 / 13) × 100% = 400/13 %

⇒ P = 30\(10\over13\)₃ %

∴ The correct answer is 30\(10\over13\) %.

Additional Information

Constant Product Rule

If A × B = Constant, an increase in A by x/y leads to a decrease in B by x/(x+y).

Important Percentage Fractions

11.11% = 1/9, 22.22% = 2/9, 33.33% = 1/3, 44.44% = 4/9.

Price-Consumption Relation

If the price of a commodity increases by R%, reduction in consumption to keep expenditure constant is [R / (100+R)] × 100%.

Given:

Raju sold a watch to Sanjeev at a 20% gain.

Sanjeev sold it to Mukesh at a loss of 10%.

Mukesh bought it for ₹4,860.

Formula Used:

Let the purchase price of the watch for Raju be ₹P.

Raju's selling price to Sanjeev = P + 20% of P = P + 0.2P = 1.2P

Sanjeev's selling price to Mukesh = 1.2P - 10% of 1.2P = 1.2P - 0.12 × 1.2P = 1.2P × 0.9 = 1.08P

Calculation:

Mukesh bought the watch for ₹4,860.

So, 1.08P = 4,860

⇒ P = 4,860 / 1.08

⇒ P = 4,500

The price at which Raju purchased the watch is ₹4,500.

Given:

The ratio of Aman's age to Shivam's age is 4 : 3.

The product of their ages is 588.

Formula Used:

Let Aman's age be 4x and Shivam's age be 3x.

4x × 3x = 588

Ratio of their ages after 5 years = (Aman's age + 5) : (Shivam's age + 5)

Calculation:

4x × 3x = 588

12x² = 588

⇒ x² = 588 / 12

⇒ x² = 49

⇒ x = 7

Aman's age = 4x = 4 × 7 = 28

Shivam's age = 3x = 3 × 7 = 21

Aman's age after 5 years = 28 + 5 = 33

Shivam's age after 5 years = 21 + 5 = 26

Ratio of their ages after 5 years = 33 : 26

The correct answer is option 4.

Given:

Population of a village 4 years ago = 3,20,000

Rate of decrease per year = 5%

Formula Used:

Population after n years = P × (1 - r/100)ⁿ

Decrease in population = Initial Population - Population after n years

Calculation:

Initial Population, P = 3,20,000

Rate of decrease, r = 5%

Number of years, n = 4

Population after 4 years = 3,20,000 × (1 - 5/100)⁴

⇒ Population after 4 years = 3,20,000 × (0.95)⁴

⇒ Population after 4 years = 3,20,000 × 0.8145

⇒ Population after 4 years ≈ 2,60,642

Decrease in population = 3,20,000 - 2,60,642

⇒ Decrease in population = 59,358

The decrease in the population in the last 4 years is 59,358.

Given:

Distance between thief and policeman = 300 m.

Speed of thief = 12 km/h.

Speed of policeman = 13 km/h.

Time = 12 minutes.

Formula Used:

Relative speed = Speed of policeman - Speed of thief

Distance = Relative speed × Time

Calculation:

Relative speed = 13 km/h - 12 km/h

Relative speed = 1 km/h

Time = 12 minutes = 12/60 hours = 1/5 hours

Distance covered in 1/5 hours at 1 km/h = 1 km/h × 1/5 hours

⇒ Distance = 1/5 km

1 km = 1000 m

⇒ Distance = (1/5) × 1000 m

⇒ Distance = 200 m

Initial distance between them was 300 m

⇒ Distance after 12 minutes = Initial distance - Distance covered

⇒ Distance after 12 minutes = 300 m - 200 m

⇒ Distance after 12 minutes = 100 m

The distance between them after 12 minutes is 100 m.

Shortcut Trick

Check divisibility for each number up to √143 ≈ 12 (i.e., 2, 3, 5, 7, 11).

143 = 11 × 13 (Not prime)

141 = 3 × 47 (Not prime)

133 = 7 × 19 (Not prime)

137 is not divisible by 2, 3, 5, 7, or 11.

