Quantitative Aptitude PYQs

Shortcut Trick

Let the breadth of the field be x m, then the length is 3x m.

Perimeter of the rectangular field = Total Fencing Cost ÷ Fencing Rate = 960 ÷ 7.5 = 128 m.

Perimeter = 2 × (Length + Breadth) ⇒ 2 × (3x + x) = 128 ⇒ 8x = 128 ⇒ x = 16 m.

Difference between length and breadth = 3x − x = 2x = 2 × 16 = 32 m.

∴ The correct answer is 32 m.

Alternate Method

Given:

Ratio of length to breadth = 3 : 1

Total cost of fencing = ₹960

Rate of fencing = ₹7.5 per meter

Formula Used:

Perimeter of a rectangle = 2 × (Length + Breadth)

Perimeter = Total Cost of Fencing ÷ Rate of Fencing

Calculations:

⇒ Let breadth = x m and length = 3x m

⇒ Perimeter = 960 ÷ 7.5 = 128 m

⇒ 2 × (3x + x) = 128

⇒ 2 × 4x = 128

⇒ 8x = 128

⇒ x = 16 m

⇒ Length = 3 × 16 = 48 m

⇒ Breadth = 16 m

⇒ Difference = 48 − 16 = 32 m

∴ The correct answer is 32 m.

Additional Information

Perimeter vs Area Fencing/Ploughing

Fencing is always done along the boundary (Perimeter), whereas ploughing or leveling is done on the surface (Area). Fencing Rate is in per meter, while Leveling Rate is in per sq. meter.

Rectangular Mensuration Formulas

Area of rectangle = Length × Breadth, and Diagonal of rectangle = √(Length² + Breadth²).

Shortcut Trick

Speed of Train B = 20 ÷ 2 = 10 m/sec.

Length of Train B = Speed × Time = 10 × 28 = 280 m.

Ratio of lengths A : B = 10 : 7. Since 7 units = 280 m, then 1 unit = 40 m.

Length of Train A = 10 × 40 = 400 m.

∴ The correct answer is 400 m.

Alternate Method

Given:

Speed of Train A (S_A) = 20 m/sec

Ratio of Speed A to B = 2 : 1

Ratio of Length A to B = 10 : 7

Time taken by Train B to cross a pole = 28 seconds

Calculations:

⇒ Speed of Train B = S_A ÷ 2 = 20 ÷ 2 = 10 m/sec

⇒ Length of Train B (L_B) = Speed_B × Time = 10 × 28 = 280 m

⇒ According to ratio, 7 units = 280 m

⇒ 1 unit = 280 ÷ 7 = 40 m

⇒ Length of Train A (L_A) = 10 units = 10 × 40 = 400 m

∴ The correct answer is 400 m.

Additional Information

Crossing a Pole or Point Object

When a train crosses a stationary pole or person, the distance covered is equal to the length of the train itself. Formula: Length = Speed × Time.

Crossing a Platform or Bridge

When a train crosses a platform or another train, the total distance covered is the sum of the lengths of the train and the object. Total Distance = L₁ + L₂.

Unit Conversion

To convert km/hr to m/sec, multiply by 5/18. To convert m/sec to km/hr, multiply by 18/5.

Common Solution:

Let the quantity of white rice sold by A be 4x

The quantity of brown rice sold by B = (x × {125/100} = 5x

The quantity of white rice sold by B = (5x + 15)

The quantity of brown rice sold by A = {5x/4} × 3 = {15x/4})

The quantity of brown rice sold by C = (70 × 3 – 5x - {15x/4})

= (210 - {35x/4})

The quantity of white rice sold by C = [100 - (210 - {35x/4})]

= ({35x/4} – 110)

Given, 260 = {35x/4} – 110 + 5x + 15 + 4x

355 = {71x/4})

So, x = 20

 

Required difference = 75 − 65 = 10 kg

Common Solution:

Let the quantity of white rice sold by A be 4x

The quantity of brown rice sold by B = (x × {125/100} = 5x

The quantity of white rice sold by B = (5x + 15)

The quantity of brown rice sold by A = {5x/4} × 3 = {15x/4})

The quantity of brown rice sold by C = (70 × 3 – 5x - {15x/4})

= (210 - {35x/4})

The quantity of white rice sold by C = [100 - (210 - {35x/4})]

= ({35x/4} – 110)

Given, 260 = {35x/4} – 110 + 5x + 15 + 4x

355 = {71x/4})

