Quantitative Aptitude PYQs

Given:

A sum was invested at a certain rate of simple interest per annum for 2 years. Had it been invested at 3% per annum more rate than the existing rate, the simple interest accrued in the 2 years would have been ₹240 more.

Formula Used:

Simple Interest = (Principal × Rate × Time) / 100

Calculation:

Let the principal be ₹P and the original rate of interest be R%.

Simple interest at the original rate for 2 years = (P × R × 2) / 100

Simple interest at the increased rate (R+3)% for 2 years = (P × (R + 3) × 2) / 100

According to the question, the difference in interest is ₹240:

\(\Rightarrow \frac{P \times (R + 3) \times 2}{100} - \frac{P \times R \times 2}{100} = 240\)

\(\Rightarrow \frac{P \times (2R + 6 - 2R)}{100} = 240\)

\(\Rightarrow \frac{2P \times 3}{100} = 240\)

\(\Rightarrow \frac{6P}{100} = 240\)

\(\Rightarrow 6P = 240 \times 100\)

\(\Rightarrow P = \frac{240 \times 100}{6}\)

\(\Rightarrow P = 4000\)

The sum invested was ₹4,000.

Given:

Cost price of the bike for Deepak = ₹8,000

Profit made by Deepak = 40%

Loss incurred by Sharath = 30%

Formula Used:

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

Selling Price (SP) = Cost Price (CP) × (1 - Loss%)

Calculation:

Let's find the selling price of the bike when Deepak sold it to Sharath.

SP_Sharath = CP_Deepak × (1 + 40%)

SP_Sharath = 8000 × 1.40

SP_Sharath = ₹11,200

Now, let's find the cost price of the bike for Sivaji when Sharath sold it at a loss of 30%.

CP_Sivaji = SP_Sharath × (1 - 30%)

CP_Sivaji = 11,200 × 0.70

CP_Sivaji = ₹7,840

The cost price of the bike for Sivaji is ₹7,840.

Given:

Marked price (MP) of a book = ₹767

Discount = 20%

Profit = 30%

Formula Used:

Selling Price (SP) = MP × (1 - Discount%)

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

Calculation:

Discount = 20% of ₹767 = 0.20 × 767 = ₹153.40

SP = MP - Discount

SP = ₹767 - ₹153.40 = ₹613.60

Profit = 30%

CP = SP / (1 + Profit)

CP = ₹613.60 / 1.30

CP = ₹472

The cost price of the book is ₹472.

Given:

P's target marks = 360.

Total marks = 500.

P gets 5% less than the total marks, not the target marks.

We need to calculate how much P actually gets.

Solution:

Step 1: Calculate 5% of the total marks

5% of 500 = (5/100) × 500 = 25 marks.

Step 2: Subtract this from the target marks

P's actual marks = Target marks - 5% of total marks.

P's actual marks = 360 - 25 = 335 marks.

P gets 335 marks.

Given:

A and B together can complete a work in 15 days.

B and C together can complete a work in 12 days.

C and A together can complete a work in 10 days.

Formula Used:

1/A + 1/B + 1/C = 1/AB + 1/BC + 1/CA

Calculation:

Let the total work be W.

Work done by A and B in one day = W/15

Work done by B and C in one day = W/12

Work done by C and A in one day = W/10

Adding all three equations:

(W/15) + (W/12) + (W/10) = 2(1/A + 1/B + 1/C)

⇒ (4W + 5W + 6W) / 60 = 2(1/A + 1/B + 1/C)

⇒ 15W / 60 = 2(1/A + 1/B + 1/C)

⇒ 1/4 = 2(1/A + 1/B + 1/C)

⇒ 1/8 = 1/A + 1/B + 1/C

From the individual pairs:

1/A + 1/B = 1/15

1/B + 1/C = 1/12

1/C + 1/A = 1/10

Adding the three pairs:

2(1/A + 1/B + 1/C) = 1/15 + 1/12 + 1/10

⇒ 2(1/8) = 1/15 + 1/12 + 1/10

⇒ 1/4 = (4 + 5 + 6) / 60

⇒ 1/4 = 15/60

⇒ 1/4 = 1/4 (Verified)

Now, isolating 1/B:

1/B = (1/8) - (1/10)

⇒ 1/B = (5 - 4) / 40

⇒ 1/B = 1/40

Therefore, B alone can complete the work in 40 days.

