Quantitative Aptitude PYQs

Given:

Inradius of the triangle = 7 cm

Area of the triangle = 343 cm²

Formula Used:

Area of triangle = (Inradius × Perimeter) / 2

Calculation:

Let the perimeter of the triangle be P.

Using the formula:

Area = (Inradius × Perimeter) / 2

⇒ 343 = (7 × P) / 2

⇒ 343 = 7P / 2

Multiplying both sides by 2:

⇒ 686 = 7P

Dividing both sides by 7:

⇒ P = 686 / 7

⇒ P = 98 cm

The perimeter of the triangle is 98 cm.

Given:

Principal (P) = Rs. 2,52,500

Rate of interest (R) = 4.5% per annum

Time (T) = 2.5 years

Formula Used:

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

Amount (A) = Principal (P) + Simple Interest (SI)

Calculation:

SI = (2,52,500 × 4.5 × 2.5)/(100)

⇒ SI = (2,52,500 × 11.25)/(100)

⇒ SI = 28,406.25

Amount (A) = 2,52,500 + 28,406.25

⇒ Amount (A) = 2,80,906.25

The amount will be Rs. 2,80,906.25.

Given:

Length of the first train = 210 m

Speed of the first train = 97 km/h

Length of the second train = 240 m

Speed of the second train = 79 km/h

Formula Used:

Time = Distance / Relative Speed

Calculation:

Relative Speed = Speed of the first train - Speed of the second train

Relative Speed = 97 km/h - 79 km/h

Relative Speed = 18 km/h

Convert relative speed to m/s:

Relative Speed = 18 × (1000 / 3600) m/s

Relative Speed = 5 m/s

Distance to be covered = Length of the first train + Length of the second train

Distance = 210 m + 240 m

Distance = 450 m

Time = Distance / Relative Speed

Time = 450 / 5

Time = 90 seconds

The train will take 90 seconds to completely cross the second train.

Given:

Marked price of the book = Rs. 700

Retailer gets a discount of 20%

Retailer sells the book at marked price

Formula Used:

Discounted Price = Marked Price × (1 - Discount Percentage)

Profit Percentage = [(Selling Price - Cost Price) / Cost Price] × 100

Calculation:

Discount Percentage = 20% = 20/100 = 0.2

Discounted Price = 700 × (1 - 0.2)

⇒ Discounted Price = 700 × 0.8

⇒ Discounted Price = 560

Selling Price = Marked Price = Rs. 700

Cost Price = Discounted Price = Rs. 560

Profit Percentage = [(Selling Price - Cost Price) / Cost Price] × 100

⇒ Profit Percentage = [(700 - 560) / 560] × 100

⇒ Profit Percentage = [140 / 560] × 100

⇒ Profit Percentage = 0.25 × 100

⇒ Profit Percentage = 25%

The profit percentage is 25%.

Given:

A person loses 18% of his money and after spending 87% of the remainder, he is left with Rs. 122.

Formula Used:

Let the original amount be x .

After losing 18%, remaining amount = x - 0.18x = 0.82x

After spending 87% of the remainder, amount left = 0.82x - 0.87 × 0.82x = 0.13 × 0.82x = 0.1066x

Calculation:

Given that the amount left is Rs. 122.

⇒ 0.1066x = 122

⇒ x = 122 / 0.1066

⇒ x ≈ 1144.5

The original amount he had was approximately Rs. 1,144.5.

Given:

Principal (P) = Rs. 8,000

Compound Interest (CI) = Rs. 1,261

Time (T) = 18 months = 1.5 years

Interest is compounded half-yearly

Formula Used:

CI = P(1 + r/n)^nt - P

Where,

P = Principal amount

r = annual interest rate

n = number of times interest is compounded per year

t = time in years

Calculation:

Given CI = Rs. 1,261

So, CI + P = Amount (A)

A = 8,000 + 1,261

A = 9,261

n = 2 (since interest is compounded half-yearly)

t = 1.5 years

We need to find the rate (r). Using the formula for compound interest:

A = P(1 + r/n)^nt

9,261 = 8,000(1 + r/2)³

Divide both sides by 8,000:

⇒ 9,261 / 8,000 = (1 + r/2)³

⇒ 1.157625 = (1 + r/2)³

Take the cube root on both sides to solve for r:

⇒ (1.157625)^1/3 = 1 + r/2

⇒ 1.05 = 1 + r/2

⇒ 1.05 - 1 = r/2

⇒ 0.05 = r/2

⇒ r = 0.05 × 2

⇒ r = 0.1

The rate of interest is 0.1 or 10%.

