Quantitative Aptitude PYQs

Given:

Prime numbers between 20 and 30: 23, 29

Formula Used:

Sum of the squares of prime numbers = p₁² + p₂²

Calculation:

Prime numbers between 20 and 30 are 23 and 29.

Sum of their squares = 23² + 29²

23² = 529

29² = 841

⇒ 529 + 841

⇒ 1370

The correct answer is option 2.

Given:

The numbers to consider are: 1², 2², 3², 4², 5², 6², 7², 8², and 9²

Formula used:

The formula for the average is:

\(\text{Average} = \frac{\text{Sum of the terms}}{\text{Number of terms}}\)

Calculations:

First, calculate the squares of the numbers:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

Now, calculate the sum of these squares:

⇒ Sum = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = 285

Next, calculate the average:

⇒ Average = \(\frac{285}{9} = \frac{95}{3}\)

Converting into mixed fraction: \(31 \frac{2}{3} \)

Therefore, the average is \(31 \frac{2}{3} \)

Given:

Numbers: 15, 35, 42, 72

Formula Used:

The least common multiple (LCM) of multiple numbers is the smallest number that is a multiple of each of the numbers.

Calculation:

Prime factorization of each number:

15 = 3 × 5

35 = 5 × 7

42 = 2 × 3 × 7

72 = 2³ × 3²

LCM is found by taking the highest power of each prime factor:

LCM = 2³ × 3² × 5 × 7

⇒ LCM = 8 × 9 × 5 × 7

⇒ LCM = 72 × 5 × 7

⇒ LCM = 360 × 7

⇒ LCM = 2520

The least common multiple of 15, 35, 42, and 72 is 2520.

Given:

The train approaches a tunnel MN, with a dog inside the tunnel at a point located at a distance of \(\frac{1}{3}\) of the total length of the tunnel (i.e., the distance from M to N).

When the train whistles, the dog runs. If the dog runs towards the entrance M, the train catches the dog exactly at M. If the dog runs towards the exit N, the train catches the dog exactly at N.

Calculations:

Let the length of the tunnel be x.

The distance of the train and entrance M be y.

Let the speeds of the train and dog be t and d respectively.

Hence, when the dog runs 2x/5, the train covers y.

⇒ (2x/5)/d = y/t      ----(i)

When the dog runs 3x/5 to the other end, the train covers x + y

⇒ (3x/5)/d = (x + y)/t      ----(ii)

∴ taking ratio of (ii) to (i) ⇒ 3/2 = (x + y)/y

⇒ 3y = 2x + 2y

⇒ y = 2x       ----(iii)

Substituting (iii) in (i)

(2x/5)/d = 2x/t

⇒ t = 5d

The ratio of the speed of dog and train is 1 : 5

Given:

The numbers are 256, 136, and 436.

Formula Used:

To find the Highest Common Factor (HCF) of three numbers, we use the Euclidean algorithm.

Calculation:

First, find the HCF of 256 and 136:

256 = 2⁸

136 = 2³ × 17

Common factor = 2³ = 8

Next, find the HCF of 8 and 436:

436 = 2² × 109

Common factor with 8 = 2² = 4

Therefore, the HCF of 256, 136, and 436 is 4.

The correct answer is option 2.

Given:

Inside radius of the bowl (r) = 3 cm.

Thickness of the steel (t) = 2 cm.

Formula Used:

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

Volume of steel used = Volume of outer hemisphere - Volume of inner hemisphere

Calculation:

Outer radius of the bowl (R) = Inside radius + Thickness = 3 cm + 2 cm = 5 cm

Volume of outer hemisphere = \( \frac{2}{3} \pi R^3 \)

Volume of outer hemisphere = \( \frac{2}{3} \times \frac{22}{7} \times 5^3 \)

Volume of outer hemisphere = \( \frac{2}{3} \times \frac{22}{7} \times 125 \)

Volume of outer hemisphere = \( \frac{2 \times 22 \times 125}{3 \times 7} \)

Volume of outer hemisphere = \( \frac{5500}{21} \)

Volume of outer hemisphere ≈ 261.90 cm³

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

Volume of inner hemisphere = \( \frac{2}{3} \times \frac{22}{7} \times 3^3 \)

Volume of inner hemisphere = \( \frac{2}{3} \times \frac{22}{7} \times 27 \)

Volume of inner hemisphere = \( \frac{2 \times 22 \times 27}{3 \times 7} \)

Volume of inner hemisphere = \( \frac{1188}{21} \)

Volume of inner hemisphere ≈ 56.57 cm³

Volume of steel used = Volume of outer hemisphere - Volume of inner hemisphere

Volume of steel used ≈ 261.90 cm³ - 56.57 cm³

Volume of steel used ≈ 205.33 cm³

The volume of the steel used in making the bowl is 205.33 cm³.