The correct answer is option 4.

Given:

Interest rate per period (4 months) = 3%

Number of periods in a year = 12 / 4 = 3

Formula Used:

Effective annual rate (EAR) = \((1 + \frac{r}{n})^{n} - 1\)

where:

r = nominal annual interest rate

n = number of compounding periods per year

Calculation:

Given r = 3% and n = 3:

Effective annual rate (EAR)

⇒ EAR = \((1 + \frac{0.03}{3})^{3} - 1\)

⇒ EAR = (1 + 0.01)³  - 1

⇒ EAR = (1.01)³ - 1

⇒ EAR = 1.030301 - 1

⇒ EAR = 0.030301

⇒ EAR = 3.03%

The effective rate of interest is 3.03%.

Given:

Numbers: 6, 7, 12, (x - 1), and 10

Their average = 10

Formula used:

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

Calculation:

Step 1: Write the formula for average:

(6 + 7 + 12 + (x - 1) + 10) / 5 = 10

Step 2: Simplify the numerator:

(6 + 7 + 12 + x - 1 + 10) / 5 = 10

(34 + x) / 5 = 10

Step 3: Multiply both sides by 5:

34 + x = 50

Step 4: Solve for x:

x = 50 - 34

x = 16

∴ The value of x is 16.

Given:

a : b = 5 : 3

We need to find the ratio:

(a³ - b³) : (a³ + b³)

Formula used:

The cube ratio of two terms a³ and b³ can be calculated directly using:

If a : b = m : n, then:

(a³ - b³) : (a³ + b³) = (m³ - n³) : (m³ + n³)

Calculation:

Step 1: Substitute the given ratio a : b = 5 : 3 into the formula.

m = 5, n = 3

Step 2: Calculate m³ and n³:

m³ = 5³ = 125

n³ = 3³ = 27

Step 3: Find (m³ - n³) and (m³ + n³):

(m³ - n³) = 125 - 27 = 98

(m³ + n³) = 125 + 27 = 152

Step 4: Write the ratio:

(a³ - b³) : (a³ + b³) = 98 : 152

Step 5: Simplify the ratio:

98 : 152 = 49 : 76

∴ The ratio is 49 : 76.

Given:

Two numbers a and b such that:

1. a and (b - a) are co-primes
2. b and (b + a) are co-primes

Formula Used:

If two numbers are co-prime, they have no common divisors other than 1.

Calculation:

Given that a and (b - a) are co-primes:

⇒ gcd(a, b - a) = 1

Given that b and (b + a) are co-primes:

⇒ gcd(b, b + a) = 1

Since gcd(b, b + a) = gcd(b, a), we have:

⇒ gcd(b, a) = 1

This implies that a and b are co-prime.

Therefore, the correct statement is that a and b are co-prime.

Given:

The shares rose successively by 20% and 30% on two consecutive days and fell by 25% and 11% in the next two days.

Formula Used:

Percentage increase or decrease = ((Final Value - Initial Value) / Initial Value) × 100 %

Calculation:

Let the initial value of the shares be 100.

After the first day, the value increases by 20%:

Value = 100 × 1.20 = 120

After the second day, the value increases by 30%:

Value = 120 × 1.30 = 156

After the third day, the value decreases by 25%:

Value = 156 × 0.75 = 117

After the fourth day, the value decreases by 11%:

Value = 117 × 0.89 = 104.13

Percentage change = ((104.13 - 100) / 100) × 100 %

Percentage change = 4.13%

Therefore, the percentage increase in the value of the shares in four days is 4.13%.

