Quantitative Aptitude PYQs

Given:

LCM of two numbers = 1728

HCF of two numbers = 4

One of the numbers = 64

Formula Used:

Product of two numbers = LCM × HCF

Calculation:

Let the other number be x.

Product of the two numbers = 64 × x

According to the formula,

64 × x = 1728 × 4

⇒ 64 × x = 6912

⇒ x = 6912 / 64

⇒ x = 108

The other number is 108.

Given:

The trader sells goods at a loss of 12.5% on the selling price.

Formula used:

Loss percentage on cost price (CP):

\(\text{Loss %} = \frac{\text{Loss}}{\text{Cost Price}} \times 100\)

Relation between SP and CP under loss:

\(\text{Selling Price (SP)} = \text{Cost Price (CP)} - \text{Loss}\)

Calculations:

Let the selling price (SP) = ₹100.

Loss = 12.5% of SP = \(12.5% \times 100 = ₹12.5\).

Cost Price (CP) = SP + Loss = ₹100 + ₹12.5 = ₹112.5.

Loss percentage on CP:

⇒ \(\text{Loss %} = \frac{12.5}{112.5} \times 100\)

⇒ \(\text{Loss %} = 11 %\)(\frac{1}{9}\)%

∴ The trader's loss percentage on cost price is 11\(\frac{1}{9}\)%.

Given:

A's income is 70% more than B's income.

Formula Used:

Percentage less = \(\frac{(A - B)}{A} \times 100\)

Calculation:

Let B's income = 100.

A's income = 100 + 70% of 100

A's income = 100 + 70

A's income = 170

Percentage less B earns than A = \(\frac{(170 - 100)}{170} \times 100\)

⇒ Percentage less B earns than A = \(\frac{70}{170} \times 100\)

⇒ Percentage less B earns than A = \(\frac{7}{17} \times 100\)

⇒ Percentage less B earns than A = \(41\frac{3}{17}\)%

The correct answer is option 2.

Shortcut Trick

Rate = 12.5% = 1/8 per year.

For 2 years, Total Interest = 2 × (1/8) = 1/4 of Principal.

Amount = Principal + Interest = 1 + 1/4 = 5/4 units.

5 units = ₹8,250 ⇒ 1 unit (Interest) = 8,250 ÷ 5 = ₹1,650.

∴ The correct answer is ₹1,650.

Alternate Method

Given:

Rate (R) = 12.5% per annum

Time (T) = 2 years

Amount (A) = ₹8,250

Formula Used:

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

Amount (A) = P + SI = P × [1 + (R × T) ÷ 100]

Calculations:

⇒ 8,250 = P × [1 + (12.5 × 2) ÷ 100]

⇒ 8,250 = P × [1 + 25 ÷ 100]

⇒ 8,250 = P × [1 + 0.25]

⇒ 8,250 = 1.25P

⇒ P = 8,250 ÷ 1.25 = ₹6,600

⇒ SI = Amount − Principal

⇒ SI = 8,250 − 6,600 = ₹1,650

∴ The correct answer is ₹1,650.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, the interest is calculated only on the Principal amount, whereas in Compound Interest, interest is calculated on the Principal plus accumulated interest.

Effective Rate of Interest

For Simple Interest, the effective rate for 'n' years is simply (R × n)%. For 12.5% over 2 years, the effective rate is 25%.

Fractional Conversion

Converting percentages to fractions (e.g., 12.5% = 1/8) often simplifies calculations in Aptitude exams.