Given:

Length of the first wire = 18 m.

Length of the second wire = 24 m.

Formula Used:

To find the maximum length of each piece, we need to determine the Greatest Common Divisor (GCD) of the lengths of the two wires.

Calculation:

To find the GCD of 18 and 24:

Prime factorization of 18 = 2 × 3²

Prime factorization of 24 = 2³ × 3

GCD is the product of the smallest power of each common prime factor.

Common prime factors are 2 and 3.

Smallest power of 2 = 2¹

Smallest power of 3 = 3¹

⇒ GCD = 2¹ × 3¹

⇒ GCD = 2 × 3

⇒ GCD = 6

The maximum length of each piece is 6 m.

Given:

The average of 5 consecutive odd numbers is 63.

Formula Used:

Average of n consecutive odd numbers = Middle number

Calculation:

Let the 5 consecutive odd numbers be: x, x+2, x+4, x+6, x+8

Given that their average is 63.

Average = (x + (x+2) + (x+4) + (x+6) + (x+8))/(5)

⇒ (5x + 20)/(5) = 63

⇒ 5x + 20 = 315

⇒ 5x = 295

⇒ x = 59

The 5 consecutive odd numbers are: 59, 61, 63, 65, 67

The largest number among them is 67.

Given:

Numbers: 90, 180, 240

Formula Used:

To find the greatest number that divides multiple numbers exactly, use the Greatest Common Divisor (GCD).

Calculation:

Prime factorization:

90 = 2 × 3² × 5

180 = 2² × 3² × 5

240 = 2⁴ × 3 × 5

Find the minimum power of each common prime factor:

2: min(1, 2, 4) = 1

3: min(2, 2, 1) = 1

5: min(1, 1, 1) = 1

GCD = 2¹ × 3¹ × 5¹

GCD = 2 × 3 × 5

⇒ GCD = 30

The greatest number that divides 90, 180, and 240 exactly is 30.

Given:

Initial base = \(b\), initial height = \(h\), initial area = \(\frac{1}{2} \times b \times h\).

New base = \(b + 60%\text{ of }b = 1.6b\).

The area remains the same.

New height = \(h_1\).

Formula used:

Area of triangle = \(\frac{1}{2} \times \text{Base} \times \text{Height}\).

Calculation:

To retain the same area:

\(\frac{1}{2} \times b \times h = \frac{1}{2} \times 1.6b \times h_1\).

Equating the areas:

⇒ \(b \times h = 1.6b \times h_1\)

⇒ \(h = 1.6h_1\)

⇒ \(h_1 = \frac{h}{1.6}\)

Percentage decrease in height:

⇒ \(\text{Decrease} = h - h_1 = h - \frac{h}{1.6} = h \times \left(1 - \frac{1}{1.6}\right)\)

⇒ \(\text{Decrease} = h \times \frac{0.6}{1.6} = h \times 0.375\)

\(\text{Percentage decrease} = 0.375 \times 100 = 37.5%\).

∴ The height should be decreased by 37.5% to retain the same area.

Given:

Interest rate for first 3 years = 4% p.a.

Interest rate for next 4 years = 8% p.a.

Interest rate for period beyond 7 years = 12% p.a.

Simple interest obtained in 11 years = Rs. 27,600

Formula Used:

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

Calculation:

Let the principal amount be Rs. P.

SI for first 3 years at 4% = (P × 4 × 3) / 100 = 12P / 100

SI for next 4 years at 8% = (P × 8 × 4) / 100 = 32P / 100

SI for the last 4 years at 12% = (P × 12 × 4) / 100 = 48P / 100

Total SI for 11 years = 12P / 100 + 32P / 100 + 48P / 100

Total SI for 11 years = (12P + 32P + 48P) / 100

Total SI for 11 years = 92P / 100

Given total SI = Rs. 27,600

Therefore, 92P / 100 = 27,600

⇒ P = (27,600 × 100) / 92

⇒ P = 30,000

The sum lent is Rs. 30,000.