∴ The correct answer is 137.

Alternate Method

Given: Numbers 143, 137, 141, 133.

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

⇒ For 143: Sum of digits = 1+4+3 = 8. Divisible by 11 (1+3 − 4 = 0). Not prime.

⇒ For 137: √137 ≈ 11.7. Primes to check: 2, 3, 5, 7, 11. None divide 137. Prime.

⇒ For 141: Sum of digits = 1+4+1 = 6. Divisible by 3. Not prime.

⇒ For 133: 133 ÷ 7 = 19. Not prime.

∴ The correct answer is 137.

Additional Information

Prime Number Test

To check if N is prime, test divisibility by all prime numbers less than or equal to √N.

Co-prime Numbers

Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

Smallest Prime

2 is the only even prime number and the smallest prime number.

Given:

4 × 0.4 × 0.04 × 0.004 × 40

Formula Used:

Product of numbers

Calculation:

4 × 0.4 = 1.6

1.6 × 0.04 = 0.064

0.064 × 0.004 = 0.000256

0.000256 × 40 = 0.01024

The value of 4 × 0.4 × 0.04 × 0.004 × 40 is 0.01024.

Given:

Markup on price = 60%

First discount = 10%

Second discount = 15%

Formula Used:

Profit % = (Selling Price - Cost Price)/Cost Price × 100

Calculation:

Let the Cost Price (CP) of the article be ₹100.

Markup Price = CP + 60% of CP

Markup Price = 100 + 60% of 100

Markup Price = 100 + 60

Markup Price = ₹160

First Discount = 10% of ₹160

First Discount = 10/100 × 160

First Discount = ₹16

Selling Price after first discount = 160 - 16

Selling Price after first discount = ₹144

Second Discount = 15% of ₹144

Second Discount = 15/100 × 144

Second Discount = ₹21.6

Selling Price after second discount = 144 - 21.6

Selling Price after second discount = ₹122.4

Profit = Selling Price - Cost Price

Profit = 122.4 - 100

Profit = ₹22.4

Profit % = (Profit / Cost Price) × 100

⇒ Profit % = (22.4 / 100) × 100

⇒ Profit % = 22.4%

The value of X is 22.4%.

Given:

Expression: (0.4 × 0.4 + 0.2 × 0.02) / 0.016

Formula Used:

Simplification of the given expression

Calculation:

First, calculate the numerator:

0.4 × 0.4 = 0.16

0.2 × 0.02 = 0.004

Numerator = 0.16 + 0.004

Numerator = 0.164

Now, calculate the division:

Denominator = 0.016

⇒ (0.164) / (0.016)

⇒ 10.25 = \(10 \frac{1}{4} \)

The correct answer is option 2.

Given:

The average of 7 consecutive integers is 7.

Formula Used:

If n is the middle integer in 7 consecutive integers, then the integers are n - 3, n - 2, n - 1, n, n + 1, n + 2, n + 3.

Average of these integers = n.

Calculation:

Given average = 7

n = 7

Greatest number = n + 3

⇒ Greatest number = 7 + 3

⇒ Greatest number = 10

The greatest number in these 7 consecutive integers is 10.

Given:

Marks in first term:

Mathematics = 60, Science = 84, Social Studies = 76

Marks in second term:

Mathematics = 78, Science = 82, Social Studies = 64

Formula Used:

Percentage change in marks = [(Total marks in second term - Total marks in first term) / Total marks in first term] × 100

Calculation:

Step 1: Calculate total marks in first term:

Total marks in first term = 60 + 84 + 76 = 220

Step 2: Calculate total marks in second term:

Total marks in second term = 78 + 82 + 64 = 224

Step 3: Calculate percentage change:

Percentage change = [(224 - 220) / 220] × 100

⇒ Percentage change = (4 / 220) × 100

⇒ Percentage change = 1.81%

∴ The percentage change in his marks is 1.81%.

