Quantitative Aptitude PYQs

Given:

The ratio of Arun's expenditure to savings is 8 ∶ 2.

Formula Used:

Percentage of salary spent = \(\frac{\text{Expenditure}}{\text{Total Salary}} \times 100\)

Calculation:

Let Arun's expenditure be 8x and his savings be 2x.

Total salary = Expenditure + Savings

Total salary = 8x + 2x = 10x

Percentage of salary spent = \(\frac{\text{Expenditure}}{\text{Total Salary}} \times 100\)

Percentage of salary spent = \(\frac{8x}{10x} \times 100\)

⇒ Percentage of salary spent = \(\frac{8}{10} \times 100\)

⇒ Percentage of salary spent = 0.8 × 100

⇒ Percentage of salary spent = 80%

Arun spends 80% of his salary.

Given:

A man can complete a work in 15 days.

A woman can complete the same work in 25 days.

Concept Used:

If M1 persons can do a work in D1 days and M2 persons can do the same work in D2 days, then the relationship between them is:

Work = M1 × D 1  = M2 × D 2

Calculation:

1 man can do a work in 15 days or 1 woman can do a piece of work in 25 days,

⇒ (1 m × 15) = (1 w × 25)

⇒ m = (353) w

Now, let the required work be done in ‘x’ days by 3 men and 5 women.

⇒ (1 w × 25) = (3 m + 5 w) × x

Eliminating one variable by putting the value of ‘m’ from our result,

⇒ (1 w × 25) = (3 (5 w/3) + 5 w) × x

⇒ 25w = 10w × x

⇒ x = 2.5 days

∴ The required time to finish the work is 2.5 days.

Given:

P ∶ Q = 5 ∶ 3

Formula Used:

(P⁴ - Q⁴) ∶ (P⁴ + Q⁴)

Calculation:

Let P = 5x and Q = 3x

⇒ (P⁴ - Q⁴) = (5x)⁴ - (3x)⁴

⇒ (P⁴ - Q⁴) = 625x⁴ - 81x⁴

⇒ (P⁴ - Q⁴) = 544x⁴

⇒ (P⁴ + Q⁴) = (5x)⁴ + (3x)⁴

⇒ (P⁴ + Q⁴) = 625x⁴ + 81x⁴

⇒ (P⁴ + Q⁴) = 706x⁴

Therefore, (P⁴ - Q⁴) ∶ (P⁴ + Q⁴) = 544 ∶ 706

⇒ (P⁴ - Q⁴) ∶ (P⁴ + Q⁴) = 272 ∶ 353

The correct answer is option 4.

Given:

Radius of the cone (r) = 7 units

Curved surface area (CSA) = 308 square units

π = 22/7

Formula used:

Curved surface area of a cone = πrℓ

Volume of a cone = (1/3)πr²h

Using Pythagoras theorem: ℓ² = r² + h²

Calculation:

Curved surface area = πrℓ

⇒ (22/7) × 7 × ℓ = 308

⇒ 22 × ℓ = 308

⇒ ℓ = 308 / 22 = 14 units

Using Pythagoras theorem to find height (h):

ℓ² = r² + h²

⇒ 14² = 7² + h²

⇒ 196 = 49 + h²

⇒ h² = 196 - 49

⇒ h² = 147

⇒ h = √147 = 7√3 units

Volume = (1/3)πr²h

⇒ Volume = (1/3) × (22/7) × 7² × 7√3

⇒ Volume = (1/3) × 22 × 49 × √3

⇒ Volume = (1/3) × 1078 × √3

⇒ Volume = 1079/√3

The volume of the tent is \(\frac{1078}{\sqrt 3}\) cubic units.

Given:

Marked price of the pencil box = Rs. 80

Discount given = 10%

Gain = 20%

Formula Used:

Selling Price (SP) = Marked Price (MP) - Discount

Cost Price (CP) = SP / (1 + Gain%)

Calculation:

Discount = 10% of MP = 10% of 80 = 0.10 × 80 = 8 Rs.

Selling Price = MP - Discount = 80 - 8 = 72 Rs

Cost Price = SP / (1 + Gain%)

⇒ CP = 72 / (1 + 20%)

⇒ CP = 72 / 1.20 = 60

The cost price of the pencil box is Rs. 60.

