Quantitative Aptitude PYQs

Given:

Ratios to compare:

\(\frac{4}{5}\), \(\frac{12}{14}\), \(\frac{21}{23}\), \(\frac{16}{19}\), \(\frac{79}{85}\) 

Formula Used:

To compare ratios, we convert them to decimal form.

Calculations:

Convert each ratio to decimal form:

\(\frac{4}{5}\) = 0.8

\(\frac{12}{14}\) = 0.857

\(\frac{21}{23}\) = 0.913

\(\frac{16}{19}\) = 0.842

\(\frac{79}{85}\) = 0.929

Comparing these decimals, the smallest ratio is 0.8.

The smallest ratio is \(\frac{4}{5}\).

Given:

Expression: 4.5 × 2.8 + 3.6 × 2.5 - 1.7 × 12

Formula Used:

Use the order of operations (BODMAS/BIDMAS):

1. Brackets
2. Orders (i.e. powers and square roots, etc.)
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Calculation:

4.5 × 2.8 + 3.6 × 2.5 - 1.7 × 12

⇒ 12.6 + 9.0 - 20.4

⇒ 21.6 - 20.4

⇒ 1.2

The result of the expression is 1.2.

Given:

Radius of the base and height of the cylinder are in the ratio 7 : 3.

Volume of the cylinder = 12474 cm³

Use π = 22/7

Formula Used:

Total surface area of the cylinder = 2πr(r + h)

Volume of the cylinder = πr²h

Calculation:

Let the radius of the base be 7x and the height be 3x.

Volume of the cylinder = πr²h

⇒ (22/7) × (7x)² × (3x) = 12474

⇒ (22/7) × 49x² × 3x = 12474

⇒ (22/7) × 147x³ = 12474

⇒ 22 × 21x³ = 12474

⇒ 462x³ = 12474

⇒ x³ = 27

⇒ x = 3

So, the radius of the base = 7x = 7 × 3 = 21 cm

Height of the cylinder = 3x = 3 × 3 = 9 cm

Total surface area of the cylinder = 2πr(r + h)

⇒ 2 × (22/7) × 21 × (21 + 9)

⇒ 2 × (22/7) × 21 × 30

⇒ 2 × 22 × 3 × 30

⇒ 2 × 22 × 90

⇒ 3960 cm²

The total surface area of the cylinder is 3960 cm².

Given:

Annual salary in 2008 = ₹48,580

This was 25% more than his salary in 2007.

Formula Used:

If the salary in 2007 is X, then:

2008 Salary = 2007 Salary + 25% of 2007 Salary

₹48,580 = ₹X + 0.25 × ₹X

Calculations:

₹48,580 = ₹X + 0.25 × ₹X

₹48,580 = ₹X(1 + 0.25)

₹48,580 = ₹X × 1.25

→ ₹X = ₹48,580 / 1.25

→ ₹X = ₹38,864

∴ Ram's salary in 2007 was ₹38,864.

Given:

Price reduction = 25% = 0.25

Increase in sale = 30% = 0.30

Formula used:

Net effect (%) = (x + y + xy/100)%

Where x = -25% (reduction) and y = 30% (increase)

Calculation:

Net effect = (-25 + 30 + (-25 × 30)/100)%

⇒ Net effect = (5 - 7.5)%

⇒ Net effect = - 2.5%

∴ The net effect on the daily sale receipts is a decrease of 2\(\frac{1}{2}\)%.

Given:

Speed 1 = 10 km/h

Speed 2 = 14 km/h

Extra distance covered at 14 km/h = 20 km

Formula used:

Time is constant, so:

Time = Distance/Speed

Calculation:

Let actual distance = D km

According to the question:

(D/10) = (D + 20)/14

Cross-multiplying:

14 × D = 10 × (D + 20)

14D = 10D + 200

14D - 10D = 200

4D = 200

D = 200/4

D = 50 km

∴ The actual distance travelled by him is 50 km.

Given:

The numbers are 72, 81, 90, 68, 61.

Formula Used:

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

Calculation:

Sum of all numbers = 72 + 81 + 90 + 68 + 61

Sum = 372

Total count of numbers = 5

Average = Sum / Total count

Average = 372 / 5

∴ Average = 74.4

Given:

Population 4 years ago = 10,000

Annual growth rate for the first 2 years = 20%

Decrease rate in the third year = 10%

Decrease rate in the fourth year = 5%

Formula Used:

Population after n years = Initial Population × (1 + growth rate/100)^{number of years}

Calculation:

Population after first year = 10,000 × (1 + 20/100)

⇒ Population after first year = 10,000 × 1.2

⇒ Population after first year = 12,000

Population after second year = 12,000 × (1 + 20/100)

⇒ Population after second year = 12,000 × 1.2

⇒ Population after second year = 14,400

Population after third year = 14,400 × (1 - 10/100)

⇒ Population after third year = 14,400 × 0.9

⇒ Population after third year = 12,960

Population after fourth year = 12,960 × (1 - 5/100)

⇒ Population after fourth year = 12,960 × 0.95

⇒ Population after fourth year = 12,312

The current population of the city after four years is 12,312.