So, x = 20

 

Calculations:

Required percentage = (115/75) x 100 = 153.33%

Common Solution:

Let the quantity of white rice sold by A be 4x

The quantity of brown rice sold by B = (x × {125/100} = 5x

The quantity of white rice sold by B = (5x + 15)

The quantity of brown rice sold by A = {5x/4} × 3 = {15x/4})

The quantity of brown rice sold by C = (70 × 3 – 5x - {15x/4})

= (210 - {35x/4})

The quantity of white rice sold by C = [100 - (210 - {35x/4})]

= ({35x/4} – 110)

Given, 260 = {35x/4} – 110 + 5x + 15 + 4x

355 = {71x/4})

So, x = 20

 

Calculations:

The average quantity of brown rice sold by A and C = (75 + 35)/2 = 55 kg

The quantity of white rice sold by D = 55 + 5 = 60 kg

The ratio of brown rice sold by D and B = 3 : 4

The quantity of brown rice sold by B = 100 kg

The quantity of brown rice sold by D = 100 x (3/4) = 75 kg

The total quantity of rice sold by D = 75 + 60 = 135 kg

Common Solution:

Let the quantity of white rice sold by A be 4x

The quantity of brown rice sold by B = (x × {125/100} = 5x

The quantity of white rice sold by B = (5x + 15)

The quantity of brown rice sold by A = {5x/4} × 3 = {15x/4})

The quantity of brown rice sold by C = (70 × 3 – 5x - {15x/4})

= (210 - {35x/4})

The quantity of white rice sold by C = [100 - (210 - {35x/4})]

= ({35x/4} – 110)

Given, 260 = {35x/4} – 110 + 5x + 15 + 4x

355 = {71x/4})

So, x = 20

 

Calculations:

The total revenue earned by selling white rice = 80 x 20 = Rs. 1600

The revenue earned by selling brown rice = 2500 – 1600 = Rs. 900

The price of each kg of brown rice = 900/75 = Rs. 12

Common Solution:

Let the quantity of white rice sold by A be 4x

The quantity of brown rice sold by B = (x × {125/100} = 5x

The quantity of white rice sold by B = (5x + 15)

The quantity of brown rice sold by A = {5x/4} × 3 = {15x/4})

The quantity of brown rice sold by C = (70 × 3 – 5x - {15x/4})

= (210 - {35x/4})

The quantity of white rice sold by C = [100 - (210 - {35x/4})]

= ({35x/4} – 110)

Given, 260 = {35x/4} – 110 + 5x + 15 + 4x

355 = {71x/4})

So, x = 20

 

Calculations:

The quantity of white rice sold by B = 115 kg

The total quantity of brown rice sold by A and B together = 75 + 100 = 175 kg

Required ratio = 115:75 = 23: 35

Given:

√(21.98 × 30.09 + 123.90) = ? ÷ 4

Formula used:

Approximation Method: Rounding decimals to the nearest whole numbers for easier calculation.

Calculations:

To find the approximate value, we round the given numbers:

21.98 ≈ 22

30.09 ≈ 30

123.90 ≈ 124

Now, substitute these approximate values into the expression:

√(22 × 30 + 124) = ? ÷ 4

⇒ √(660 + 124) = ? ÷ 4

⇒ √784 = ? ÷ 4

Since the square root of 784 is 28:

⇒ 28 = ? ÷ 4

Now, solve for ? by performing multiplication:

⇒ ? = 28 × 4

⇒ ? = 112

∴ The correct answer is option (5).

Given:

29.98% of 560 + 120.11 of 39 = ?

Formula used:

Approximation method: Rounding values to the nearest integers to simplify calculation.

Calculation:

For approximate value:

29.98% ≈ 30%

120.11 ≈ 120

⇒ ? ≈ 30% of 560 + 120 × 39

⇒ ? = (30/100) × 560 + 120 × 39

⇒ ? = 3 × 56 + 4680

⇒ ? = 168 + 4680

⇒ ? = 4848

∴ The correct answer is option (2).

Given:

22.02² + (1 / 9) of 44 = ? − 45.09% of 100.9

Formula used:

Approximation method: Round off the decimal numbers to the nearest integers to simplify the calculation.

Calculations:

To find the approximate value, let:

22.02 ≈ 22

44 ≈ 45 (since 45 is a multiple of 9)

45.09% ≈ 45%

100.9 ≈ 100

Now, substitute these values into the expression:

22² + (1 / 9) × 45 = ? − 45% of 100

⇒ 484 + 5 = ? − (45 / 100) × 100

⇒ 489 = ? − 45

⇒ ? = 489 + 45

⇒ ? = 534

∴ The correct answer is option (1).

