Quantitative Aptitude PYQs

Given:

The sides of the triangle are 5 cm, 6 cm, and 8 cm.

Formula Used:

Area of a triangle with sides a, b, and c is calculated using Heron's formula:

Area = \( \sqrt{s(s-a)(s-b)(s-c)} \)

where s is the semi-perimeter of the triangle, s = (a + b + c) / 2

Calculation:

First, calculate the semi-perimeter (s):

s = (5 + 6 + 8) / 2

s = 19 / 2

s = 9.5 cm

Then, apply Heron's formula:

Area = \( \sqrt{9.5(9.5-5)(9.5-6)(9.5-8)} \)

Area = \( \sqrt{9.5 \times 4.5 \times 3.5 \times 1.5} \)

Area = \( \sqrt{224.4375} \)

Area ≈ 14.98 cm²

The area of the triangle is approximately 14.98 cm², which is option 4.

Given:

A completes 2/3 of the job in 8 days.

B completes the remaining 1/3 of the job in 20 days.

We need to find how many days A and B can complete the entire job if they work together.

Calculation:

Step 1: Find A's work rate

A completes 2/3 of the job in 8 days. Therefore, A's work rate per day is:

A's rate = (2/3) ÷ 8 = 2/24 = 1/12

Step 2: Find B's work rate

B completes 1/3 of the job in 20 days. Therefore, B's work rate per day is:

B's rate = (1/3) ÷ 20 = 1/60

Step 3: Combined work rate of A and B

When A and B work together, their combined work rate is:

Combined rate = A's rate + B's rate

Combined rate = 1/12 + 1/60

Find the LCM of 12 and 60:

Combined rate = 5/60 + 1/60 = 6/60 = 1/10

Step 4: Total time to complete the job

The time to complete the entire job when working together is the reciprocal of their combined rate:

Time = 1 ÷ (1/10) = 10 days

A and B can complete the job together in 10 days.

Given:

Radius of the cone (r) = 4 cm

Height of the cone (h) = 3 cm

Formula Used:

Volume of a cone = (1/3) × π × r² × h

Calculation:

Volume of the cone

⇒ (1/3) × π × 4² × 3

⇒ (1/3) × π × 16 × 3

⇒ (1/3) × π × 48

⇒ 16π cm³

The volume of the cone is 16π cm³.

Given:

Discount received by Ram = 12%

Savings = Rs. 600

Formula Used:

Discount = (Discount Percentage × Marked Price) / 100

Marked Price = Savings / (Discount Percentage / 100)

Amount Paid = Marked Price - Discount

Calculation:

Let the marked price be M.

Given, 12% of M = Rs. 600

⇒ 12/100 × M = 600

⇒ M = 600 × 100 / 12

⇒ M = 5000

The discount is 12% of M , i.e. Rs. 600, so:

Amount Paid = Marked Price - Discount

⇒ Amount Paid = 5000 - 600

⇒ Amount Paid = 4400

Ram paid Rs. 4400 for the fan.

Given:

Twin prime numbers are prime numbers whose difference is always equal to 2.

We need to find the number of twin primes between 35 and 100.

Formula Used:

Identify the twin prime pairs between the given range.

Calculation:

Prime numbers between 35 and 100 are: 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Identify twin primes:

(37, 39) - not primes

(41, 43) - twin primes

(43, 45) - not primes

(47, 49) - not primes

(53, 55) - not primes

(59, 61) - twin primes

(61, 63) - not primes

(67, 69) - not primes

(71, 73) - twin primes

(73, 75) - not primes

(79, 81) - not primes

(83, 85) - not primes

(89, 91) - not primes

(97, 99) - not primes

Therefore, the twin prime pairs are: (41, 43), (59, 61), (71, 73)

The number of twin primes between 35 and 100 is 3.

Given:

Length of the train = 300 m

Length of the platform = 5 × length of the train = 5 × 300 m = 1500 m

Total distance to be covered = Length of the train + Length of the platform = 300 m + 1500 m = 1800 m

Time taken = 7 minutes 40 seconds = 7 × 60 + 40 = 460 seconds

Formula Used:

Speed = Total distance / Time

Calculation:

Total distance = 1800 m

Time = 460 seconds

Speed = 1800 / 460

Speed = \( \frac{1800}{460} \) m/s

Speed = \( \frac{1800}{460} = \frac{1800 \div 20}{460 \div 20} = \frac{90}{23} \) m/s

Speed = \( 3 \frac{21}{23} \mathrm{~m} / \mathrm{s} \)

The speed of the train is \( 3 \frac{21}{23} \mathrm{~m} / \mathrm{s} \).