Shortcut Trick Population after fourth year = 10,000 × (1 + 20/100) × (1 + 20/100) × (1 - 10/100) × (1 - 5/100)

⇒ 10,000 × 1.2 × 1.2 × 0.9 × 0.95

⇒ 12,312

Given:

8 men can complete a work in 6 hours.

Formula Used:

Work = Men × Time

Calculation:

Let the total work be W.

Work done by 8 men in 6 hours:

W = 8 × 6

Work done by 12 men in t hours:

W = 12 × t

Since the work is the same:

8 × 6 = 12 × t

⇒ t = (8 × 6) / 12

⇒ t = 48 / 12

⇒ t = 4

12 men can complete the same work in 4 hours.

Given:

Cost Price (CP) of each water bottle = ₹18

Selling Price (SP) of each water bottle = ₹20

Number of water bottles = 10

Formula used:

Total CP = Number of bottles × CP per bottle

Total SP = Number of bottles × SP per bottle

Profit % = (Profit / Total CP) × 100

Calculation:

Total CP = 10 × 18 = ₹180

Total SP = 10 × 20 = ₹200

Profit = Total SP - Total CP

⇒ Profit = ₹200 - ₹180 = ₹20

Profit % = (20 / 180) × 100

⇒ Profit % = 11\(\frac{1}{9}\)%

∴ Lalita’s overall profit percentage is 11\(\frac{1}{9}\)%

Given:

Market Price (MP) = 25% above Cost Price (CP)

Discount = 12%

Formula used:

Gain % = [(SP - CP) / CP] × 100

Calculation:

Let the Cost Price be ₹100.

Market Price = CP + 25% of CP = 1.25 × CP

Market Price = 1.25 × 100 = ₹125

Selling Price (SP) = MP - 12% of MP = MP × 0.88

Selling Price = 0.88 × 125 = ₹110

Gain = SP - CP = 110 - 100 = ₹10

Gain % = (10/100) × 100 = 10%

∴ The gain in the deal is 10%.

Shortcut Trick

The difference between the Total Surface Area (3πr²) and the Curved Surface Area (2πr²) of a hemisphere is simply the area of its circular base (πr²).

Difference = Area of base = πr²

Difference = (22/7) × 21 × 21 = 22 × 3 × 21

Difference = 66 × 21 = 1386 cm²

∴ The correct answer is 1386 cm².

Alternate Method

Given:

Radius of the hemisphere (r) = 21 cm

Value of π = 22/7

Formula Used:

Curved Surface Area (CSA) of hemisphere = 2πr²

Total Surface Area (TSA) of hemisphere = 3πr²

Calculations:

⇒ Curved Surface Area (CSA) = 2 × (22/7) × 21 × 21

⇒ CSA = 2 × 22 × 3 × 21 = 2772 cm²

⇒ Total Surface Area (TSA) = 3 × (22/7) × 21 × 21

⇒ TSA = 3 × 22 × 3 × 21 = 4158 cm²

⇒ Difference = TSA − CSA

⇒ Difference = 4158 − 2772 = 1386 cm²

∴ The correct answer is 1386 cm².

Additional Information

Surface Area of a Sphere

A sphere has only one surface area, which is 4πr². A hemisphere is half of a sphere but adds a flat circular base.

Volume of a Hemisphere

The volume of a hemisphere is exactly half the volume of a sphere: V = (2/3)πr³.

Base Area Relationship

The TSA of a hemisphere is always 1.5 times its CSA because 3πr² = 1.5 × (2πr²).

Given:

Total amount = ₹4000

Rate of interest for first part = 3% p.a.

Rate of interest for second part = 5% p.a.

Total annual interest = ₹144

Formula used:

Total interest = Interest on first part + Interest on second part

SI = (P × R × T) ÷ 100

Calculation:

Let the first part invested at 3% be ₹x.

Then, the second part invested at 5% will be ₹(4000 - x).

According to the question:

(x × 3 × 1)/100 + ((4000 - x) × 5 × 1)/100 = 144

⇒ 0.03x + 0.05(4000 - x) = 144

⇒ 0.03x + 200 - 0.05x = 144

⇒ -0.02x = 144 - 200

⇒ -0.02x = -56

⇒ x = 56 ÷ 0.02

⇒ x = 2800

∴ The amount invested at 3% per annum simple interest is ₹2800.

Given:

Smallest prime number between 1 and 100 = 2

Largest prime number between 1 and 100 = 97

Formula Used:

Difference = Largest prime number - Smallest prime number

Calculation:

Smallest prime number = 2

Largest prime number = 97

Difference = 97 - 2

∴ The difference between the smallest and the largest prime numbers between 1 to 100 is 95.