Shortcut Trick

Total Work = LCM(15, 30) = 30 units.

Efficiency: A = 2, B = 1. Combined (A+B) = 3 units/day.

6 days work of (A+B) = 6 × 3 = 18 units. Remaining = 30 − 18 = 12 units.

B + C finish 12 units in 5 days → Efficiency of (B+C) = 12/5 = 2.4.

Efficiency of C = 2.4 − 1 = 1.4 units/day (or 7/5).

Time for C alone = 30 ÷ (7/5) = 150/7 days.

∴ The correct answer is 150/7 days.

Alternate Method

Given: A takes 15 days, B takes 30 days. They work together for 6 days. B and C finish remaining in 5 days.

Formula: Total Work = Efficiency × Time

⇒ Work done by A and B in 6 days = 6 × (2 + 1) = 18 units

⇒ Remaining work = 30 − 18 = 12 units

⇒ Let Efficiency of C be 'c'. Work done by B and C in 5 days = 5 × (1 + c) = 12

⇒ 1 + c = 12/5

⇒ c = 12/5 − 1 = 7/5

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time = 30 ÷ (7/5) = (30 × 5) / 7 = 150/7 days

∴ The correct answer is 150/7 days.

Additional Information

Combined Efficiency

When multiple people work together, their individual efficiencies are added to find the total work done per day.

Work & Wages

Wages are generally distributed in the ratio of the total work done by each individual (Efficiency × Time).

Alternate Day Work

In alternate day problems, calculate the work done in one complete cycle (e.g., 2 days) and then divide total work by cycle work.

Shortcut Trick

Ratio of Father : Son currently is 2 : 1 and 10 years ago was 3 :

1. Make the difference between ratios equal: (2 : 1) × 2 → 4 : 2 and (3 : 1) × 1 → 3 :
2. Difference of 1 unit (4 − 3 or 2 − 1) corresponds to 10 years.

Father age = 4 × 10 = 40 years; Son age = 2 × 10 = 20 years.

Mother : Father = 9 : 8 ⇒ Since 8 units = 40, then 9 units (Mother) = 45 years.

Difference = 45 − 20 = 25 years.

∴ The correct answer is 25 years.

Alternate Method

Given: Present age Father (F) = 2 × Son (S); Ratio Mother (M) : Father (F) = 9 : 8.

⇒ Let Son's present age be x. Then Father's present age = 2x.

⇒ 10 years ago: (2x − 10) = 3 × (x − 10)

⇒ 2x − 10 = 3x − 30

⇒ x = 20 (Son's present age)

⇒ Father's present age = 2 × 20 = 40 years.

⇒ Mother's age = (9/8) × 40 = 45 years.

⇒ Difference = 45 − 20 = 25 years.

∴ The correct answer is 25 years.

Additional Information

Age Difference Constant

The numerical difference between the ages of two persons remains the same throughout their lives, regardless of the number of years passed.

Ratio Balancing Method

To equate parts in age ratios, multiply the ratios such that the difference between the ratio terms of the two individuals becomes equal.

Cross Multiplication Method

For problems involving two different time periods, use (Initial Ratio) → (Change) → (Final Ratio) to solve for units directly.

Shortcut Trick

Total investment ≈ ₹45000.

Ratio of interest is SI_A : SI_B = 15 : 8.

Since SI = P × R × T, then P ∝ SI ÷ (R × T). For equal time T = 2 years, P_A : P_B = (15 ÷ 7.5) : (8 ÷ 8) = 2 :

1. Investment in Scheme B = 1 ÷ (2 + 1) × ₹45000 = ₹15000.

∴ The correct answer is ₹15000.

Alternate Method

Given:

Total investment (P) ≈ ₹45000

Rate of interest for Scheme A (R_A) = 7.5% p.a.

Rate of interest for Scheme B (R_B) = 8% p.a.

Time (T) = 2 years

Ratio of interest received = SI_A : SI_B = 15 : 8

Formula Used:

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

Calculations:

Let the amount invested in Scheme A be P_A and in Scheme B be P_B.