Shortcut Trick

Rate = 20% = 1/5 → (Principal : Amount) = 5 : 6

For 2 years, equate instalments: (5 × 6 : 6 × 6) and (5² : 6²)

Year 1: 30 : 36 | Year 2: 25 : 36

Total Principal = 30 + 25 = 55 units

55 units = ₹66,550 ⇒ 1 unit = ₹1,210

Each instalment = 36 units = 36 × 1,210 = ₹43,560

∴ The correct answer is ₹43,560.

Alternate Method

Given:

Principal (P) = ₹66,550

Rate of Interest (R) = 20% p.a.

Number of instalments (n) = 2

Formula Used:

P = x / (1 + R/100)¹ + x / (1 + R/100)² where x = instalment value.

Calculation:

⇒ 66,550 = x / (1 + 20/100) + x / (1 + 20/100)²

⇒ 66,550 = x / 1.2 + x / 1.44

⇒ 66,550 = (1.2x + x) / 1.44

⇒ 66,550 = 2.2x / 1.44

⇒ x = (66,550 × 1.44) / 2.2

⇒ x = 30,250 × 1.44

⇒ x = 43,560

∴ The correct answer is ₹43,560.

Additional Information

Compound Interest Instalment Formula

For n years, Principal P = x [1/k + 1/k² + ... + 1/kⁿ], where k = (1 + R/100) and x is the annual instalment.

Simple Interest Instalment

In SI, the instalment x = (100 × A) / [100n + RTn(n−1)/2], where A is the total debt amount to be paid.

Effective Rate of Interest

For CI, the effective rate for 2 years is R + R + (R × R)/100. For 20%, it is 20 + 20 + 4 = 44%.

Given:

Price this year (P1) = ₹120

Price after this year (P2) = ₹140

Overall price increase in two years = 25%

Formula used:

The formula for percentage increase is:

\(\text{Percentage Increase} = \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}} \times 100 \)

Calculations:

First, calculate the percentage increase this year:

⇒ \(\text{Percentage Increase} = \frac{140 - 120}{120} \times 100 = 16.67% \)

Now, the total percentage increase after two years is 25%. The remaining percentage increase for the next year is:

⇒ \(\text{Remaining Increase} = 25% - 16.67% = 8.33% \)

However, the percentage increase for the next year should be calculated on the new price (₹140). Let the percentage increase for the next year be \(\text{r}%\):

⇒ \(\text{New Price after two years} = 140 \times \left(1 + \frac{r}{100}\right) = 120 \times 1.25 \)

⇒ \(\text{140} \times \left(1 + \frac{r}{100}\right) = 150 \)

⇒ \(\left(1 + \frac{r}{100}\right) = \frac{150}{140} = 1.0714 \)

⇒ \(\frac{r}{100} = 1.0714 - 1 = 0.0714 \)

⇒ \(\text{r} = 7.14% \)

Thus, the percentage increase in the next year is 7.14%.

Given:

Surface area of the sphere = 2500π cm²

Formula Used:

Surface area of a sphere = 4πr²

Diameter of the sphere = 2r

Calculation:

Let the radius of the sphere be r cm.

Surface area = 4πr²

2500π = 4πr²

Dividing both sides by π:

2500 = 4r²

Dividing both sides by 4:

625 = r²

Taking the square root of both sides:

r = 25 cm

Diameter = 2r

Diameter = 2 × 25

Diameter = 50 cm

The diameter of the sphere is 50 cm.

Given:

The price of an article has been reduced by 20%.

Formula Used:

Restoration Percentage = \(\frac{Reduction Percentage}{(100 - Reduction Percentage)} \times 100\)

Calculation:

Reduction Percentage = 20%

New Price = 100% - 20% = 80%

Restoration Percentage = \(\frac{20}{80} \times 100\)

Restoration Percentage = \( \frac{1}{4} \times 100\)

Restoration Percentage = 25%

Restoration Factor = 25/100 = 0.25

To restore the original price, the new price must be increased by 0.25 or 25%.