Given:

We need to find the number between 2800 and 3000 that leaves no remainder when divided by 24, 32, and 45.

Formula Used:

We need to find the Least Common Multiple (LCM) of 24, 32, and 45.

Calculation:

Prime factorization of 24: 24 = 2³ × 3

Prime factorization of 32: 32 = 2⁵

Prime factorization of 45: 45 = 3² × 5

LCM = 2⁵ × 3² × 5

LCM = 32 × 9 × 5

LCM = 1440

Now, we need to find the multiple of 1440 between 2800 and 3000.

Multiples of 1440: 1440, 2880, 4320, ...

The number between 2800 and 3000 is 2880.

The correct answer is option 4: 2880

Given:

x = 6/7 and y = 1/x

We need to find how much more y is than x.

Formula used:

The difference between y and x is:

Difference = y - x

Percentage more is:

Percentage more = [(y - x) / x] × 100

Calculation:

Step 1: Calculate y:

y = 1/x = 1 / (6/7) = 7/6

Step 2: Find the difference between y and x:

Difference = y - x = (7/6) - (6/7)

Difference = [(49 - 36) / 42] = 13/42

Step 3: Calculate the percentage more:

Percentage more = [(y - x) / x] × 100

Percentage more = [(13/42) / (6/7)] × 100

Percentage more = [(13/42) × (7/6)] × 100

Percentage more = [(13 × 7) / (42 × 6)] × 100

Percentage more = [91 / 252] × 100

Percentage more ≈ 36.11%

∴ y is approximately 36.11% more than x.

Given:

Anand can do a piece of work in 9 days.

Vinod can complete the same work in 6 days.

Formula Used:

Work rate = 1 / Number of days

Combined work rate = Sum of individual work rates

Time to complete work together = 1 / Combined work rate

Calculation:

Anand's work rate = 1 / 9

Vinod's work rate = 1 / 6

Combined work rate = (1 / 9) + (1 / 6)

⇒ Combined work rate = (2 + 3) / 18

⇒ Combined work rate = 5 / 18

Time to complete work together = 1 / (5 / 18)

⇒ Time to complete work together = 18 / 5

The correct answer is option 3.

Given:

Radii of the three spherical balls: 30 cm, 40 cm, 50 cm

Cost of paint per cm²: ₹0.60

Formula Used:

Surface area of a sphere = 4πr²

Total cost = Surface area × Cost per cm²

Calculation:

Surface area of the first ball = 4π(30)²

Surface area of the first ball = 4π(900)

Surface area of the first ball = 3600π cm²

Surface area of the second ball = 4π(40)²

Surface area of the second ball = 4π(1600)

Surface area of the second ball = 6400π cm²

Surface area of the third ball = 4π(50)²

Surface area of the third ball = 4π(2500)

Surface area of the third ball = 10000π cm²

Total surface area = 3600π + 6400π + 10000π cm²

Total surface area = 20000π cm²

Cost for one π cm2  = ₹0.60

Total cost for 20000π cm2 = 0.60 × 20000

Total cost = ₹12000

The total cost of paint required to paint the surfaces of all three balls is ₹12000.

Given:

Simplify the following:

\(\sqrt{\frac{0.009 \times 0.025 \times 0.016 \times 0.08}{0.002 \times 0.0008 \times 0.0002}}\)

Calculation:

First, calculate the numerator:

0.009 × 0.025 = 0.000225

0.000225 × 0.016 = 0.0000036

0.0000036 × 0.08 = 0.000000288

Next, calculate the denominator:

0.002 × 0.0008 = 0.0000016

0.0000016 × 0.0002 = 0.00000000032

Now, divide the numerator by the denominator:

0.000000288 / 0.00000000032 = 900

Finally, take the square root of the result:

√900 = 30

The correct answer is option 4.