Given:

First length = 5 meters 95 centimeters = 595 centimeters

Second length = 8 meters = 800 centimeters

Third length = 16 meters 65 centimeters = 1665 centimeters

Formula used:

The maximum possible length for exact measurement = HCF (Highest Common Factor) of the three lengths

Calculation:

Finding the HCF of 595, 800, and 1665:

1. HCF of 595 and 800:

595 = 5 x 7 x 17

800 = 2⁵ x 5²

HCF(595, 800) = 5

2. Now, HCF of 5 and 1665:

1665 = 3 x 5 x 7 x 19

HCF(5, 1665) = 5

Thus, HCF(595, 800, 1665) = 5

The maximum possible length that can be used for exact measurement is 5 centimeters.

Given:

S.P of two articles = Rs. 5000 each

Loss on one article = 50/3%

Formula used:

C.P = S.P × [100/(100 – loss%)]

Calculation:

C.P of one article = Rs. 5000 × [100/(250/3)]

⇒ Rs. 5000 × 100 × (3/250)

⇒ Rs. 6000

Now, As the there is no profit or no loss

So, C.P of other article = Rs. (10,000 – 6000)

⇒ Rs. 4000

S.P of other article = Rs. 5000

Gain = Rs. (5000 – 4000)

⇒ Rs. 1000

Gain% = (1000/4000) × 100%

⇒ 25%

∴ The other article is sold at 25% profit

Given:

Cash price of music player = Rs. 4,510

Down payment = Rs. 1,200

Rate of compound interest = 10% per annum

Number of installments = 3

Formula used:

Price to be paid in installments = x/[1 + (r/100)] + x/[1 + (r/100)]² + x/[1 + (r/100)]³

Where, x = Value of each installment to be paid

r = Rate of compound interest p.a.

Calculation:

Remaining price to be paid in installments = Total price - Down payment

⇒ Remaining price = 4510 - 1200 = 3310

⇒ 3310 = x/[1 + (1/10)] + x/[1 + (1/10)]² + x/[1 + (1/10)]³

⇒ 3310 = x × [(10/11) + (10/11)² + (10/11)³]

⇒ 3310 = x × [(10/11) + (100/121) + (1000/1331)]

⇒ 3310 = x × [(1210/1331) + (1100/1331) + (1000/1331)]

⇒ 3310 = x × (3310/1331)

⇒ x = 1331

∴ The amount of each installment is Rs. 1331.

Given:

Initial population (2017) = 3,68,000

Final population (2020) = 3,27,200

Formula Used:

Percentage Decrease = \(\frac{Initial Population - Final Population}{Initial Population} × 100\)

Calculation:

Initial Population = 3,68,000

Final Population = 3,27,200

Decrease in Population = 3,68,000 - 3,27,200

Decrease in Population = 40,800

\(Percentage Decrease = \frac{40,800}{3,68,000} × 100\)

Percentage Decrease = 0.110869565 × 100

Percentage Decrease ≈ 11

The correct answer is option 4.

Given:

a, b, c, d, and e are distinct prime numbers.

The sum of these five numbers is odd.

Concept Used:

If the sum of five distinct prime numbers is odd, then all of them must be odd.

Calculation:

If the sum of five distinct prime numbers is odd, then all of them must be odd.

So all 5 prime numbers are odd, When we remove any prime number from these 5 other 4 numbers will also be odd numbers.

A sum of 4 odd numbers will always be an even number.

example: 3 + 5 + 7 + 11 = 26

⇒ a + b + c + d must be even.

The correct answer is option 4.

Formula Used:

Two numbers are relatively prime if their greatest common divisor (GCD) is 1.

Calculation:

1. For (51, 119):

Prime factors of 51: 3 × 17

Prime factors of 119: 7 × 17

GCD = 17

Not relatively prime.

2. For (98, 567):

Prime factors of 98: 2 × 7²

Prime factors of 567: 3 × 3 × 3 × 3 × 7

GCD = 7

Not relatively prime.

3. For (27, 51):

Prime factors of 27: 3³

Prime factors of 51: 3 × 17

GCD = 3

Not relatively prime.

