Quantitative Aptitude PYQs

Given:

Find the average of all the even numbers from 22 to 30 (inclusive both).

Formula used:

Average of n even numbers = Sum of even numbers / Number of even numbers

Calculation:

Even numbers from 22 to 30: 22, 24, 26, 28, 30

Sum of these even numbers = 22 + 24 + 26 + 28 + 30

⇒ Sum = 130

Number of even numbers = 5

⇒ Average = 130 / 5

⇒ Average = 26

∴ The correct answer is option 1.

Given:

200 men have provisions for 30 days.

50 men leave.

Formula used:

Initial total consumption days = Total men × Number of days

New number of days = Initial total consumption days / Remaining men

Calculation:

Initial total consumption days = 200 × 30 = 6000

Remaining men = 200 - 50 = 150

⇒ New number of days = 6000 / 150

⇒ New number of days = 40

∴ The correct answer is option (3).

Shortcut Trick

Ratio of Boys : Girls = 7 : 9

Given, 7 units (Boys) = 1183

1 unit = 1183 ÷ 7 = 169

Total units = 7 + 9 = 16 units

Total Students = 16 × 169 = 2704

∴ The correct answer is 2704.

Alternate Method

Given:

Ratio of Boys to Girls = 7 : 9

Number of Boys in the school = 1183

Formula Used:

Total = (Sum of ratio parts ÷ Given ratio part) × Given value

Calculations:

⇒ Let the number of boys be 7x and the number of girls be 9x.

⇒ 7x = 1183

⇒ x = 1183 ÷ 7

⇒ x = 169

⇒ Total number of students = 7x + 9x = 16x

⇒ Total = 16 × 169

⇒ Total = 2704

The total number of boys and girls in the school is 2704, which corresponds to option (3).

∴ The correct answer is 2704.

Additional Information

Ratio and Proportion Basics

A ratio a : b indicates that for every 'a' parts of one quantity, there are 'b' parts of another. The multiplier 'x' helps find the actual values.

Unitary Method

To find the total value, first calculate the value of one single part (unit) by dividing the given total by its respective ratio part.

Percentage of a Part

The percentage of any part in a ratio a : b is calculated as [a ÷ (a + b)] × 100.

Given:

If \(\frac{52416}{168}=312 \)

Calculation:

\(\frac{52416}{168}=312 \)

⇒ \(\frac{52.416}{0.0168}\)

⇒ (52416/1000)/(168/10000)

⇒ (52416/1000) × (10000/168)

⇒ (52416 / 168) × 10

⇒ 312 × 10

⇒ 3120

∴ The correct answer is option (1).

Given:

\(\dfrac{b}{a} = 0.75\)

Formula used:

Substitute \(\dfrac{b}{a} = 0.75\) into the expression \(\dfrac{2a - b}{2a + b} + \dfrac{8}{9}\)

Calculation:

\(b = 0.75a\)

⇒ \(\dfrac{2a - 0.75a}{2a + 0.75a} + \dfrac{8}{9}\)

⇒ \(\dfrac{1.25a}{2.75a} + \dfrac{8}{9}\)

⇒ \(\dfrac{125}{275} + \dfrac{8}{9}\)

⇒ \(\dfrac{5}{11} + \dfrac{8}{9}\)

⇒ \(\dfrac{45 + 88}{99}\)

⇒ \(\dfrac{133}{99}\)

∴ The correct answer is option (1).

Given:

A woman gains 20% by selling an article for a certain price. 

Formula used:

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

Calculation:

Let Cost Price (CP) = 100

Selling Price (SP) with 20% profit = 100 + 20% of 100

SP = 100 + 20 = 120

If she sells at double the price:

New SP = 2 × 120 = 240

Profit% = (240 - 100) / 100 × 100

⇒ Profit% = 140%

∴ The correct answer is option (2).

Given:

Lucas's Monthly Income = ₹42,500

Income Increase Percentage = 18%

Formula used:

New Income = Old Income × (1 + Increase Percentage/100)

Calculation:

New Income = 42,500 × (1 + 18/100)

⇒ New Income = 42,500 × 1.18

⇒ New Income = 50,150

∴ The correct answer is option 4.

Given:

Volume (V) of sphere = Surface Area (S) of sphere

Formula used:

Volume (V) = \(\dfrac{4}{3}\pi r^3\)

Surface Area (S) = \({4\pi r^2}\)

Calculation:

\(\dfrac{4}{3}\pi r^3 = 4\pi r^2\)

⇒ \(\dfrac{r}{3} = 1\)

⇒ r = 3 units

∴ The correct answer is option (1).

Given:

125 identical small solid spheres are made out of a big solid sphere of radius 5 cm.

Formula used:

Volume of sphere = \(\dfrac{4}{3}\pi r^3\)

Surface area of sphere = \(4\pi r^2\)

Calculation:

Let the radius of each small sphere be r cm.