Given:

Marked price (MP) = (9/8) of Cost Price (CP)

Profit desired = 3%

Formula Used:

Discount = (MP - Selling Price)/MP × 100

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

Calculation:

Let CP = 100

MP = (9/8) × 100 = 112.5

SP = 100 × (1 + 3/100) = 100 × 1.03 = 103

Discount = (112.5 - 103)/112.5 × 100

⇒ Discount = 9.5/112.5 × 100

⇒ Discount = 0.0844 × 100

⇒ Discount = 8.44%

The correct answer is option 3.

Given:

The circumference of the great circle of a hemisphere = 9π cm.

Formula Used:

1. Circumference of a circle = 2πr
2. Total surface area of a hemisphere = 3πr²

Calculation:

Circumference of the great circle of the hemisphere = 2πr

⇒ 2πr = 9π

⇒ r = 9π / 2π

⇒ r = 4.5 cm

Total surface area of the hemisphere = 3πr²

⇒ Total surface area = 3π(4.5)²

⇒ Total surface area = 3π × 20.25

⇒ Total surface area = 60.75π cm²

The total surface area of the hemisphere is 60.75π cm².

Shortcut Trick

For 2 years, the difference between CI and SI is calculated directly using the formula: Difference = P × (R / 100)²

⇒ 98 = 80000 × (R / 100)²

⇒ R² = (98 × 10000) ÷ 80000

⇒ R² = 98 ÷ 8 = 12.25

⇒ R = √12.25 = 3.5% = 3½%

∴ The correct answer is 3½%.

Alternate Method

Given:

Principal (P) = ₹80,000

Time (T) = 2 years

Difference between CI and SI (D) = ₹98

Let the rate of interest be R% per annum.

Formula Used:

Difference (D) = P × (R / 100)²

Calculations:

⇒ 98 = 80000 × (R / 100)²

⇒ 98 = 80000 × (R² / 10000)

⇒ 98 = 8 × R²

⇒ R² = 98 ÷ 8

⇒ R² = 49 ÷ 4

⇒ R = √(49 ÷ 4)

⇒ R = 7 ÷ 2

⇒ R = 3.5% or 3½%

∴ The correct answer is 3½%.

Additional Information

Difference between CI and SI for 3 years

The difference between compound interest and simple interest for 3 years is calculated using the formula: D = P × (R / 100)² × (300 + R) ÷ 100.

Effective Rate for 2 years

The effective compound interest rate for 2 years at R% per annum is (2R + R² ÷ 100)%. The effective simple interest rate is 2R%. The difference is (R² ÷ 100)% of the principal.

Tree Method for Compound Interest

In the tree method, the second year's CI includes the interest on the principal plus the interest on the first year's interest. This interest on interest is exactly the difference between CI and SI for 2 years.

Given:

Selling price (SP) = ₹4,800

Gain% = 20%

New Selling price (SP_new) = ₹4,500

Formula Used:

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

Gain% or Loss% = [(SP_new - CP) / CP] × 100

Calculation:

Cost Price (CP) = ₹4,800 / (1 + 0.20)

⇒ CP = ₹4,800 / 1.20

⇒ CP = ₹4,000

Gain% or Loss% = [(₹4,500 - ₹4,000) / ₹4,000] × 100

⇒ Gain% or Loss% = [₹500 / ₹4,000] × 100

⇒ Gain% or Loss% = 12.5%

Therefore, the gain percentage if it had been sold for ₹4,500 is 12.5%.

Given:

The train covers a distance of 54 km from station A to station G in an hour, stopping at stations B, C, D, E, and F for 72 seconds each.

Formula Used:

Speed = Total distance / Total time taken

Calculation:

Time taken for stops:

There are 5 stops, each for 72 seconds.

Total stopping time = 5 × 72 seconds

Total stopping time = 360 seconds

Total stopping time in hours = 360 / 3600

Total stopping time in hours = 0.1 hours

Total time taken to cover the distance:

Total travel time = 1 hour - 0.1 hour

Total travel time = 0.9 hours

Speed of the train:

Distance = 54 km

Time = 0.9 hours

Speed = Distance / Time

Speed = 54 / 0.9

Speed = 60 km/h

The speed of the train is 60 km/h.