Quantitative Aptitude PYQs

Calculation

Given: Average (B + C + D)/3 = 42

→ B + C + D = 42 × 3 = 126

B + C = 50 + 48 = 98 → D = 126 – 98 = 28

Shop A on Tuesday = 56

Difference = 56 – 28 = 28 cakes

Calculation

Given: Shop C on Wednesday = 48

Ratio C (Wed) : C (Thu) = 4 : 5

→ C (Thu) = (48 × 5)/4 = 60

Shop B (Thu) = C (Thu) + 19 = 60 + 19 = 79 cakes

Calculation

Shop A (Tue + Wed) = 56 + 70 = 126

Shop C Wed = 48

Ratio = 126 : 48

→ 21 : 8

Calculation

Shop C Monday = 32, Wednesday = 48

Difference = 48 – 32 = 16

Percentage less = (16 ÷ 48) × 100 = 33.33%

Shortcut Trick

Let the total distance be 3x km. Bus distance = 2x, Bicycle distance = x.

Total time = (Distance₁ ÷ Speed₁) + (Distance₂ ÷ Speed₂)

⇒ 3 = (2x ÷ 60) + (x ÷ 30) ⇒ 3 = (x ÷ 30) + (x ÷ 30) ⇒ 3 = 2x ÷ 30

⇒ x = (3 × 30) ÷ 2 = 45 km.

∴ The correct answer is 45 km.

Alternate Method

Given: Bus Speed = 60 km/hr, Bicycle Speed = 30 km/hr, Total Time = 3 hours.

Formula: Time = Distance ÷ Speed

Let total distance be D.

⇒ Time taken by bus = (2D/3) ÷ 60 = 2D ÷ 180 = D ÷ 90

⇒ Time taken by bicycle = (D/3) ÷ 30 = D ÷ 90

⇒ Total Time = (D ÷ 90) + (D ÷ 90) = 3

⇒ 2D ÷ 90 = 3

⇒ D = (3 × 90) ÷ 2 = 135 km

⇒ Distance by bicycle = (1/3) × 135

∴ The correct answer is 45 km.

Additional Information

Average Speed Formula

When a distance is divided into parts, Average Speed = Total Distance ÷ Total Time.

Time, Speed, Distance Relationship

For a constant distance, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Unit Conversion

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

Shortcut Trick

Difference in percentage votes = 52% − (100% − 52%) = 52% − 48% = 4%.

Since 4% of valid votes = 98 votes, then 100% of valid votes = (98 ÷ 4) × 100 = 2450.

Total votes polled = Valid votes + Invalid votes = 2450 + 68 = 2518.

∴ The correct answer is 2518 votes.

Alternate Method

Given: Invalid votes = 68, Winner share = 52%, Victory margin = 98 votes.

Let the total valid votes be V.

⇒ Winner = 52% of V

⇒ Loser = (100 − 52)% of V = 48% of V

⇒ Margin = 52% of V − 48% of V = 4% of V

⇒ 4% of V = 98

⇒ V = (98 × 100) ÷ 4 = 2450

⇒ Total votes polled = V + 68 = 2450 + 68 = 2518

∴ The correct answer is 2518 votes.

Additional Information

Percentage Change Base

Always ensure the percentage is calculated on the Valid Votes base unless the question specifies it is on the Total Polled Votes.

Successive Elections

If voters didn't cast votes, subtract them from Total Voters before calculating Invalid Votes: Total → Cast → Valid → Result.

Shortcut Trick

Market Price (MP) represents 100%.

Discount = 12% ⇒ Selling Price (SP) = 100% − 12% = 88%.

Given SP = Rs. 22,880.

Using unitary method: 88% = 22,880.

1% = 22,880 ÷ 88 = 260.

100% = 260 × 100 = 26,000.

∴ The correct answer is Rs. 26,000.