Given:

Principal (P) = Rs. 10,000

Time (T) = 1.5 years

Rate of interest (R) = 10% p.a.

Formula Used:

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

Compound Interest (CI) when interest is compounded semi-annually:

Amount (A) = P (1 + (R/n)/100)^nT

CI = A - P

where n = number of times interest is compounded per year

Calculation:

Simple Interest:

SI = (10,000 × 10 × 1.5) / 100

SI = 1,500

Compound Interest:

Here, n = 2 (since interest is compounded semi-annually)

Amount (A) = 10,000 (1 + (10/2)/100)^2×1.5

Amount (A) = 10,000 (1 + 5/100)³

Amount (A) = 10,000 (1 + 0.05)³

Amount (A) = 10,000 (1.05)³

Amount (A) = 10,000 × 1.157625

Amount (A) = 11,576.25

CI = 11,576.25 - 10,000

CI = 1,576.25

Difference between CI and SI:

Difference = CI - SI

Difference = 1,576.25 - 1,500

Difference = 76.25

The correct answer is option 2.

Given:

We need to find the smallest 4-digit prime number.

Formula Used:

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

Calculation:

We need to check the primality of the smallest 4-digit numbers.

1000: Not a prime number (divisible by 2, 5).

1001: Not a prime number (divisible by 7, 11, 13).

1002: Not a prime number (divisible by 2).

1003: Not a prime number (divisible by 17, 59).

1004: Not a prime number (divisible by 2).

1005: Not a prime number (divisible by 3, 5).

1006: Not a prime number (divisible by 2).

1007: Not a prime number (divisible by 19, 53).

1008: Not a prime number (divisible by 2, 3).

1009: Check for primality.

1009 is not divisible by any prime number ≤ \( \sqrt{1009} \approx 31.76 \).

Divisors checked: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.

Since 1009 is not divisible by any of these, it is a prime number.

The smallest 4-digit prime number is 1009.

Given:

Work by 14 men in 6 days = Total work

Work by 20 women in 8 days = Total work

7 men worked for 3 days, remaining work to be done by women in 6 days.

Formula used:

Work = Number of workers × Days × Efficiency

Efficiency ratio of men and women:

Work by 14 men in 6 days = Work by 20 women in 8 days

⇒ 14 × 6 × Efficiency of men = 20 × 8 × Efficiency of women

⇒ Efficiency of men : Efficiency of women = 20 × 8 : 14 × 6

⇒ Efficiency of men : Efficiency of women = 160 : 84 = 80 : 42 = 40 : 21

Calculations:

Step 1: Total work = Work done by 14 men in 6 days

⇒ Total work = 14 × 6 × Efficiency of men = 14 × 6 × 40 = 3360 units

Step 2: Work done by 7 men in 3 days

⇒ Work done = 7 × 3 × Efficiency of men = 7 × 3 × 40 = 840 units

Step 3: Remaining work

⇒ Remaining work = Total work - Work done by 7 men = 3360 - 840 = 2520 units

Step 4: Work to be completed by women in 6 days

⇒ Required number of women × 6 × Efficiency of women = Remaining work

⇒ Required number of women × 6 × 21 = 2520

⇒ Required number of women = 2520 / (6 × 21)

⇒ Required number of women = 20

∴ 20 women are required to complete the remaining work in 6 days.

Given:

Lengths to be measured: 1 m, 20 cm, 3 m, 90 cm.

Formula Used:

Greatest Common Divisor (GCD) of the lengths

Calculation:

First, convert all lengths to cm:

1 m = 100 cm

20 cm = 20 cm

3 m = 300 cm

90 cm = 90 cm

Lengths in cm: 100 cm, 20 cm, 300 cm, 90 cm

Find the GCD of the lengths:

GCD(100, 20, 300, 90)

Prime factorization of each length:

100 = 2² x 5²

20 = 2² x 5

300 = 2² x 3 x 5²

90 = 2 x 3² x 5

Common factors are 2 and 5:

Smallest power of 2 is 2¹

Smallest power of 5 is 5¹

GCD = 2¹ x 5¹ = 2 x 5 = 10

The greatest possible length of scale to measure exactly the given lengths is 10 cm.