Given:

We need to find the smallest four-digit number that is divisible by 15, 25, 40, and 75.

Formula Used:

The smallest number divisible by multiple numbers is the Least Common Multiple (LCM) of those numbers.

Calculation:

First, find the LCM of 15, 25, 40, and 75.

Prime factorization:

15 = 3 × 5

25 = 5²

40 = 2³ × 5

75 = 3 × 5²

LCM is obtained by taking the highest powers of all prime factors.

LCM = 2³ × 3 × 5²

⇒ LCM = 8 × 3 × 25

⇒ LCM = 600

Now, find the smallest four-digit number divisible by 600.

Smallest four-digit number = 1000

Divide 1000 by 600 and round up to the next whole number.

⇒ 1000 / 600 ≈ 1.67

⇒ Next whole number = 2

Multiply LCM by 2:

⇒ 600 × 2 = 1200

The smallest four-digit number divisible by 15, 25, 40, and 75 is 1200.

Shortcut Trick

List the prime numbers less than 28: 2, 3, 5, 7, 11, 13, 17, 19, 23.

Group the numbers to simplify the addition: (2 + 3 + 5) + (7 + 13) + (11 + 19) + (17 + 23).

Step 1: 10 + 20 + 30 + 40 = 100.

∴ The correct answer is 100.

Alternate Method

Given: We need to find the sum of all prime numbers smaller than 28.

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

⇒ Identify prime numbers between 1 and 28: 2, 3, 5, 7, 11, 13, 17, 19, 23.

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23

⇒ Sum = (2 + 3 + 5) + (7 + 11 + 13) + (17 + 19 + 23)

⇒ Sum = 10 + 31 + 59

⇒ Sum = 100.

∴ The correct answer is 100.

Additional Information

Definition of Prime Numbers

Numbers that have exactly two factors, 1 and the number itself. Example: 2, 3, 5, 7, 11...

Even Prime Number

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

Composite Numbers

Numbers having more than two factors are called composite numbers. 1 is neither prime nor composite.

Twin Primes

Pairs of prime numbers that have a difference of 2 are called twin primes (e.g., 3 and 5, 5 and 7).

Given:

Repeating decimal: \(0.\overline{63}\)

Calculation:

Let x = 0.636363...

Multiply by 100:

100x = 63.636363...

Subtract x from 100x:

100x - x = 63.636363... - 0.636363...

99x = 63

⇒ x = 63 / 99 = 7 / 11

0.63 expressed in the form p/q is 7/11.

Given:

Formulas for Loss% and Profit%

Formula used:

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

Loss = Cost Price (CP) - Selling Price (SP)

Profit% = (Profit / CP) × 100

Loss% = (Loss / CP) × 100

Calculation:

1. Analyze each

Statement:

1) Loss% = (SP - CP) / CP × 100

This formula is incorrect.

Loss% should be (CP - SP) / CP × 100.

2) Profit% = (SP - CP) / CP × 100

This formula is correct.

Profit% = (Profit / CP) × 100, and Profit = SP - CP.

3) Profit% = (CP - SP) / CP × 100

This formula is incorrect.

Profit% should be (SP - CP) / CP × 100.

4) Loss% = (SP + CP) / CP × 100

This formula is incorrect.

Loss% should be (CP - SP) / CP × 100.

∴ The correct statement is: Profit% = (SP - CP) / CP × 100

Given:

x = 6/5

Numbers: 6x, 37/6, 6 - x, 5 + x

Calculation:

6x = 6 × (6/5) = 36/5 = 7.2

6 - x = 6 - (6/5) = (30/5) - (6/5) = 24/5 = 4.8

5 + x = 5 + (6/5) = (25/5) + (6/5) = 31/5 = 6.2

37/6 = 6.1666...

The numbers are in ascending order:

4.8 < 6.1666... < 6.2 < 7.2

24/5 < 37/6 < 31/5 < 36/5

6 - x < 37/6 < 5 + x < 6x

∴ The ascending order is: 6 - x, 37/6, 5 + x, 6x.

Given:

Fractions: 9/10, 12/15, 18/35, 21/40

Formula used:

LCM of fractions = (LCM of numerators) / (HCF of denominators)

Calculation:

LCM of the numerators (9, 12, 18, 21):

9 = 3 × 3

12 = 2 × 2 × 3

18 = 2 × 3 × 3

21 = 3 × 7

LCM(9, 12, 18, 21) = 2 × 2 × 3 × 3 × 7 = 252

HCF of the denominators (10, 15, 35, 40):

10 = 2 × 5

15 = 3 × 5

35 = 5 × 7

40 = 2 × 2 × 2 × 5

HCF(10, 15, 35, 40) = 5

LCM of the fractions is,

LCM = (LCM of numerators) / (HCF of denominators)

LCM = 252 / 5

∴ The LCM of 9/10, 12/15, 18/35, 21/40 is 252/5.