Quantitative Aptitude PYQs

Given:

\(\frac{0.061 \times 0.061 \times 0.061+0.031 \times 0.031 \times 0.031}{0.061 \times 0.061-0.061 \times 0.031+0.031 \times 0.031} \)

Formula Used:

a³ + b³ = (a + b)(a² - ab + b²)

Calculation:

⇒ \(\frac{0.061 \times 0.061 \times 0.061+0.031 \times 0.031 \times 0.031}{0.061 \times 0.061-0.061 \times 0.031+0.031 \times 0.031} \)

⇒ \(\frac{(0.061)^3+(0.031)^3}{(0.061)^2 -0.061 \times 0.031+(0.031)^2 } \)

⇒ \(\frac{(0.061)^3+(0.031)^3}{(0.061)^2 -0.061 \times 0.031+(0.031)^2 } \)

Using identity,

⇒ \(\frac{(0.061+0.031)((0.061)^2 -0.061 \times 0.031+(0.031)^2)}{(0.061)^2 -0.061 \times 0.031+(0.031)^2 } \)

⇒ 0.061 + 0.031

⇒ 0.092

∴ Option 1 is the correct answer.

Given:

A can complete the work in 8 days.

B can complete the work in 10 days.

C can complete the work in 12 days.

Formula Used:

Work done per day by all working together = \(\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right)\)

Time taken to complete the work together = 1/(Work done per day by all working together)

Calculation:

Work done per day by A = 1/8

Work done per day by B = 1/10

Work done per day by C = 1/12

Work done per day by all working together:

\( \left(\frac{1}{8} + \frac{1}{10} + \frac{1}{12}\right) \)

Finding LCM of 8, 10, and 12 = 120

\( \left(\frac{15}{120} + \frac{12}{120} + \frac{10}{120}\right) \)

\( \left(\frac{15 + 12 + 10}{120}\right) \)

\( \left(\frac{37}{120}\right) \)

Time taken to complete the work together:

\( \frac{1}{\left(\frac{37}{120}\right)} \)

\( \left(\frac{120}{37}\right) \)

All of them together will complete the work in \(\left(\frac{120}{37}\right)\) days.

Given:

Diameter of the sphere = 30 cm

Radius of the sphere (r₁) = 30 ÷ 2 = 15 cm

Diameter of the wire = 10 cm

Radius of the wire (r₂) = 10 ÷ 2 = 5 cm

Formula used:

Volume of sphere = Volume of wire

\(\dfrac{4}{3}\pi r_1^3 = \pi r_2^2 \times \text{length of wire (L)}\)

Calculation:

\(\dfrac{4}{3} \pi (15)^3 = \pi (5)^2 \times L\)

⇒ \(\dfrac{4}{3} \times 15^3 = 5^2 \times L\)

⇒ \(\dfrac{4}{3} \times 3375 = 25 \times L\)

⇒ \(\dfrac{13500}{3} = 25 \times L\)

⇒ \(4500 = 25 \times L\)

⇒ \(\dfrac{4500}{25} = L\)

⇒ L = 180 cm

∴ The length of the wire is 180 cm.

Shortcut Trick

According to the question, 45% of Ram's money added to 50% of Soham's money equals 86% of Soham's money.

Equating the parts: 45% of Ram = (86% − 50%) of Soham.

45 × Ram = 36 × Soham.

Ratio of Soham to Ram (S/R) = 45 ÷ 36 = 5 ÷ 4.

Required Percentage = (5 ÷ 4) × 100 = 125%.

∴ The correct answer is 125%.

Alternate Method

Given:

Let Ram's money be R and Soham's money be S.

Equation: 45% of R + 50% of S = 86% of S.

Calculations:

⇒ (45/100) × R + (50/100) × S = (86/100) × S

⇒ 0.45R + 0.50S = 0.86S

⇒ 0.45R = 0.86S − 0.50S

⇒ 0.45R = 0.36S

⇒ S/R = 0.45 ÷ 0.36

⇒ S/R = 45 ÷ 36 = 5/4

⇒ Percentage of Soham's money with respect to Ram's = (S/R) × 100

⇒ Required Percentage = (5/4) × 100 = 125%

∴ The correct answer is 125%.

Additional Information

Percentage Representation

To find what percentage A is of B, use the formula: (A ÷ B) × 100. Here, Ram's money is the base (B).

Ratio to Percentage

A ratio x : y can be converted to a percentage by multiplying the fraction (x/y) by 100. Common fractions like 5/4 = 1.25 = 125%.

Percentage Change

Percentage Increase/Decrease = (Difference ÷ Original Value) × 100.

Given:

The 5^th prime number and the prime number immediately greater than 100.