Alternate Method

Given → Selling Price (SP) = Rs. 22,880 | Discount Percentage = 12%
Formulas: SP = MP × (100 − Discount%) ÷ 100 | MP = (SP × 100) ÷ (100 − Discount%)

⇒ Let the Market Price (MP) be x.
⇒ SP = MP − Discount
⇒ 22,880 = x − (12% of x)
⇒ 22,880 = x − 0.12x
⇒ 22,880 = 0.88x
⇒ x = 22,880 ÷ 0.88
⇒ x = 26,000

∴ The correct answer is Rs. 26,000.

Additional Information

Market Price & Discount

Market Price (List Price) is the price printed on the product. Discount is always calculated on the Market Price.

Ratio Method

12% discount = 12/100 = 3/25. If MP = 25 units, Discount = 3 units, then SP = 22 units. 22 units = 22,880 ⇒ 1 unit = 1,040 ⇒ 25 units = 26,000.

Shortcut Trick

Ratio of investments P : Q : R = 35000 : 65000 : 75000 = 7 : 13 : 15.

Total ratio parts = 7 + 13 + 15 = 35 units.

Total loss =  ₹ 70000.

Value of 1 unit = 70000 ÷ 35 =  ₹ 2000.

Loss share of P = 7 units × 2000 =  ₹ 14000.

∴ The correct answer is  ₹ 14000.

Alternate Method

Given: Capital of P =  ₹ 35000, Q =  ₹ 65000, R =  ₹ 75000. Total loss =  ₹ 70000. Time = 2 years (constant).

Formula: Profit/Loss Share = Investment × Time Period.

⇒ Since time is the same for all, the loss is divided in the ratio of their capitals.

⇒ Ratio of loss = 35000 : 65000 : 75000 = 7 : 13 : 15.

⇒ Loss of P = [7 ÷ (7 + 13 + 15)] × 70000.

⇒ Loss of P = (7 ÷ 35) × 70000.

⇒ Loss of P = 7 × 2000 = 14000.

∴ The correct answer is  ₹ 14000.

Additional Information

Partnership Ratios

When the time period for all investments is the same, the profit or loss is shared strictly in the ratio of the capital invested.

Variable Time Investment

If partners invest for different durations, the ratio is calculated as: (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Profit ratio is proportional to investment × time:

[A × 12] / [B × n] = 15/14

[10000 × 12] / [14000 × n] = 15/ 14

[120000 × 14 = 15 × 14000 × n]

n = 8 months

Common calculation

Profit ratio is proportional to investment × time:

[A × 12] / [B × n] = 15/14

[10000 × 12] / [14000 × n] = 15/ 14

[120000 × 14 = 15 × 14000 × n]

n = 8 months

Now, New case A invest for n = 8 months and B for 12 months

A invests = 10,000 for 8 months → A × time = 10,000 × 8 = 80,000

B invests = 14,000 for 12 months → B × time = 14,000 × 12 = 168,000

Profit ratio A : B = 80,000 : 168,000 = 10 : 21

Total ratio sum = 10 + 21 = 31

Total profit = 7750

B’s share z = [21/31] × 7750 = 5250

General Solution:

Let

Males: A = x, B = y, C = z

Total males: x + y + z = 86

Given: y = 1.5 × x ⇒ y = 3x/2

z = y + 6 = 3x/2 + 6

Solve for x:

x + y + z = 86 ⇒ x + 3x/2 + (3x/2 + 6) = 86

x + 1.5x + 1.5x + 6 = 86

4x + 6 = 86 ⇒ 4x = 80 ⇒ x = 20

Then y = 1.5 × 20 = 30

z = 30 + 6 = 36

Females:

Let females in B = fB

Females in A = fA = 1.25 × fB

Females in C = fC = 20 = 0.8 × fA ⇒ fA = 25

Then fB = fA/1.25 = 25/1.25 = 20

Thus,

Males: A = 20, B = 30, C = 36

Females: A = 25, B = 20, C = 20

Calculations:

Males in A = 20, Males in C = 36

Average = (20 + 36) ÷ 2 = 56 ÷ 2 = 28

Hence, the correct answer is 28.