4. For (103, 113):

Prime factors of 103: 103 (prime)

Prime factors of 113: 113 (prime)

GCD = 1

Relatively prime.

The correct answer is option 4.

Shortcut Trick

Distance = [(S₁ × S₂) ÷ (S₂ − S₁)] × Total Time Difference

S₁ = 8 km/h, S₂ = 10 km/h, Time Difference = 7 + 5 = 12 mins = 1 / 5 hr.

Distance = [(8 × 10) ÷ (10 − 8)] × (1 / 5)

Distance = (80 ÷ 2) × (1 / 5) = 40 × (1 / 5) = 8 km.

∴ The correct answer is 8 km.

Alternate Method

Given:

Speed of Ram on day 1 (S₁) = 8 km/h, time late = 7 minutes.

Speed of Ram on day 2 (S₂) = 10 km/h, time early = 5 minutes.

Total time difference = 7 + 5 = 12 minutes.

Time difference in hours = 12 / 60 = 1 / 5 hours.

Formula Used:

Time = Distance ÷ Speed

Calculation:

Let the distance between Ram's house and the station be D km.

⇒ Time taken at 8 km/h = D / 8 hours

⇒ Time taken at 10 km/h = D / 10 hours

⇒ Difference in time = 1 / 5 hours

⇒ (D / 8) − (D / 10) = 1 / 5

⇒ (5D − 4D) / 40 = 1 / 5

⇒ D / 40 = 1 / 5

⇒ D = 40 / 5

⇒ D = 8 km

∴ The correct answer is 8 km.

Additional Information

Constant Distance Rule

When the distance is constant, the ratio of speeds is inversely proportional to the ratio of times taken. If the speed ratio is a : b, then the time ratio will be b : a.

Average Speed

If a certain distance is covered at speed x and the same distance is covered at speed y, the average speed for the whole journey is calculated as (2 × x × y) ÷ (x + y).

Relative Speed

When two bodies move in the same direction, their relative speed is the difference of their individual speeds. When moving in opposite directions, it is the sum of their speeds.

Given:

Height of the cylindrical box (h) = 1.08 m

Base radius of the cylindrical box (r) = 42 cm = 0.42 m

Cost of metal sheet per m² = Rs. 90

\(π = \frac{22}{7}\)

Formula Used:

Surface area of the cylindrical box = 2πr(h + r)

Cost of the material used = Surface area × Cost per m²

Calculation:

Surface area of the cylindrical box

⇒ 2π r(h + r)

⇒ 2 × \(\frac{22}{7}\) × 0.42 × (1.08 + 0.42)

⇒ 2 × \(\frac{22}{7}\) × 0.42 × 1.5

⇒ 3.96 m²

Cost of the material used

⇒ Surface area × Cost per m²

⇒ 3.96 × 90

⇒ Rs. 356.4

The cost of the material used in the box is Rs. 356.4.

Given:

Average score in 10 innings = 52 runs

Total innings = 11

Desired average score = 58 runs

Formula Used:

Average score = Total runs / Number of innings

Calculation:

Total runs in 10 innings = 52 × 10

Total runs in 10 innings = 520

Let the number of runs to be scored in the 11^th inning be x

Total runs in 11 innings = 520 + x

Desired average = (Total runs in 11 innings) / 11

58 = (520 + x ) / 11

⇒ 58 × 11 = 520 + x

⇒ 638 = 520 + x

⇒ x = 638 - 520

⇒ x = 118

The number of runs to be scored by him in the 11^th inning to raise the average score to 58 is 118.

Given:

Principal amount borrowed (P) = Rs. 12,000

Time (T) = 3 years

Interest rate for borrowing (R₁) = 2% per annum

Interest rate for lending (R₂) = 4% per annum

Formula Used:

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

Gain per year = (SI_lending - SI_borrowing) / T

Calculation:

SI_borrowing = (12000 × 2 × 3) / 100 = 720

SI_lending = (12000 × 4 × 3) / 100 = 1440

Gain in the transaction for 3 years:

Gain = SI_lending - SI_borrowing

Gain = 1440 - 720 = 720

Gain per year = 720 / 3 = 240

The correct answer is option 1: Rs. 240

Given:

Base length of the parallelogram = 4 cm.