Given:

Radius of the larger sphere (R) = 6 cm

Radius of each smaller wax ball (r) = 2 cm

Formula Used:

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

Calculation:

Volume of the larger sphere:

⇒ \(\frac{4}{3} \pi R^3\)

⇒ \(\frac{4}{3} × \frac{22}{7} × 6^3\)

⇒ \(\frac{4}{3} × \frac{22}{7} × 216\)

⇒ \(\frac{4 × 22 × 216}{3 × 7}\)

⇒ \(\frac{19008}{21}\)

⇒ 904 cm³

Volume of one smaller wax ball:

⇒ \(\frac{4}{3} \pi r^3\)

⇒ \(\frac{4}{3} × \frac{22}{7} × 2^3\)

⇒ \(\frac{4}{3} × \frac{22}{7} × 8\)

⇒ \(\frac{4 × 22 × 8}{3 × 7}\)

⇒ \(\frac{704}{21}\)

⇒ 33.52 cm³

Number of smaller wax balls:

⇒ \(\frac{\text{Volume of the larger sphere}}{\text{Volume of one smaller wax ball}}\)

⇒ \(\frac{904}{33.52}\)

⇒ 26.97

≈ 27

The number of wax balls that can be moulded out of a sphere of radius 6 cm is 27.

Given:

The price of sugar falls by 30%.

Formula Used:

Percentage increase in consumption = (Percentage decrease in price) / (1 - Fractional decrease in price)

Calculation:

Percentage decrease in price = 30%

Fractional decrease in price = 30 / 100 = 0.3

Percentage increase in consumption = (30 / (1 - 0.3))

Percentage increase in consumption = 30 / 0.7

Percentage increase in consumption = 42⁶⁄₇%

The householder must increase its consumption by 42⁶⁄₇% to not decrease its expenditure on sugar.

Given:

The ratio of the ages of Ram and Shyam is 5 ∶ 3.

Ram is 12 years older than Shyam.

Formula used:

If the ratio of ages is \(a ∶ b\) and the difference is given, then:

\(\text{Difference} = \text{(Difference in ratio)} \times \text{Age multiplier}\).

Find the multiplier to calculate actual ages.

Calculations:

Let the common multiplier of the ratio be x.

Ram's age = 5x, Shyam's age = 3x.

Ram is 12 years older than Shyam:

⇒ \(5x - 3x = 12\)

⇒ \(2x = 12\)

⇒ \(x = 6\)

Shyam's age = \(3x = 3 \times 6 = 18\).

∴ Shyam's age is 18 years.

Given:

Speed of the thief = 70 km/h.

Speed of the policeman = 84 km/h.

The policeman started chasing after 1 hour.

Formula Used:

Time taken by the policeman to catch the thief = Distance / Relative Speed

Calculation:

Distance covered by the thief in 1 hour = 70 km

Relative Speed = Speed of the policeman - Speed of the thief

Relative Speed = 84 km/h - 70 km/h = 14 km/h

Time taken by the policeman to catch the thief = Distance / Relative Speed

⇒ Time taken by the policeman to catch the thief = 70 km / 14 km/h

⇒ Time taken by the policeman to catch the thief = 5 hours

The policeman had to drive for 5 hours before catching the thief.

Given:

Diameter of the cylinder = 32 m

Radius of the cylinder (r) = 32 / 2 = 16 m

Height of the cylinder (h_cyl) = 14 m

Height of the cone from the ground (h_total) = 26 m

Height of the cone (h_cone) = h_total - h_cyl = 26 - 14 = 12 m

Formula Used:

Surface area of the cylinder = 2πr h_cyl

Surface area of the cone = πr√(r² + h_cone²)

Total area of the canvas = Surface area of the cylinder + Surface area of the cone

Calculation:

Surface area of the cylinder:

⇒ 2πr h_cyl

⇒ 2 × 3.14 × 16 × 14

⇒ 1406.72 m²

Surface area of the cone:

⇒ πr√(r² + h_cone²)

⇒ 3.14 × 16 × √(16² + 12²)

⇒ 3.14 × 16 × √(256 + 144)

⇒ 3.14 × 16 × √400

⇒ 3.14 × 16 × 20

⇒ 1004.80 m²

Total area of the canvas:

⇒ Surface area of the cylinder + Surface area of the cone

⇒ 1406.72 + 1004.80

⇒ 2411.52 m²

The area of the canvas required for the tent is 2411.52 m².