General Solution:

Let

Males: A = x, B = y, C = z

Total males: x + y + z = 86

Given: y = 1.5 × x ⇒ y = 3x/2

z = y + 6 = 3x/2 + 6

Solve for x:

x + y + z = 86 ⇒ x + 3x/2 + (3x/2 + 6) = 86

x + 1.5x + 1.5x + 6 = 86

4x + 6 = 86 ⇒ 4x = 80 ⇒ x = 20

Then y = 1.5 × 20 = 30

z = 30 + 6 = 36

Females:

Let females in B = fB

Females in A = fA = 1.25 × fB

Females in C = fC = 20 = 0.8 × fA ⇒ fA = 25

Then fB = fA/1.25 = 25/1.25 = 20

Thus,

Males: A = 20, B = 30, C = 36

Females: A = 25, B = 20, C = 20

Calculations:

Original females:

A = 25

B = 20

C = 20

Increase females in C by 10%:

New females in C = 20 × 1.10 = 22

Total females now = 25 + 20 + 22 = 67

Hence, the correct answer is 67.

General Solution:

Let

Males: A = x, B = y, C = z

Total males: x + y + z = 86

Given: y = 1.5 × x ⇒ y = 3x/2

z = y + 6 = 3x/2 + 6

Solve for x:

x + y + z = 86 ⇒ x + 3x/2 + (3x/2 + 6) = 86

x + 1.5x + 1.5x + 6 = 86

4x + 6 = 86 ⇒ 4x = 80 ⇒ x = 20

Then y = 1.5 × 20 = 30

z = 30 + 6 = 36

Females:

Let females in B = fB

Females in A = fA = 1.25 × fB

Females in C = fC = 20 = 0.8 × fA ⇒ fA = 25

Then fB = fA/1.25 = 25/1.25 = 20

Thus,

Males: A = 20, B = 30, C = 36

Females: A = 25, B = 20, C = 20

Calculations:

From previous solution:

Males in C = 36

Females in B = 20

Ratio (Males in C : Females in B) = 36 : 20 = 9 : 5

Hence, the correct answer is 9 : 5.

Females with B = 20

Males with A = 20

Required percentage = (Females B ÷ Males A) × 100

= (20 ÷ 20) × 100 = 100%

Hence, the correct answer is 100%.

General Solution:

Let

Males: A = x, B = y, C = z

Total males: x + y + z = 86

Given: y = 1.5 × x ⇒ y = 3x/2

z = y + 6 = 3x/2 + 6

Solve for x:

x + y + z = 86 ⇒ x + 3x/2 + (3x/2 + 6) = 86

x + 1.5x + 1.5x + 6 = 86

4x + 6 = 86 ⇒ 4x = 80 ⇒ x = 20

Then y = 1.5 × 20 = 30

z = 30 + 6 = 36

Females:

Let females in B = fB

Females in A = fA = 1.25 × fB

Females in C = fC = 20 = 0.8 × fA ⇒ fA = 25

Then fB = fA/1.25 = 25/1.25 = 20

Thus,

Males: A = 20, B = 30, C = 36

Females: A = 25, B = 20, C = 20

Calculations:

From general solution, Person C has:

Males = 36

Females = 20

Difference = Males − Females = 36 − 20 = 16

Hence, the correct answer is 16.

General Solution:

Let

Males: A = x, B = y, C = z

Total males: x + y + z = 86

Given: y = 1.5 × x ⇒ y = 3x/2

z = y + 6 = 3x/2 + 6

Solve for x:

x + y + z = 86 ⇒ x + 3x/2 + (3x/2 + 6) = 86

x + 1.5x + 1.5x + 6 = 86

4x + 6 = 86 ⇒ 4x = 80 ⇒ x = 20

Then y = 1.5 × 20 = 30

z = 30 + 6 = 36

Females:

Let females in B = f_B

Females in A = f_A = 1.25 × f_B

Females in C = f_C = 20 = 0.8 × f_A ⇒ f_A = 25

Then f_B = f_A/1.25 = 25/1.25 = 20

Thus,

Males: A = 20, B = 30, C = 36

Females: A = 25, B = 20, C = 20

Calculations:

Females with A = 25, Females with B = 20

Total females with A and B = 25 + 20 = 45

Hence, the correct answer is 45.