Given:

Three numbers are co-prime to each other.

Product of the first two numbers = 391

Product of the last two numbers = 943

Formula Used:

Since the numbers are co-prime, they have no common factors other than 1.

Prime factorization of the product will give the individual numbers.

Calculation:

Prime factorization of 391:

391 = 17 × 23

Prime factorization of 943:

943 = 23 × 41

Since 23 is common in both factorizations, it is the middle number.

The middle number is 23.

Given:

Numbers: 4, 3, 7, -3, 0, -5, -7, 5, -6, 0, -4, 6

Formula Used:

Average = (Sum of the numbers) / (Total number of elements)

Calculation:

Sum of the numbers = 4 + 3 + 7 + (-3) + 0 + (-5) + (-7) + 5 + (-6) + 0 + (-4) + 6

Sum of the numbers = 4 + 3 + 7 - 3 + 0 - 5 - 7 + 5 - 6 + 0 - 4 + 6

Sum of the numbers = 0

Total number of elements = 12

Average = (Sum of the numbers) / (Total number of elements)

Average = 0 / 12

⇒ Average = 0

The correct answer is option 4.

Given:

We need to find the smallest number which, when divided by 15, 17, and 19, leaves remainders 4, 6, and 8, respectively.

Calculation:

Step 1: General form of the problem

If a number leaves remainders r1, r2, r3 when divided by divisors d1, d2, d3, the number satisfies:

N ≡ r1 (mod d1)

N ≡ r2 (mod d2)

N ≡ r3 (mod d3)

Here:

N ≡ 4 (mod 15)

N ≡ 6 (mod 17)

N ≡ 8 (mod 19)

Step 2: Adjust the congruences

Rewrite the congruences by subtracting the remainder from the divisor:

N - 4 is divisible by 15, so N ≡ -11 (mod 15).

N - 6 is divisible by 17, so N ≡ -11 (mod 17).

N - 8 is divisible by 19, so N ≡ -11 (mod 19).

In simplified terms, N ≡ -11 (mod LCM(15, 17, 19)).

Step 3: Calculate the LCM

The LCM of 15, 17, and 19 (all are coprime) is:

LCM(15, 17, 19) = 15 × 17 × 19 = 4845.

Step 4: Solve for N

Since N ≡ -11 (mod 4845), the smallest positive N is:

N = 4845 - 11 = 4834.

The smallest number is 4834.

Given:

Initial investment amount = 100 (assuming 100 for simplicity)

Share rose by 25% on the first day

Share fell by 8% on the second day

Share fell by 10% on the third day

Formula Used:

Final Amount = Initial Amount × (1 + Percentage Change)

Percentage Profit/Loss = \(\frac{(Final\ Amount - Initial\ Amount)}{Initial\ Amount} \times 100\)

Calculation:

Initial Amount = 100

After first day rise of 25%:

Final Amount = 100 × (1 + 25/100)

⇒ Final Amount = 100 × 1.25 = 125

After second day fall of 8%:

Final Amount = 125 × (1 - 8/100)

⇒ Final Amount = 125 × 0.92 = 115

After third day fall of 10%:

Final Amount = 115 × (1 - 10/100)

⇒ Final Amount = 115 × 0.90 = 103.5

Percentage Profit/Loss:

Percentage Profit/Loss = \(\frac{(103.5 - 100)}{100} \times 100\)

⇒ Percentage Profit/Loss = 3.5%

Venu made a 3.5% profit.

Shortcut Trick

Use the weighted average deviation method from the target 30%.

Concentration A (15%) is −15% from 30%. Concentration B (30%) is 0% from 30%. Concentration C (45%) is +15% from 30%.

Net deviation = p × (−15) + 5 × (0) + 3 × (+15) = 0

⇒ −15p + 45 = 0 ⇒ 15p = 45 ⇒ p = 3

∴ The correct answer is 3.