Volume of the big sphere = Volume of 125 small spheres

\(\dfrac{4}{3}\pi (5)^3 = 125 \times \dfrac{4}{3}\pi r^3\)

⇒ \(5^3 = 125 \times r^3\)

⇒ \(125 = 125r^3\)

⇒ \(r^3 = 1\)

⇒ \(r = 1\)

Surface area of each small sphere = \(4\pi (1)^2\)

⇒ Surface area = \(4\pi cm^2\)

∴ The correct answer is option (4).

Given:

A man completes his journey in 6 h. 

Total Distance (D) = d1 + d2

Where, d1 = Distance covered at 30 km/h

d2 = Distance covered at 50 km/h

Time (t) = t1 + t2

Where, t1 = Time taken to cover d1

t2 = Time taken to cover d2

Calculation:

Let d1 = d2 = D/2

t₁ = d₁ / 30 = (D/2) / 30

t₂ = d₂ / 50 = (D/2) / 50

Total time = 6 h

⇒ 6 = (D/2) / 30 + (D/2) / 50

⇒ 6 = (5D + 3D) / 300

⇒ 6 = 8D / 300

⇒ 6 × 300 = 8D

⇒ D = 225 km

∴ The correct answer is option 2.

Given:

Principal (P) = ₹7,500

Rate (R) = 7.5% per annum

Time (T) = 4 years 9 months = 4.75 years

Formula used:

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

Calculations:

SI = \(\dfrac{7500 \times 7.5 \times 4.75}{100}\)

⇒ SI = \(\dfrac{267,187.5}{100}\)

⇒ SI = 2,671.875

∴ The correct answer is option 3.

Given:

Selling Price (SP) = ₹6,500

Discount = 20%

Formula used:

SP = MP × (1 - (Discount/100))

Where, MP = Marked Price

Calculations:

6500 = MP × (1 - (20/100))

⇒ 6500 = MP × (1 - 0.2)

⇒ 6500 = MP × 0.8

⇒ MP = 6500 / 0.8

⇒ MP = 8125

∴ The correct answer is option (1).

Shortcut Trick

Total interest percentage = (Compound Interest ÷ Principal) × 100

⇒ (8,320 ÷ 50,000) × 100 = 16.64%

Using the Successive Percentage rule for 8%: 8 + 8 + (8 × 8)/100 = 16.64%.

The effective rate for 2 years matches the interest percentage.

∴ The correct answer is 2 years.

Alternate Method

Given: Principal (P) = ₹50,000, Compound Interest (CI) = ₹8,320, Rate (R) = 8%.

Formula Used: Amount (A) = P(1 + R/100)ⁿ

⇒ Amount (A) = Principal + CI = 50,000 + 8,320 = ₹58,320

⇒ 58,320 = 50,000(1 + 8/100)ⁿ

⇒ 58,320 ÷ 50,000 = (1.08)ⁿ

⇒ 1.1664 = (1.08)ⁿ

⇒ (1.08)² = (1.08)ⁿ

⇒ n = 2

∴ The correct answer is 2 years.

Additional Information

Compound Interest Formula

The amount A after n years is A = P(1 + r/100)ⁿ, and CI = A − P.

Effective Interest Rate

For a rate of r% for 2 years, the effective compound interest rate is (2r + r²/100)%.

CI and SI Difference

The difference between Compound Interest and Simple Interest for 2 years is P(r/100)².

Rule of 72

A quick way to estimate the time to double an investment is 72 ÷ interest rate.

Shortcut Trick

Let total votes be x.

Valid votes = x × (9/10) × (4/5) = 18x/25.

Winning margin = (70% − 30%) = 40% of valid votes.

⇒ (18x/25) × (2/5) = 6480.

⇒ x = (6480 × 125) / 36 = 180 × 125 = 22500.

∴ The correct answer is 22,500.

Alternate Method

Given: Winner got 70% of valid votes, Loser got 30%.

Difference: 70% − 30% = 40% of valid votes = 6480.

⇒ Total Valid Votes = (6480 / 40) × 100 = 16200.

⇒ Voted Votes (80% are valid) = (16200 / 80) × 100 = 20250.

⇒ Total Votes (90% participated) = (20250 / 90) × 100 = 22500.

∴ The correct answer is 22,500.

Additional Information

Election Base Calculations

Always identify the base: Valid votes are calculated after removing non-voters and invalid votes from the total voting list.

Winning Margin Rule

Margin = (Winner% − Loser%) × Total Valid Votes. In a two-candidate election, Loser% = 100% − Winner%.

Successive Percentage

Effective Valid Votes = Total × (Participated%) × (Valid%). Use fractions for faster calculation (e.g., 80% = 4/5).