Given:

The profit made by selling an article is 30% of its selling price.

Formula Used:

Profit Percentage = (Profit / Cost Price) x 100

Calculation:

Let the selling price (SP) of the article be $100.

Given, Profit = 30% of SP

⇒ Profit = 30% of 100

⇒ Profit = 30

We know, Cost Price (CP) = Selling Price - Profit

⇒ CP = 100 - 30

⇒ CP = 70

Now, Profit Percentage = (Profit / CP) x 100

⇒ Profit Percentage = (30 / 70) x 100

⇒ Profit Percentage = (3 / 7) x 100

⇒ Profit Percentage = 42 (6/7)%

The correct answer is option 3.

Given:

Length is decreased by 10%.

Breadth is decreased by 20%.

Height is increased by 30%.

Formula Used:

Volume of cuboid = Length × Breadth × Height

Calculation:

Let initial dimensions be L, B, and H.

Initial volume = L × B × H

New length = L × (1 - 0.10) = 0.90L

New breadth = B × (1 - 0.20) = 0.80B

New height = H × (1 + 0.30) = 1.30H

New volume = 0.90L × 0.80B × 1.30H

New volume = 0.90 × 0.80 × 1.30 × L × B × H

New volume = 0.936 × L × B × H

Percentage change in volume = \(\left(\frac{{\text{New volume} - \text{Initial volume}}}{\text{Initial volume}}\right) × 100\)

Percentage change in volume = \(\left(\frac{0.936LBH - LBH}{LBH}\right) × 100\)

Percentage change in volume = \(\left(0.936 - 1\right) × 100\)

Percentage change in volume = -0.064 × 100

Percentage change in volume = -6.4%

Negative sign is indicating that volume will be decrese.

The total percentage change in the volume of the cuboid is 6.4%.

Given:

Discount given by Trader A = 25%

Trade discount by Trader B = 20%

Cash discount by Trader B = 4%

Discount by Trader A is ₹360 more than the discount by Trader B.

Formula Used:

Marked Price (MP) = x

Discount by Trader A = 25% of x = 0.25x

Discount by Trader B = 20% of x + 4% of (80% of x) = 0.20x + 0.04(0.80x)

Calculation:

According to the given condition:

0.25x = (0.20x + 0.04 × 0.80x) + 360

⇒ 0.25x = 0.20x + 0.032x + 360

⇒ 0.25x = 0.232x + 360

⇒ 0.25x - 0.232x = 360

⇒ 0.018x = 360

⇒ x = 360 / 0.018

⇒ x = 20000

The marked price of the item is ₹20000.

Given:

Numbers: 8, 6, 14, 7, and x

Average of these numbers = 11

Formula Used:

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

Calculation:

Let the sum of the numbers be S. The total number of values is 5.

⇒ 11 = S / 5

⇒ S = 11 × 5

⇒ S = 55

Sum of the given numbers = 8 + 6 + 14 + 7 + x

⇒ 8 + 6 + 14 + 7 + x = 55

⇒ 35 + x = 55

⇒ x = 55 - 35

⇒ x = 20

The value of x is 20.

Shortcut Trick

Relative Speed = 8 − 7 = 1 km/h.

Thief's Lead = Speed × Time = 7 km/h × (6/60) h = 0.7 km.

Time to Catch = Lead ÷ Relative Speed = 0.7 km ÷ 1 km/h = 0.7 hours.

Distance by Policeman = Speed × Time = 8 km/h × 0.7 h = 5.6 km.

∴ The correct answer is 5.6 km.

Alternate Method

Given:

Speed of Policeman (V_p) = 8 km/h

Speed of Thief (V_t) = 7 km/h

Time Delay = 6 minutes = 6/60 = 0.1 hours

Formula Used:

Distance = Speed × Time

Relative Speed (same direction) = V₁ − V₂

Calculations:

⇒ Distance covered by Thief in 6 minutes = 7 × (6/60) = 0.7 km

⇒ Relative Speed of Policeman with respect to Thief = 8 − 7 = 1 km/h

⇒ Time taken by Policeman to catch the Thief = Distance ÷ Relative Speed

⇒ Time = 0.7 ÷ 1 = 0.7 hours

⇒ Total distance covered by Policeman = Speed_p × Time

⇒ Distance = 8 × 0.7 = 5.6 km

∴ The correct answer is 5.6 km.

Additional Information

Relative Speed

When two objects move in the same direction, relative speed is the difference (u − v). In opposite directions, it is the sum (u + v).