Given:

Diameter of the pillar = 42 cm

Height of the pillar = 15 m = 1500 cm

π = $\frac{22}{7}$

Formula Used:

Volume of a cylinder = π r² h

Calculation:

Radius of the pillar = Diameter / 2 = 42 / 2 = 21 cm

Volume of the pillar = π r² h

Volume = (22/7) × 212 × 1500

⇒ Volume = (22/7) × 441 × 1500

⇒ Volume = 22 × 63 × 1500

⇒ Volume = 22 × 94500

⇒ Volume = 2079000 cm³

The correct answer is option 1: 2.079 × 10⁶ cm³.

Given:

A sum becomes two times in 4 years at a certain rate of simple interest.

Formula Used:

Simple Interest formula: \(A = P(1 + rt)\)

Where A is the amount, P is the principal, r is the rate of interest per year, and t is the time in years.

Calculation:

Let the principal amount be P and the rate of interest be r.

After 4 years, the amount becomes 2P.

Using the simple interest formula:

2P = P(1 + 4r)

⇒ 2 = 1 + 4r

⇒ 4r = 1

⇒ r = 1/4

We need to find the time (t) in which the amount becomes seven times the principal (7P):

Using the simple interest formula again:

7P = P(1 + rt)

⇒ 7 = 1 + (1/4)t

⇒ 7 - 1 = (1/4)t

⇒ 6 = (1/4)t

⇒ t = 6 × 4

⇒ t = 24

The time in which the same amount will be seven times at the same rate of interest is 24 years.

Given:

Ram and Shyam together can complete the work in 12 days.

Shyam alone can complete the work in 15 days.

Shyam alone worked for 5 days.

Formula Used:

Work done by both together in 1 day = 1/12

Work done by Shyam alone in 1 day = 1/15

Work done by Ram alone in 1 day = 1/12 - 1/15

Remaining work after Shyam worked for 5 days = Total work - Work done by Shyam in 5 days

Time taken by Ram to complete the remaining work = Remaining work / Work done by Ram in 1 day

Calculation:

Work done by both together in 1 day = 1/12

Work done by Shyam alone in 1 day = 1/15

Work done by Ram alone in 1 day = 1/12 - 1/15

⇒ Work done by Ram alone in 1 day = (15 - 12) / (12 × 15)

⇒ Work done by Ram alone in 1 day = 3 / 180

⇒ Work done by Ram alone in 1 day = 1/60

Work done by Shyam in 5 days = 5 × (1/15)

⇒ Work done by Shyam in 5 days = 5/15

⇒ Work done by Shyam in 5 days = 1/3

Remaining work = 1 - 1/3

⇒ Remaining work = 2/3

Time taken by Ram to complete the remaining work = (2/3) / (1/60)

⇒ Time taken by Ram to complete the remaining work = (2/3) × 60

⇒ Time taken by Ram to complete the remaining work = 40 days

The correct answer is option 4.

Given:

The price of CNG is increased by 12%.

Formula Used:

If expenditure remains the same, then:

New Price × New Consumption = Old Price × Old Consumption

New Consumption = Old Consumption × (Old Price / New Price)

Percentage Reduction = (Old Consumption - New Consumption) / Old Consumption × 100

Calculation:

Let the old price be 1 unit and the old consumption be 1 unit.

New Price = Old Price + 12% of Old Price

New Price = 1 + 0.12

New Price = 1.12

New Consumption = 1 × (1 / 1.12)

New Consumption = 1 / 1.12

New Consumption = 0.892857

Percentage Reduction = (1 - 0.892857) / 1 × 100

Percentage Reduction = 0.107143 × 100

Percentage Reduction = 10.71%

The percentage reduction in consumption is 10.71%.

Given:

Radius of the hemispherical bowl = 3 cm

π = 3.14

Formula Used:

Volume of a hemisphere = (2/3) π r³

Calculation:

Volume of the bowl

⇒ (2/3) × 3.14 × 3³

⇒ (2/3) × 3.14 × 27

⇒ 56.52 cm³

Since 1 cm³ = 0.001 liters

⇒ 56.52 cm³ = 56.52 × 0.001 liters

⇒ 0.05652 liters

The hemispherical bowl can hold 0.05652 liters of water.