Height of the parallelogram = 5 cm.

Formula Used:

Area of a parallelogram = Base × Height

Calculation:

Area = 4 cm × 5 cm

⇒ Area = 20 cm²

The area of the parallelogram is 20 cm².

Given:

Sushil's income is 22% less than Abhishek's income.

Formula Used:

Percentage increase = (Difference / Original) × 100

Calculation:

Let Abhishek's income be A .

Sushil's income = A - 0.22A = 0.78A

Difference in income = A - 0.78A = 0.22A

Percentage increase = (0.22A / 0.78A) × 100

Percentage increase = (0.22 / 0.78) × 100

Percentage increase = (22 / 78) × 100

Percentage increase = 0.2820 × 100

Percentage increase = 28.20%

Abhishek's income is 28.20% more than Sushil's income.

Given:

Principal (P) = Rs. 2,000

Rate (R) = 8% p.a.

Time (T) = 1 year

Formula Used:

Compound Interest (CI) for annually: \(CI = P \left(1 + \frac{R}{100}\right)^T - P\)

Compound Interest (CI) for half-yearly: \(CI = P \left(1 + \frac{R/2}{100}\right)^{2T} - P\)

Calculation:

For annually:

CIannually  = \(2000 \left(1 + \frac{8}{100}\right)^1 - 2000\)

⇒ CI_annually = 2000(1 + 0.08) - 2000

⇒ CIannually = 2000(1.08) - 2000

⇒ CIannually = 2160 - 2000

⇒ CIannually = 160

For half-yearly:

CIhalf-yearly = \( 2000 \left(1 + \frac{8/2}{100}\right)^{2×1} - 2000\)

⇒ CIhalf-yearly = \(2000 \left(1 + \frac{4}{100}\right)^2 - 2000\)

⇒ CIhalf-yearly = 2000(1 + 0.04)² - 2000

⇒ CIhalf-yearly = 2000(1.04)² - 2000

⇒ CIhalf-yearly = 2000(1.0816) - 2000

⇒ CIhalf-yearly = 2163.2 - 2000

⇒ CI_half-yearly = 163.2

Difference in Compound Interest:

Difference = ⇒ CIhalf-yearly - CIannually 

⇒ Difference = 163.2 - 160

⇒ Difference = 3.2

The difference between the compound interests is Rs. 3.20.

Given:

Sum of two numbers = 1904

HCF = 56

Formula Used:

Let the two numbers be 56a and 56b, where a and b are co-prime.

Sum of numbers = 56a + 56b = 1904

Number of pairs is the number of integer solutions for the equation a + b = 34

Calculation:

56a + 56b = 1904

⇒ a + b = 34

The pairs of (a, b) are:

(1, 33), (3, 31), (5, 29), (7, 27), (9, 25), (11, 23), (13, 21), (15, 19)

There are 8 such pairs.

The correct answer is option 3.

Shortcut Trick

Increase in surface area = 4π[(r + 3)² − r²] = 156π

⇒ (r + 3)² − r² = 39

⇒ r² + 6r + 9 − r² = 39

⇒ 6r = 30 ⇒ r = 5 cm

∴ The correct answer is 5 cm.

Alternate Method

Given:

Increase in radius = 3 cm

Increase in surface area = 156π cm²

Formula Used:

Surface area of a sphere = 4πr²

Calculations:

Let the original radius of the sphere be r cm.

⇒ Original surface area = 4πr²

⇒ New radius = (r + 3) cm

⇒ New surface area = 4π(r + 3)²

According to the given condition:

⇒ 4π(r + 3)² − 4πr² = 156π

Dividing the entire equation by 4π:

⇒ (r + 3)² − r² = 39

⇒ (r² + 6r + 9) − r² = 39

⇒ 6r + 9 = 39

⇒ 6r = 30

⇒ r = 5 cm

∴ The correct answer is 5 cm.

Additional Information

Volume of a Sphere

The volume of a sphere is given by V = (4÷3)πr³, where r is the radius.

Hemisphere Surface Area

The curved surface area of a solid hemisphere is 2πr², and its total surface area is 3πr².

Volume of a Hemisphere

The volume of a hemisphere is exactly half that of a full sphere, given by V = (2÷3)πr³.