Catching Distance

The distance covered by the chaser to catch the target is always (Speed of Chaser × Time to Catch).

Unit Conversion

To convert km/h to m/s, multiply by 5/18. To convert minutes to hours, divide by 60.

Given:

Expression to simplify: 1.6 + 1.2 × (1)/(4) - {(1)/(5) + (1 - 0.8)}

Formula Used:

Follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Calculation:

First, simplify the parentheses:

1 - 0.8 = 0.2

So, the expression inside the curly braces becomes:

{(1)/(5) + 0.2}

Next, simplify the fraction and addition inside the curly braces:

(1)/(5) = 0.2

0.2 + 0.2 = 0.4

Now, the expression is:

1.6 + 1.2 × (1)/(4) - 0.4

Next, simplify the multiplication:

(1)/(4) = 0.25

1.2 × 0.25 = 0.3

Now, the expression is:

1.6 + 0.3 - 0.4

Finally, perform the addition and subtraction:

1.6 + 0.3 = 1.9

1.9 - 0.4 = 1.5

The correct answer is option 1 (1.5).

Given:

Loan Amount = ₹1,200

Simple Interest paid = ₹432

Time = Rate of Interest (R) years

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) / 100

Calculation:

Here, SI = ₹432, Principal = ₹1,200, Time = Rate (R)

Using the formula,

₹432 = (₹1,200 × R × R) / 100

⇒ 432 = 12R²

⇒ R² = 432 / 12

⇒ R² = 36

⇒ R = √36

⇒ R = 6

The rate of interest per annum is 6%.

Given:

Number of scouts = 350

Radius of the cylindrical glass (r) = 10 cm

Height of the cylindrical glass (h) = 15 cm

Cost of milk = ₹50 per liter

Formula Used:

Volume of a cylinder (V) = πr²h

1 liter = 1000 cm³

Total cost = (Total volume in liters) × (Cost per liter)

Calculation:

Volume of one glass:

V = πr²h

⇒ V = (22/7) × 10² × 15

⇒ V = (22/7) × 100 × 15

⇒ V = 22 × 100 × 15 / 7

⇒ V = 33000 / 7 cm³

⇒ V ≈ 4714.29 cm³

Total volume for 350 scouts:

Total volume = 4714.29 × 350

⇒ Total volume = 1650000 cm³

Convert to liters:

Total volume in liters = 1650000 / 1000

⇒ Total volume in liters = 1650 liters

Total cost:

Total cost = 1650 × 50

⇒ Total cost = 82500

The correct answer is option 2 (₹82,500).

Given:

The ratio of Ajay's marks in Mathematics to Science = 9:13

Marks in Science - Marks in Mathematics = 16

Formula Used:

Let marks in Mathematics be 9x and marks in Science be 13x.

According to the given condition: 13x - 9x = 16

Calculation:

13x - 9x = 16

⇒ 4x = 16

⇒ x = 4

Marks in Mathematics = 9x = 9 × 4 = 36

The marks in Mathematics are 36.

Shortcut Trick

Let the original amount of money be P.

Using the successive percentage multiplying factor method:

P × 75% × 40% = 360

P × (3/4) × (2/5) = 360

P = 360 × (10/3) = 1200

∴ The correct answer is ₹1200.

Alternate Method

Given:

Percentage of money lost = 25%

Percentage of remaining money spent = 60%

Amount left = ₹360

Formula Used:

Remaining Amount = Total × (1 − L₁/100) × (1 − L₂/100)

Calculations:

⇒ Let the original amount be 100 units.

⇒ After losing 25%, remaining amount = 100 − 25 = 75 units.

⇒ He then spends 60% of the remaining amount.

⇒ Amount spent = 60% of 75 units = (60 ÷ 100) × 75 = 45 units.

⇒ Remaining amount left with him = 75 − 45 = 30 units.

⇒ According to the question, 30 units = ₹360.

⇒ 1 unit = 360 ÷ 30 = ₹12.

⇒ Original amount = 100 units = 100 × 12 = ₹1200.

∴ The correct answer is ₹1200.

Additional Information

Successive Percentage Change

When a value is subjected to multiple percentage changes sequentially, the final value can be calculated directly using: Final = Initial × (1 − a/100) × (1 − b/100).

Base of Percentage

It is critical to identify the correct base for any percentage. Words like 'of the remaining' indicate the new percentage must be calculated on the newly reduced value, not the original starting total.