Given:

The numbers are 1.7, 2.4, \(\frac{3}{5}\) , 4.34, 2.6, and 5.

Formula Used:

Average = \(\frac{\text{Sum of all numbers}}{\text{Total count of numbers}}\)

Calculation:

Sum of all numbers = 1.7 + 2.4 + \(\frac{3}{5}\) + 4.34 + 2.6 + 5

First, calculate \(\frac{3}{5}\) :

\(\frac{3}{5}\) = 0.6

Now, Sum of all numbers = 1.7 + 2.4 + 0.6 + 4.34 + 2.6 + 5

Sum of all numbers = 16.64

Total count of numbers = 6

Average = \(\frac{16.64}{6}\)

⇒ Average = 2.773

Rounded off to 3 decimal places, the average is 2.773.

The average of the given numbers, rounded off to 3 decimal places, is 2.773.

Given:

Number of workers = 56

Number of days = 5

Wages = Rs. 4080

Formula used:

M1D1/Wages1 = M2D2/Wages2  

M1, M2, D1, D2 are number of men and number of days  

Calculation:

M1 = 56; Wages1 = Rs. 4080, D1 = 5; M2 = 48; and D2 = 14

⇒ (56 × 5)/4080 = (48 × 14)/wages2

⇒ Wages = (48 × 14 × 4080)/(56 × 5)

⇒ Wages = Rs. 9792

∴ 48 workers earn Rs. 9792 in 14 days

Concept used:

A prime number is a whole number greater than 1, whose only factor is 1 and the number itself.

Calculation:

The number between 301 and 320 are:

302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319

Prime numbers between 301 and 320 are:

307, 311, 313,  and 317.

So there are total 4 prime numbers between 301 and 320.

Given:

Marked price of the article = ₹420

Selling price of the article = ₹368

Loss = 8%

Formula Used:

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

Marked Price % above Cost Price = (Marked Price - Cost Price)/(Cost Price) × 100

Calculation:

Let Cost Price be CP.

Loss % = 8%

Loss = CP - Selling Price

⇒ 8% of CP = CP - ₹368

⇒ (8)/(100) × CP = CP - 368

⇒ 0.08CP = CP - 368

⇒ CP - 0.08CP = 368

⇒ 0.92CP = 368

⇒ CP = (368)/(0.92)

⇒ CP = 400

Marked Price % above Cost Price = (Marked Price - Cost Price)/(Cost Price) × 100

⇒ (420 - 400)/(400) × 100

⇒ (20)/(400) × 100

⇒ (1)/(20) × 100

⇒ 5%

The marked price of the article is 5% above the cost price.

Given:

We need to find the sum of all the prime numbers between 57 and 75.

Formula Used:

Sum of prime numbers = Sum of all prime numbers in the given range

Calculation:

Prime numbers between 57 and 75 are: 59, 61, 67, 71, 73

Sum of prime numbers = 59 + 61 + 67 + 71 + 73

⇒ Sum of prime numbers = 59 + 61 = 120

⇒ Sum of prime numbers = 120 + 67 = 187

⇒ Sum of prime numbers = 187 + 71 = 258

⇒ Sum of prime numbers = 258 + 73 = 331

The sum of all the prime numbers between 57 and 75 is 331.

Given:

A sum of money doubles itself in 10 years at compound interest.

Formula Used:

Compound Interest Formula: A = P(1 + r/n)^nt

Where A = amount, P = principal, r = rate of interest, n = number of times interest applied per time period, t = time.

To find the number of years to become sixteen times, we use the formula A = P(1 + r)^t because interest is compounded annually.

Calculation:

Given: A = 2P, t = 10 years

⇒ 2P = P(1 + r)¹⁰

⇒ 2 = (1 + r)¹⁰

We need to find the time when the amount becomes 16 times the principal.