Formula Used:

Product of two numbers = Number1 × Number2

Calculation:

The 5^th prime number = 11

The prime number immediately greater than 100 = 101

Product of these numbers = 11 × 101

⇒ 11 × 101 = 1111

The product of the 5^th prime number and the prime number immediately greater than 100 is 1111.

Given:

Principal (P) = ₹5000

Time (t) = 2 years

Rate of interest (borrowed) = 4% p.a.

Rate of interest (lent) = 25/4% p.a.

Formula used:

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

Gain = Interest received - Interest paid

Calculation:

Interest paid:

⇒ SI = \(\frac{5000 \times 4 \times 2}{100}\)

⇒ SI = ₹400

Interest received:

⇒ SI = \(\frac{5000 \times \frac{25}{4} \times 2}{100}\)

⇒ SI = ₹625

Gain:

⇒ Gain = ₹625 - ₹400

⇒ Gain = ₹225

∴ The gain in the transaction is ₹225.

Given:

Initial population = 1,87,500

Birth rate = 13%

Death rate = 9%

Formula Used:

Population after n years = Initial population × (1 + net growth rate)ⁿ

Calculation:

Net growth rate = Birth rate - Death rate

Net growth rate = 13% - 9% = 4% = 0.04

Population after 2 years = 1,87,500 × (1 + 0.04)²

⇒ Population after 2 years = 1,87,500 × 1.04²

⇒ Population after 2 years = 1,87,500 × 1.0816

⇒ Population after 2 years = 2,02,800

The population of the country after 2 years is 2,02,800.

Shortcut Trick

Net growth rate = Birth rate - Death rate = 13% - 9% = 4%

For 2 years, successive change:

⇒ Population after 2 years = 1,87,500 × (100 + 4) / 100 × (100 + 4) / 100 = 2,02,800

∴ Population of the country after 2 years = 2,02,800.

Given:

Train: 550 km at 55 km/h

Plane: 4500 km at 500 km/h

Boat: 400 km at 25 km/h

Auto: 55 km at 35 km/h

Formula Used:

Average speed = Total distance traveled / Total time taken

Calculation:

Total Distance = 550 + 4500 + 400 + 55

Total Distance = 5505 km

Time taken by Train = 550 / 55

Time taken by Train = 10 hours

Time taken by Plane = 4500 / 500

Time taken by Plane = 9 hours

Time taken by Boat = 400 / 25

Time taken by Boat = 16 hours

Time taken by Auto = 55 / 35

Time taken by Auto = \( \frac{55}{35} \)

Time taken by Auto = \( \frac{11}{7} \) hours

Total Time = 10 + 9 + 16 + \( \frac{11}{7} \)

Total Time = 35 + \( \frac{11}{7} \)

Total Time = \( \frac{245 + 11}{7} \)

Total Time = \( \frac{256}{7} \)

Average Speed = Total Distance / Total Time

Average Speed = 5505 / \( \frac{256}{7} \)

\( \text{Average Speed} = \frac{5505 \times 7}{256} \)

\( \text{Average Speed} = \frac{38535}{256} \)

Average Speed = 150¹³⁵⁄₂₅₆ km/h

The average speed for the entire journey is 150¹³⁵⁄₂₅₆ km/h.

Shortcut Trick

Check the conditions using the given prime number values for k.

Substitute k = 2:

k = 2 (Prime), k² + 3 = 7 (Prime), k³ + 5 = 13 (Prime).

Since all values are prime numbers, the condition is satisfied.

∴ The correct answer is 2.

Alternate Method

Given: k, (k² + 3) and (k³ + 5) are prime numbers.

We test the value of k starting from the smallest prime number.

Case 1: Let k = 2

⇒ First number = 2 (It is a prime number)

⇒ Second number = k² + 3 = 2² + 3 = 4 + 3 = 7 (It is a prime number)

⇒ Third number = k³ + 5 = 2³ + 5 = 8 + 5 = 13 (It is a prime number)

Case 2: Let k = 3

⇒ First number = 3 (Prime)

⇒ Second number = k² + 3 = 3² + 3 = 12 (Not a prime number)

Case 3: Let k = 7

⇒ First number = 7 (Prime)

⇒ Second number = k² + 3 = 7² + 3 = 52 (Not a prime number)

⇒ Only k = 2 satisfies the condition that all three expressions are prime.

∴ The correct answer is 2.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.

Even Prime Number

2 is the only even prime number. All other prime numbers are odd.

Smallest Prime

The smallest prime number is 2. Note that 1 is neither prime nor composite.

Composite Numbers

Numbers that have more than two factors are called composite numbers (e.g., 4, 6, 8, 9, 10).