Shortcut Trick

Rate = 10% = 1/10. For Compound Interest instalments, ratio of Principal to Amount for 1^st year is 10 : 11 and for 2^nd year is 10² : 11² = 100 : 121.

To make instalments equal, multiply 1^st year ratio by 11: (10 × 11) : (11 × 11) = 110 : 121.

Total Principal = 110 + 100 = 210 units. Given instalment (121 units) = ₹121.

∴ 1 unit = ₹1. Total sum = 210 × 1 = ₹210.

∴ The correct answer is ₹210.

Alternate Method

Given: Instalment (I) = ₹121, Rate (r) = 10%, n = 2 years.

Formula Used: Principal (P) = [I / (1 + r/100)¹] + [I / (1 + r/100)²]

⇒ P = [121 / (1 + 10/100)] + [121 / (1 + 10/100)²]

⇒ P = [121 / (1.1)] + [121 / (1.1)²]

⇒ P = [121 / 1.1] + [121 / 1.21]

⇒ P = 110 + 100

⇒ P = ₹210

∴ The correct answer is ₹210.

Additional Information

CI Instalment Formula

The total principal P for n instalments of value I is P = I × [((1+R)ⁿ − 1) / (R(1+R)ⁿ)], where R is the decimal rate.

Present Value Method

Total sum borrowed is the sum of the present values of each instalment: P = PV₁ + PV₂ + ... + PV_n.

SI vs CI Instalment

In Simple Interest, instalments are calculated on a linear basis, whereas in Compound Interest, each instalment reduces the principal exponentially.

Shortcut Trick

Since Area of Circle = Area of Rectangle, simply multiply the dimensions of the rectangle.

Area = Length × Breadth = 44 × 56.

Calculation: 44 × 56 = 2464 cm².

∴ The correct answer is 2464 cm².

Alternate Method

Given:

Length of rectangle (L) = 44 cm

Breadth of rectangle (B) = 56 cm

Formula Used:

Area of Rectangle = Length × Breadth

Calculations:

⇒ Area of rectangle = 44 × 56

⇒ Area of rectangle = 2464 cm²

⇒ According to the question, Area of Circle = Area of Rectangle

⇒ Area of Circle = 2464 cm²

∴ The correct answer is 2464 cm².

Additional Information

Area of Circle

The area of a circle is calculated using the formula A = πr², where r is the radius.

Perimeter of Rectangle

The perimeter of a rectangle is the total distance around the outside, calculated as P = 2 × (Length + Breadth).

Circumference of Circle

The boundary of a circle is called the circumference, given by the formula C = 2πr or πd.

Follow BODMAS rule to solve this question, as per the order given below,

Step - 1 - Parts of an equation enclosed in the ‘BRACKETS’ must be solved first

Step - 2 - Any mathematical ‘OF’ or ‘EXPONENTS’ must be solved next

Step - 3 - Next the part of the equation that contains ‘DIVISION; and ‘MULTIPLICATION’ are calculated

Step - 4 - Last but not least, the parts of the equation that contains ‘ADDITION’ and ‘SUBTRACTION’ should be calculated

Now, the given expression:

⇒ 410 × ? × 26 = 67250 + 50010

⇒ 10660 × ? = 117260

⇒ \(?; = ;\frac{{117260}}{{10660}}; = ;11\)

Follow BODMAS rule to solve this question, as per the order given below,

Step - 1 - Parts of an equation enclosed in the ‘BRACKETS’ must be solved first

Step - 2 - Any mathematical ‘OF’ or ‘EXPONENTS’ must be solved next

Step - 3 - Next the part of the equation that contains ‘DIVISION; and ‘MULTIPLICATION’ are calculated

Step - 4 - Last but not least, the parts of the equation that contains ‘ADDITION’ and ‘SUBTRACTION’ should be calculated

Now, the given expression:

⇒ 207 ÷ 3 × 2.25 + 43.5 – 9 = ?

⇒ 69 × 2.25 + 43.5 – 9 = ?

⇒ 155.25 + 43.5 – 9 = ?

⇒ 198.75 – 9 = ?

⇒ 189.75 = ?