⇒ 16P = P(1 + r)^t

⇒ 16 = (1 + r)^t

Since 2 = (1 + r)¹⁰, we can write:

⇒ 16 = 2⁴ = ((1 + r)¹⁰)^4/10

⇒ 16 = (1 + r)⁴

So, t = 4 × 10 = 40 years.

In 40 years, the sum of money will become sixteen times.

Given:

Population in 2021 = 245000

Population in 2022 = 294000

Formula used:

Percentage Change = \(\dfrac{(New Value - Old Value)}{Old Value} \times 100 \)

Calculation:

Percentage Change = \(\dfrac{(294000 - 245000)}{245000} \times 100 \)

⇒ Percentage Change = \(\dfrac{49000}{245000} \times 100 \)

⇒ Percentage Change = \(\dfrac{1}{5} \times 100 \)

⇒ Percentage Change = 20%

∴ The correct answer is option (1).

Given:

Mohit purchased a table for ₹3,500

Mohit sold the table for ₹3,200

Formula used:

Loss Percentage = \(\frac{Loss}{Cost Price} \times 100\)

Calculation:

Loss = Cost Price - Selling Price

Loss = ₹3,500 - ₹3,200

⇒ Loss = ₹300

Loss Percentage = \(\frac{300}{3500} \times 100 \)

⇒ Loss Percentage = (300 / 3,500) × 100

⇒ Loss Percentage = 8.57%

∴ The correct answer is option (2).

Given:

Radius (r) = 7 cm

Cost per 4 cm² = Rs. 6

Formula used:-

Curved Surface Area of Hemisphere = 2πr²

Total surface area to be painted = Surface area (inside) + Surface area (outside)

Calculations:

Surface area (inside) = 2π(7)² = 2π(49) = 2 × (22/7) × 49

⇒ Surface area (inside) = 308 cm²

Total Surface Area = Inside Surface Area + Outside Surface Area

Total Surface Area = 308 cm2  + 308 cm2             

Total Surface Area = 616 cm2 

The cost of painting this area at the given rate of Rs. 6 per 4 cm2

Cost = (Total Surface Area / 4 cm²)   × Rs. 6

Cost = (616 / 4) × 6 = Rs. 924    

∴The cost of painting of the hemispherical Rs.924

Given:

A alone can finish a work in 12 days

B alone can finish a work in 18 days

Total amount = Rs. 6,000

Formula used:

Work done by A in 1 day = 1/12

Work done by B in 1 day = 1/18

Share of A = (Work done by A in 1 day) / (Work done by A in 1 day + Work done by B in 1 day)

Calculation:

Work done by A in 1 day = 1/12

Work done by B in 1 day = 1/18

⇒ Total work done by A and B in 1 day = 1/12 + 1/18

⇒ Total work done by A and B in 1 day = (3 + 2) / 36

⇒ Total work done by A and B in 1 day = 5/36

Share of A = (1/12) / (5/36)

⇒ Share of A = (1/12) × (36/5)

⇒ Share of A = 3/5

⇒ Amount received by A = (3/5) × 6000

⇒ Amount received by A = 3600

∴ The correct answer is option 2.

Given:

The smallest 5-digit prime number is:

Formula used:

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

Calculations:

We need to check each option to determine the smallest 5-digit prime number.

Option 1: 10007

Check if 10007 is a prime number:

10007 is not divisible by any prime number less than or equal to √10007 (approximately 100.04).

⇒ 10007 is a prime number.

Option 2: 10003

Check if 10003 is a prime number:

Cannot be prime numbers as divisible by 7 

⇒ 10003 is not a prime number.

Option 3: 10001

Check if 10001 is a prime number:

10001 is divisible by 73 (10001 ÷ 73 = 137).

⇒ 10001 is not a prime number.

Option 4: 10009

Check if 10009 is a prime number:

10009 is not divisible by any prime number less than or equal to √10009 (approximately 100.04).

⇒ 10009 is a prime number.

∴ The correct answer is option (1).