Given:

1. 15 ∶ 20

2. 7 ∶ 21

3. 21 ∶ 25

4. 17 ∶ 25

Formula Used:

To find the smallest ratio, convert each ratio to its simplest form and compare.

Calculation:

1. 15 ∶ 20

⇒ 15/20

⇒ 3/4

⇒ 0.75

2. 7 ∶ 21

⇒ 7/21

⇒ 1/3

⇒ 0.333

3. 21 ∶ 25

⇒ 21/25

⇒ 0.84

4. 17 ∶ 25

⇒ 17/25

⇒ 0.68

Comparing all ratios:

0.75, 0.333, 0.84, 0.68

The smallest ratio is 0.333, which corresponds to option 2 (7 ∶ 21).

Given:

Cost price (CP) of guavas = ₹300 per kg.

Selling price (SP) of guavas = ₹25 per 100 gm.

Formula Used:

1 kg = 1000 gm

Profit or Loss percentage = ((SP - CP) / CP) × 100

Calculation:

SP of 1 kg of guavas = 25 × 10 = ₹250

CP of 1 kg of guavas = ₹300

Profit or Loss = SP - CP

⇒ Profit or Loss = 250 - 300

⇒ Profit or Loss = -50

Since the result is negative, it's a loss.

Loss percentage = ((-50) / 300) × 100

⇒ Loss percentage = (-50 / 300) × 100

⇒ Loss percentage = -16.67%

The correct answer is option 1: a loss of 16.67%.

Given:

Number of people = 25

Working hours per day = 15

Work completed = 5 units

Number of days = 8

Number of people to complete new work = 12

Working hours per day for new work = 20

New work to be completed = 10 units

Formula Used:

Work = Number of people × Number of hours per day × Number of days

Calculation:

Let the total work in man-hours be W.

W = 25 × 15 × 8

W = 3000 man-hours for 5 units of work

Therefore, 1 unit of work = 3000 / 5

1 unit of work = 600 man-hours

So, 10 units of work = 600 × 10

10 units of work = 6000 man-hours

Now, we have 12 people working 20 hours a day to complete 10 units of work.

Let the number of days required be D.

Work = 12 × 20 × D

Therefore, 12 × 20 × D = 6000

⇒ D = 6000 / (12 × 20)

⇒ D = 6000 / 240

⇒ D = 25

The number of days required is 25 days.

The correct answer is option 1.

Shortcut Trick

When a solid sphere is moulded into n smaller balls, the ratio of radii is the cube root of the ratio of volumes.

Volume of large sphere = 8 × Volume of small ball

⇒ R³ = 8 × r³

⇒ R = 2 × r

⇒ 20 = 2 × r ⇒ r = 10 cm

∴ The correct answer is 10 cm.

Alternate Method

Given:

Radius of large solid sphere (R) = 20 cm

Number of small spherical balls (n) = 8

Formula Used:

Volume of a sphere = (4/3) × π × r³

Calculations:

⇒ Volume of large sphere = 8 × Volume of one small ball

⇒ (4/3) × π × (20)³ = 8 × (4/3) × π × r³

⇒ 20 × 20 × 20 = 8 × r³

⇒ 8000 = 8 × r³

⇒ r³ = 1000

⇒ r = ³√1000

⇒ r = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Surface Area of Sphere

The total surface area of a solid sphere is given by the formula 4 × π × r².

Volume Conservation

When one solid object is recast or moulded into another shape, the total volume remains constant.

Hemisphere Properties

Volume of hemisphere = (2/3)πr³ and Total Surface Area = 3πr².

Given:

Marked price of the bag = Rs. 1,050

Discount offered = 10%

Profit made = 5%

Formula Used:

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

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

Calculation:

Discount = 10% of Marked Price

Discount = 10% of Rs. 1,050

Discount = Rs. 105

Selling Price (SP) = Marked Price - Discount

SP = Rs. 1,050 - Rs. 105

SP = Rs. 945

Profit% = 5% = 5/100 = 0.05

Cost Price (CP) = SP / (1 + 0.05)

⇒ CP = Rs. 945 / 1.05

⇒ CP = Rs. 900

The cost price of the bag is Rs. 900.