Given:

2.5 × (4 - (24 - 33) ÷ \(\left(\frac{11}{4}\right)\)) ÷ 8 × \((\frac{4}{3})\) 

Formula used:

Order of operations (BODMAS/BIDMAS): Brackets, Orders (i.e. powers and square roots, etc.), Division and Multiplication, Addition and Subtraction

Calculation:

2.5 × (4 - (24 - 33) ÷ \(\left(\frac{11}{4}\right)\)) ÷ 8 × \((\frac{4}{3})\) 

⇒ 2.5 × (4 - (16 - 27) ÷ (11/4)) ÷ 8 × (4/3)

⇒ 2.5 × (4 - (-11) × (4/11)) ÷ 8 × (4/3)

⇒ 2.5 × (4 + 4) ÷ 8 × (4/3)

⇒ 2.5 × (8) ÷ 8 × (4/3)

⇒ 2.5 × (4/3)

⇒ (10/3)

∴ The correct answer is option (2).

Given:

A car covers a distance of 52 km in 1 hour 20 minutes 40 seconds moving at \(\dfrac{5}{8}\) of its original speed.

Formula used:

Speed = Distance / Time

Calculation:

Time taken = 1 hour 20 minutes 40 seconds = 1 + 20/60 + 40/3600 hours

⇒ Time taken = 1 + 1/3 + 1/90 = 1.3444 hours (approx)

Let the original speed be x km/h.

Given, the car is moving at \(\dfrac{5}{8}\) of its original speed.

Therefore, the speed = \(\dfrac{5}{8} \times x\) km/h

⇒ Distance = Speed × Time

⇒ 52 = \(\dfrac{5}{8} \times x \times 1.3444\)

⇒ x = \(\dfrac{52 \times 8}{5 \times 1.3444}\)

⇒ x = \(\dfrac{416}{6.722}\)

⇒ x = 61.88 km/h

∴ The correct answer is option (2).

Given:

The average marks of all the students in a class 10^th examination is 85.

The sum of the marks of all the girls is 3500.

The sum of the marks of the boys is 2025.

Formula used:

Average = Total Marks / Number of Students

Calculation:

Let the number of students be x.

Total marks of all students = Sum of marks of girls + Sum of marks of boys

Total marks = 3500 + 2025

Total marks = 5525

Average marks = 85

⇒ 85 = 5525 / x

⇒ x = 5525 / 85

⇒ x = 65

∴ The correct answer is option 2.

Given:

Marked price % above purchase price = 45%

Desired profit % = 16%

Formula Used:

Marked Price (MP) = Cost Price (CP) × (1 + Marked price %/100)

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

Discount % = ((Marked Price (MP) - Selling Price (SP)) / Marked Price (MP)) × 100

Calculations:

Let CP = 100

MP = 100 × (1 + 45/100)

⇒ MP = 100 × 1.45 = 145

SP = 100 × (1 + 16/100)

⇒ SP = 100 × 1.16 = 116

Discount = MP - SP

⇒ Discount = 145 - 116 = 29

Discount % = (Discount / MP) × 100

⇒ Discount % = (29 / 145) × 100

⇒ Discount % = 20%

∴ The discount should be 20%.

Given:

Number = 483y718

Divisible by = 11

y = digit

Formula Used:

A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11.

Calculations:

Number = 483y718

Digits at odd places (from right) = 8, 7, 3, 4

Sum of digits at odd places = 8 + 7 + 3 + 4 = 22

Digits at even places (from right) = 1, y, 8

Sum of digits at even places = 1 + y + 8 = 9 + y

Difference = |(Sum of digits at odd places) - (Sum of digits at even places)|

⇒ Difference = |22 - (9 + y)|

⇒ Difference = |13 - y|

For divisibility by 11, the difference must be 0 or a multiple of 11.

Case 1: Difference = 0

|13 - y| = 0

⇒ 13 - y = 0

⇒ y = 13 (Not a single digit)

Case 2: Difference = 11

|13 - y| = 11

⇒ 13 - y = 11 or 13 - y = -11

⇒ y = 13 - 11 or y = 13 + 11

⇒ y = 2 or y = 24 (Not a single digit)

From Case 2, y = 2 is a single digit.

∴ The value of y is 2.

Given:

Principal (P) = ₹54,000

Time (t) = 2 years

Difference between CI and SI = ₹1,215

Formula used:

CI - SI for 2 years = P\((\dfrac{r}{100})^2\)

Where, r = rate

Calculations:

1,215 = 54,000\((\dfrac{r}{100})^2\)

⇒ \(\dfrac{1,215}{54,000} = (\dfrac{r}{100})^2\)

⇒ \(\dfrac{1,215}{54,000} = \dfrac{r^2}{10,000}\)

⇒ \(\dfrac{1,215}{54,000} \times 10,000 = r^2\)

⇒ \(\dfrac{225}{1} = r^2\)

⇒ \(r = \sqrt{225}\)

⇒ r = 15

∴ The correct answer is option (3).