Given:

Price increase of tea = 28%

Expenditure remains unchanged

Formula Used:

Percentage reduction in consumption = \(\frac{\text{Percentage increase in price}}{100 + \text{Percentage increase in price}} \times 100 \)

Calculation:

Percentage increase in price = 28%

Equation:

⇒ Percentage reduction in consumption = \(\frac{28}{100 + 28} \times 100\)

⇒ Percentage reduction in consumption = \(\frac{28}{128} \times 100\)

⇒ Percentage reduction in consumption = \(\frac{28 \times 100}{128}\)

⇒ Percentage reduction in consumption = \(\frac{2800}{128}\)

⇒ Percentage reduction in consumption ≈ 21.875

To the nearest whole number, the percentage reduction in consumption should be 22%.

Given:

Length of sides of the triangle are 24 cm, 17 cm, and 9 cm.

Formula Used:

Area of a triangle using Heron's formula:

Area = \( √{s(s-a)(s-b)(s-c)} \)

where s is the semi-perimeter of the triangle, and a , b , c are the lengths of the sides.

s = (a + b + c) / 2

Calculation:

Let a = 24 cm, b = 17 cm, and c = 9 cm.

Semi-perimeter s = (24 + 17 + 9) / 2

Semi-perimeter s = 50 / 2

Semi-perimeter s = 25 cm

Area = \( √{25(25-24)(25-17)(25-9)} \)

Area = \( √{25 \times 1 \times 8 \times 16} \)

Area = \( √{3200} \)

Area = \( 40 √{2} \)

The area of the triangle is 40√2 cm².

Given:

Marked price is 75% more than the cost price.

Discount on the marked price = 40%.

Formula Used:

Percentage Profit = \(\frac{Selling Price - Cost Price}{Cost Price} \times 100 \)

Calculation:

Let Cost Price (CP) = 100.

Marked Price (MP) = CP + 75% of CP = 100 + 75 = 175.

Discount = 40% of MP = 40% of 175 = 0.40 × 175 = 70.

Selling Price (SP) = MP - Discount = 175 - 70 = 105.

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

⇒ Percentage Profit = \(\frac{105 - 100}{100} \times 100\)

⇒ Percentage Profit = \(\frac{5}{100} \times 100\)

⇒ Percentage Profit = 5%

The percentage profit is 5%.

Given:

First five composite numbers: 4, 6, 8, 9, 10.

Formula Used:

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

Calculation:

Sum of first five composite numbers = 4 + 6 + 8 + 9 + 10

Sum = 37

Total count of numbers = 5

Average = Sum / Total count

⇒ Average = 37 / 5

⇒ Average = 7.4

The average of the first five composite numbers is 7.4.

Shortcut Trick

Under similar conditions, the ratio of an object's height to the length of its shadow is constant.

Ratio = Height of Pole ÷ Shadow of Pole = 8.3 ÷ 4.15 = 2

Height of the Tree = Shadow of Tree × Ratio

Height of the Tree = 40 × 2 = 80 m

∴ The correct answer is 80 m.

Alternate Method

Given:

Height of the pole = 8.3 m

Shadow of the pole = 4.15 m

Shadow of the tree = 40 m

Formula Used:

Since the shadows are cast under similar conditions, the angle of elevation of the sun is the same, forming similar triangles. Therefore:

Height of Object 1 ÷ Shadow of Object 1 = Height of Object 2 ÷ Shadow of Object 2

Calculation:

Let the height of the tree be h m.

⇒ 8.3 ÷ 4.15 = h ÷ 40

⇒ 2 = h ÷ 40

⇒ h = 40 × 2

⇒ h = 80

∴ The correct answer is 80 m.

Additional Information

Properties of Similar Triangles

When two right-angled triangles are formed under the same angle of elevation (such as by sun rays at the same time), they are similar, meaning the ratio of their corresponding sides is always identically equal.

Direct Proportion

The height of a vertical object and the length of its cast shadow are directly proportional. If Height ∝ Shadow, then H₁/S₁ = H₂/S₂.

Angle of Elevation

The phrase 'similar conditions' implies that the angle of elevation of the sun is identical for both locations, ensuring the shadow lengths follow the exact same geometric scale as the object heights.

Given:

Cost Price (CP) of 150 guavas = ₹525

Selling Price (SP) per dozen (12 guavas) = ₹36

Formula Used:

Profit or Loss Percentage = \(\frac{Profit or Loss}{Cost Price} × 100\)

Calculation:

Total Selling Price (SP) for 150 guavas:

Number of dozens = 150 / 12

⇒ Number of dozens = 12.5

Selling Price (SP) for 12.5 dozens:

SP = 12.5 × ₹36

⇒ SP = ₹450

Loss = CP - SP

Loss = ₹525 - ₹450

⇒ Loss = ₹75

Loss Percentage:

Loss Percentage = (Loss / CP) × 100

⇒ Loss Percentage = (75 / 525) × 100

⇒ Loss Percentage = (1 / 7) × 100

⇒ Loss Percentage = 14.29%

The correct answer is option 4.