⇒ SI_A = (P_A × 7.5 × 2) ÷ 100 = 15P_A ÷ 100

⇒ SI_B = (P_B × 8 × 2) ÷ 100 = 16P_B ÷ 100

Given that SI_A : SI_B = 15 : 8

⇒ (15P_A ÷ 100) ÷ (16P_B ÷ 100) = 15 ÷ 8

⇒ 15P_A ÷ 16P_B = 15 ÷ 8

⇒ P_A ÷ P_B = 16 ÷ 8

⇒ P_A ÷ P_B = 2 ÷ 1

⇒ Ratio of investment P_A : P_B = 2 : 1

⇒ Total investment ratio parts = 2 + 1 = 3 parts

⇒ 3 parts = ₹45000

⇒ 1 part = ₹15000

⇒ Investment in Scheme B (P_B) = 1 part = ₹15000

∴ The correct answer is ₹15000.

Additional Information

Simple Interest Principle

Simple Interest is calculated only on the principal amount, hence the annual interest remains constant every year.

Ratio of Principals in Simple Interest

When the ratio of interest is given, the ratio of the principal amounts is directly proportional to the interest and inversely proportional to the product of rate and time: P₁ : P₂ = (I₁ ÷ R₁T₁) : (I₂ ÷ R₂T₂).

Shortcut Trick

Approximate Investment: A = 8000, B = 5000. Ratio = 8 : 5.

Approximate Total Profit = 22000, B profit = 6000 ∴ A profit = 16000.

Profit Ratio A : B = 16000 : 6000 = 8 : 3.

Using Profit = Investment × Time: (8 × 365) ÷ (5 × Days_B) = 8 ÷ 3.

365 ÷ (5 × Days_B) = 1 ÷ 3 ⇒ Days_B = (365 × 3) ÷ 5 = 73 × 3 = 219 days.

Given Days_B = T + 2 ⇒ T = 219 − 2 = 217.

∴ The correct answer is 217.

Alternate Method

Given → Investment A ≈ 8000, Investment B ≈ 5000. Total Profit ≈ 22000, Profit B ≈ 6000.
Formulas: Profit Ratio = (I₁ × T₁) : (I₂ × T₂) | Profit_A = Total Profit − Profit_B

⇒ Profit share of A = 22000 − 6000 = 16000.
⇒ Ratio of profits (A : B) = 16000 : 6000 = 16 : 6 = 8 : 3.
⇒ Ratio of (Investment × Time) = (8000 × 365) : (5000 × (T + 2)).
⇒ (8000 × 365) ÷ (5000 × (T + 2)) = 8 ÷ 3.
⇒ (8 × 365) ÷ (5 × (T + 2)) = 8 ÷ 3.
⇒ 365 ÷ (T + 2) = 5 ÷ 3.
⇒ T + 2 = (365 × 3) ÷ 5 = 73 × 3 = 219.
⇒ T = 219 − 2 = 217.

∴ The correct answer is 217.

Additional Information

Partnership Principles

When investments are for different time periods, the profit is distributed in the ratio of the product of Investment and Time.

Approximation in Exams

Values like 7999.99 should be rounded to the nearest integer (8000) to simplify calculations without affecting accuracy significantly.

Shortcut Trick

Assume Total Income = 100 units.

Rent = 30% of 100 = 30 units → Remaining = 70 units.

Groceries = 25% of 70 = 17.5 units → New Remaining = 52.5 units.

Entertainment share = 52.5 × (2 ÷ 5) = 21 units.

∴ The correct answer is 21%.

Alternate Method

Given: Rent = 30%, Groceries = 25% of remaining, Ent : Savings = 2 : 3.

⇒ Let Total Income = I

⇒ After Rent, Remaining = I × (100 − 30)/100 = 0.70I

⇒ After Groceries, Remaining = 0.70I × (100 − 25)/100 = 0.70I × 0.75 = 0.525I

⇒ Entertainment Expenditure = 0.525I × [2 ÷ (2 + 3)]

⇒ Entertainment = 0.525I × (2 ÷ 5) = 0.525I × 0.4 = 0.21I

⇒ Percentage = (0.21I ÷ I) × 100 = 21%

∴ The correct answer is 21%.

Additional Information

Successive Percentage Change

When values are calculated on the remaining balance, use the multiplying factor: Final = Initial × (1 − r₁/100) × (1 − r₂/100).

Ratio Distribution

To find a specific part in a ratio a:b, multiply the total by a/(a+b) or b/(a+b).

Base Value Importance

Always identify if the percentage is of the Total Income or the Remaining Amount to avoid calculation errors.