Alternate Method

Given: Concentrations of Wine A = 15%, Wine B = 30%, Wine C = 45%. Mixing ratio = p : 5 : 3. Final concentration = 30%.

Formula Used: Final Concentration = (C₁V₁ + C₂V₂ + C₃V₃) ÷ (V₁ + V₂ + V₃)

Calculations:

⇒ 30 = (15 × p + 30 × 5 + 45 × 3) ÷ (p + 5 + 3)

⇒ 30 × (p + 8) = 15p + 150 + 135

⇒ 30p + 240 = 15p + 285

⇒ 30p − 15p = 285 − 240

⇒ 15p = 45

⇒ p = 3

∴ The correct answer is 3.

Additional Information

Weighted Average Rule

The average of a mixture depends on the quantity of each component: Average = Σ(Value_i × Quantity_i) ÷ ΣQuantity_i.

Alligation for Three Components

When mixing three items, the mean value must lie between the lowest and highest concentrations. It can be solved by pairing the mean with values above and below it.

Given:

Radius of the sphere = 1.8 cm.

Formula Used:

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

Calculation:

Let r = 1.8 cm

Volume = \(\frac{4}{3} \pi (1.8)^3\)

Volume = \(\frac{4}{3} \pi (1.8 × 1.8 × 1.8)\)

Volume = \(\frac{4}{3} \pi (5.832)\)

Volume = \(\frac{4 × 5.832}{3} \pi\)

Volume = \(\frac{23.328}{3} \pi\)

Volume = 7.776 π cm³

The exact volume of the sphere is 7.776 π cm³.

Given:

Expression: \(\frac{(4.7)^2 - (1.3)^2}{4.7 - 1.3}\)

Formula Used:

\((a^2 - b^2)\) = (a - b)(a + b)

Calculation:

\(\frac{(4.7)^2 - (1.3)^2}{4.7 - 1.3}\)

= \(\frac{(4.7 - 1.3)(4.7 + 1.3)}{4.7 - 1.3}\)

= (4.7 + 1.3) = 6

The correct answer is option 4.

Given:

(8888 ÷ 88) ÷ 5

Formula Used:

Simplification of the arithmetic expression

Calculation:

First, simplify the expression inside the parentheses:

8888 ÷ 88

⇒ 8888 ÷ 88 = 101

Next, divide the result by 5:

101 ÷ 5

⇒ 101 ÷ 5 = 20.2 = \(20\frac{1}{5}\)

Therefore, the correct answer is \(20\frac{1}{5}\).

Given:

Side of the square field = 63 m

Each horse is roped such that it can touch the middle of each side, hence the rope length = 63/2 = 31.5 m

π = 22/7

Formula Used:

Area of square = side²

Area grazed by one horse (quarter circle) = (1/4) × π × radius²

Remaining area = Area of square - 4 × Area grazed by one horse

Calculation:

Area of the square field = 63²

Area of the square field = 3969 m²

Radius of the quarter circle = 31.5 m

Area grazed by one horse = (1/4) × (22/7) × 31.5²

⇒ Area grazed by one horse = (1/4) × (22/7) × 992.25

⇒ Area grazed by one horse = (22 × 992.25) / 28

⇒ Area grazed by one horse = 21829.5 / 28

⇒ Area grazed by one horse ≈ 779.625 m²

Total area grazed by 4 horses = 4 × 779.625

Total area grazed by 4 horses ≈ 3118.5 m²

Remaining area = Area of square field - Total area grazed by 4 horses

⇒ Remaining area = 3969 - 3118.5

⇒ Remaining area ≈ 850.5 m²

The remaining area of the ground where the horses do not graze is 850.5 m².

Shortcut Trick

Convert the numbers into like decimals: 3.60, 0.54, 1.08.

Treat them as integers by removing the decimal: 360, 54, 108.

Find the HCF of 360, 54, and 108:

Factors of 54: 18 × 3 | 108: 18 × 6 | 360: 18 × 20. Thus, HCF = 18.

Apply the 2 decimal places back to the result: 0.18.

∴ The correct answer is 0.18.

Alternate Method

Given: Numbers are 3.6, 0.54, and 1.08.