Given:

4 men or 8 women can complete the work in 12 days.

Work done by 4 men in 12 days = 1 unit of work.

Work done by 8 women in 12 days = 1 unit of work.

Total work (W) = 1 unit.

Formula used:

Work rate = Total work ÷ Time

Combined rate = Work rate of men + Work rate of women

Time = Total work ÷ Combined rate

Calculation:

Work rate of 4 men in 1 day:

⇒ Work rate = Total work ÷ Time = 1 ÷ 12 = 1/12 units/day.

⇒ Rate of 1 man = (1/12) ÷ 4 = 1/48 units/day.

Work rate of 8 women in 1 day:

⇒ Work rate = Total work ÷ Time = 1 ÷ 12 = 1/12 units/day.

⇒ Rate of 1 woman = (1/12) ÷ 8 = 1/96 units/day.

Combined rate of 9 men and 6 women:

⇒ Rate = (9 × 1/48) + (6 × 1/96)

⇒ Rate = 9/48 + 6/96

⇒ Rate = 18/96 + 6/96 = 24/96 = 1/4 units/day.

Time taken by 9 men and 6 women:

⇒ Time = Total work ÷ Combined rate

⇒ Time = 1 ÷ (1/4) = 4 days.

∴ 9 men and 6 women can complete the work in 4 days.

Given:

The selling price of 32 items is equal to the cost price of 45 items.

Formula Used:

Percentage of Profit = (Profit) / (Cost Price) × 100

Calculation:

Let the cost price of each item be C and the selling price of each item be S.

According to the given condition,

32S = 45C

⇒ S = (45 / 32)C

Profit on each item = Selling Price - Cost Price

⇒ Profit = S - C

⇒ Profit = (45 / 32)C - C

⇒ Profit = ((45 / 32) - 1)C

⇒ Profit = ((45 - 32) / 32)C

⇒ Profit = (13 / 32)C

Percentage of Profit = (Profit) / (Cost Price) × 100

⇒ Percentage of Profit = \(\frac{\frac{13}{32}C}{C} × 100\)

⇒ Percentage of Profit = (13 / 32) × 100

⇒ Percentage of Profit = 40.625%

The correct answer is option 4.

Given:

The ratio of the wife's share to the daughter's share = 1 ∶ 4.

The ratio of the daughter's share to the son's share = 1 ∶ 4.

The wife gets ₹5,000 less than the son.

Calculation:

(Wife's share)/(Daughter's share) = 1/4 

(Daughter's share)/(Son's share) = 1/4

After combine the ratio, we get

Wife's share : Daughter's share : Son's share = 1 : 4 : 16

Let wife's share = Rs. x

Daughter's share = 4x

Son's share = 16x

Total Property = x + 4x + 16x = 21x

According to question,

16x - x = 5000

15x = 5000

21x = (5000/15x) × 21x = 7000

The value (in ₹) of the whole property = Rs. 7000

Shortcut Trick

Let HCF be H. Then LCM = 18H.

Sum: H + 18H = 152 ⇒ 19H = 152 ⇒ H = 8.

LCM = 18 × 8 = 144.

Using Product of Numbers = HCF × LCM:

16 × Other Number = 8 × 144 ⇒ Other Number = 144 ÷ 2 = 72.

∴ The correct answer is 72.

Alternate Method

Given:

LCM = 18 × HCF

Sum of HCF and LCM = 152

First Number (n₁) = 16

Formula Used:

Product of two numbers = HCF × LCM

Calculations:

⇒ Let HCF = x

⇒ LCM = 18x

⇒ x + 18x = 152

⇒ 19x = 152

⇒ x = 152 ÷ 19 = 8

⇒ HCF = 8 and LCM = 18 × 8 = 144

⇒ n₁ × n₂ = HCF × LCM

⇒ 16 × n₂ = 8 × 144

⇒ n₂ = (8 × 144) ÷ 16

⇒ n₂ = 144 ÷ 2 = 72

∴ The correct answer is 72.

Additional Information

HCF and LCM Relation

For any two numbers a and b, the product of the numbers is always equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

HCF Property

The HCF of two or more numbers is the greatest number that divides each of them exactly. LCM is always a multiple of HCF.

Co-prime Numbers

If the HCF of two numbers is 1, they are called co-prime numbers. For co-prime numbers, LCM is simply the product of the numbers.

Given:

In an equilateral triangle, each side is 'a' units.

Formula Used:

Altitude of an equilateral triangle = \(\frac{a √{3}}{2}\)

Calculation:

Side of the equilateral triangle = a units

Altitude = \(\frac{a √{3}}{2}\)

⇒ Altitude = \(\frac{a √{3}}{2}\)

The altitude of the equilateral triangle is \(\frac{a √{3}}{2}\) units.