Formula Used: HCF of Decimals = (HCF of numbers treated as integers) ÷ 10ⁿ (where n is the maximum decimal places).

Step-by-step calculation:

⇒ Convert numbers to fractions with same denominator: 360/100, 54/100, 108/100.

⇒ HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators).

⇒ Numerators are 360, 54, 108.

⇒ 54 = 2 × 3³

⇒ 108 = 2² × 3³

⇒ 360 = 2³ × 3² × 5

⇒ HCF (360, 54, 108) = 2¹ × 3² = 2 × 9 = 18.

⇒ Denominator LCM(100, 100, 100) = 100.

⇒ Final HCF = 18 ÷ 100 = 0.18.

∴ The correct answer is 0.18.

Additional Information

HCF of Fractions

To find the HCF of fractions, use the formula: HCF = (HCF of Numerators) ÷ (LCM of Denominators).

LCM of Decimals

Similar to HCF, convert decimals to integers by multiplying by 10ⁿ, find their LCM, and then divide by 10ⁿ.

Relationship Rule

For any two numbers A and B, the relationship is: A × B = HCF(A, B) × LCM(A, B).

Given:

Principal (P) = ₹2,196

Rate (R) = 25%

Time (T) = 3 years

Formula Used: Compound Interest Formula: \(A = P \left(1 + \frac{R}{100}\right)^T\)

Instalment Formula: \(A = \frac{P \times \left(1 + \frac{R}{100}\right)^T \times \frac{R}{100}}{ \left(1 + \frac{R}{100}\right)^T - 1}\)

Calculation:

First, calculate the amount (A) to be paid after 3 years:

Rate (R) = 25% = 0.25

Time (T) = 3 years

Amount (A) after 3 years:

\(A = 2196 \left(1 + 0.25\right)^3\)

\(A = 2196 \left(1.25\right)^3\)

\(A = 2196 \times 1.953125\)

A = 4290

Now, calculate the annual instalment:

Instalment (I):

\(I = \frac{2196 \times 1.953125 \times 0.25}{1.953125 - 1}\)

\(I = \frac{1072.5}{0.953125}\)

I = 1125

The amount of instalment is ₹1125.

Given:

If A ∶ B ∶ C = 2 ∶ 3 ∶ 5

Formula Used:

To find the ratio (A)/(B):(B)/(C):(C)/(A)

Calculation:

Given A ∶ B ∶ C = 2 ∶ 3 ∶ 5

Let A = 2k, B = 3k, and C = 5k

Then, (A)/(B) = (2k)/(3k) = 2/3

(B)/(C) = (3k)/(5k) = 3/5

(C)/(A) = (5k)/(2k) = 5/2

The ratio (A)/(B) ∶ (B)/(C) ∶ (C)/(A) is (2/3) ∶ (3/5) ∶ (5/2)

To simplify, find a common denominator:

2/3 = 2 × 10 / 3 × 10 = 20 / 30

3/5 = 3 × 6 / 5 × 6 = 18 / 30

5/2 = 5 × 15 / 2 × 15 = 75 / 30

The ratio is 20 ∶ 18 ∶ 75

The correct answer is option 1.

Given:

Length of the train = 750 m

Length of the platform = 2 × 750 m = 1500 m

Speed of the train = 81 km/h

Formula Used:

Time = Total Distance / Speed

Calculation:

Speed in m/s = 81 × (1000/3600)

Speed in m/s = 81 × (5/18)

Speed in m/s = 22.5 m/s

Total Distance = Length of the train + Length of the platform

Total Distance = 750 m + 1500 m

Total Distance = 2250 m

Time = Total Distance / Speed

⇒ Time = 2250 / 22.5

⇒ Time = 100 seconds

The time taken by the train to cross the platform is 100 seconds.

Given:

A number y, when reduced by 25%, becomes 120% of a second number z.

Formula Used:

Percentage of z that y is = (y / z) × 100

Calculation:

Let the original number y be y . When reduced by 25%, it becomes 0.75y .

Given that 0.75y = 1.2z

Solving for y in terms of z:

⇒ y = \(\frac{1.2z}{0.75}\)

⇒ y = \(\frac{1.2}{0.75}z\)

⇒ y = 1.6z

Therefore, y is 160% of z.

Hence, the correct answer is option 1.