Quantitative Aptitude PYQs

Shortcut Trick

Assume Marked Price (MP) is 100.

Given Discount = 20%, so Selling Price (SP) = 100 − 20 = 80.

Difference (MP − CP) = 50% of SP = 0.5 × 80 = 40.

Since MP − CP = 40, then 100 − CP = 40 ⇒ Cost Price (CP) = 60.

Ratio MP : CP = 100 : 60 = 5 : 3.

∴ The correct answer is 5 : 3.

Alternate Method

Given: MP − CP = 0.5 × SP and Discount% = 20%

Formulas: SP = MP × (100 − D%) ÷ 100

⇒ Let Marked Price be M and Cost Price be C.

⇒ Selling Price (S) = M − 0.20M = 0.8M.

⇒ According to the question: M − C = 0.5 × S.

⇒ M − C = 0.5 × (0.8M) = 0.4M.

⇒ M − 0.4M = C ⇒ 0.6M = C.

⇒ M ÷ C = 1 ÷ 0.6 = 10 ÷ 6 = 5 ÷ 3.

∴ The correct answer is 5 : 3.

Additional Information

Marked Price and Discount

Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP. SP = MP − Discount.

Relationship between CP, SP, and MP

Profit is SP − CP, whereas Discount is MP − SP. To find the ratio between MP and CP, we can use the formula MP ÷ CP = (100 + P%) ÷ (100 − D%) if profit percentage is known.

Shortcut Trick

Initial Ratio (Water : Milk) = 6 : 7

Final Ratio (Water : Milk) = 3 : 5. To make Water constant, multiply final ratio by 2 → 6 : 10.

Increase in Milk = 10 − 7 = 3 units. Given, 3 units = 15 litres → 1 unit = 5 litres.

Initial mixture x = (6 + 7) units = 13 units = 13 × 5 = 65 litres.

∴ The correct answer is 65 litres.

Alternate Method

Given: Initial ratio 6:7, added 15 litres milk, final ratio 3:5.

Formula Used: Ratio × constant = Quantity; Initial Value x = Sum of initial components.

⇒ Let water = 6k and milk = 7k. Total volume x = 13k.

⇒ 6k / (7k + 15) = 3 / 5

⇒ 5 × 6k = 3 × (7k + 15)

⇒ 30k = 21k + 45

⇒ 9k = 45 ⇒ k = 5

⇒ x = 13 × 5 = 65 litres.

∴ The correct answer is 65 litres.

Additional Information

Constant Variable Concept

When only one component is added or removed from a mixture, the quantity of the other component remains constant. Always equate its ratio units first.

Alligation Rule

Used to find the ratio in which two quantities are mixed to get a final concentration: (Higher value − Mean) / (Mean − Lower value).

Replacement Method

Final Quantity = Initial Quantity × (1 − amount replaced/total)ⁿ, where n is the number of times the process is repeated.

Calculations:

Total number of books issued to A in all library together

⇒ 18 + 30 + 24 = 72

Total number of books issued to C in all library together

⇒ 25 + 10 + 32 = 67

So, required difference 72 – 67 = 5

Calculations:

Number of books issued to A in library Q = 30

Number of books issued to B in library Q = 35

Required ratio 30 : 35 = 6: 7

So, x = 7 ⇒  9x = 9 × 7 = 63

Calculations:

Total number of books issued to C in library R = 32

Number of books issued to D in library R:

⇒ 32 × 25/100 + 12

⇒ 8 + 12 = 20

Calculations:

Total Number of books issued to A = 18 + 30 + 24 = 72

Total Number of fiction books issued to A = 72 × (25 /100)

Total Number of fiction books issued to A = 18

Calculations:

Total number of book issue in library P to all three people together = [18 + 20 + 25] = 63

Total number of book issue in library Q to all three people together = [30 + 35 +10] = 75

So, Total number of book issue in library P to all three people together is 75 – 63 = 12 less than the number of book issue in library Q to all three people together.

Shortcut Trick

Assume Total Population = 100 units.

Literates = 80 units, Illiterates = 20 units (Ratio 4 : 1).

Literate Females = 30% of 80 = 24 units | Illiterate Females = 20% of 20 = 4 units.

Total Females = 24 + 4 = 28 units → 56 (Given).

1 unit = 2 | Total population = 100 × 2 = 200.

∴ The correct answer is 200.

Alternate Method

Given: Ratio of Literate to Illiterate = 4 : 1. Total Females = 56.

Let Literate = 400x and Illiterate = 100x. Total Population = 500x.

⇒ Literate Females = 30% of 400x = 0.30 × 400x = 120x.

⇒ Illiterate Females = 20% of 100x = 0.20 × 100x = 20x.

⇒ Total Females = 120x + 20x = 140x.

⇒ 140x = 56

⇒ x = 56 ÷ 140 = 0.4

⇒ Total Population = 500 × 0.4 = 200.

∴ The correct answer is 200.

Additional Information

Ratio Distribution

To divide a quantity N in ratio a : b, the parts are [a/(a+b)] × N and [b/(a+b)] × N.

Percentage of a Group

If x% are males, then (100 − x)% are females, provided there are only two categories.

Unitary Method

Finding the value of a single unit (140 units = 56 → 1 unit = 0.4) to calculate the total required value.

Shortcut Trick

Circumference of Circle = π × d = (22/7) × 56 = 176 cm.

Perimeter of Rectangle = 2 × (L + B). Given L + B = diameter = 56 cm.

⇒ Perimeter = 2 × 56 = 112 cm.

Ratio = 176 : 112 = 11 : 7 (dividing both by 16).

∴ The correct answer is 11 : 7.

Alternate Method

Given:

Diameter (d) of circle = 56 cm.

Sum of length (L) and breadth (B) of rectangle (L + B) = 56 cm.

Formula Used:

⇒ Circumference of circle (C) = π × d

⇒ Perimeter of rectangle (P) = 2 × (L + B)

Calculations:

⇒ C = (22/7) × 56

⇒ C = 22 × 8 = 176 cm

⇒ P = 2 × 56 = 112 cm

⇒ Ratio = C / P = 176 / 112

⇒ Ratio = (16 × 11) / (16 × 7) = 11 / 7

∴ The correct answer is 11 : 7.

Additional Information

Circle Properties

Area of circle = πr² or (πd²)/4. Circumference = 2πr or πd.

Rectangle Mensuration

Area of rectangle = L × B. Diagonal = √(L² + B²).

Standard π Values

For calculations involving 7 or its multiples, use π ≈ 22/7 for easier cancellation.

Shortcut Trick

When distance is constant, time is inversely proportional to speed (T ∝ 1/S).

Initial speed = 100% → New speed = 100% + 100% = 200%.

Speed ratio = 1 : 2 ∴ Time ratio = 2 : 1.

2 units = 10 hours ⇒ 1 unit = 5 hours.

∴ The correct answer is 5 hr.

Alternate Method

Given: Initial Speed (S₁) = 25 km/h, Initial Time (T₁) = 10 hours.

Formula Used: Distance = Speed × Time

⇒ Total Distance = S₁ × T₁ = 25 × 10 = 250 km.

⇒ New Speed (S₂) = 25 + (100% of 25) = 25 + 25 = 50 km/h.

⇒ New Time (T₂) = Distance ÷ S₂

⇒ T₂ = 250 ÷ 50 = 5 hours.

∴ The correct answer is 5 hr.

Additional Information

Inverse Proportion Rule

If distance is constant, S₁ × T₁ = S₂ × T₂. This allows for quick calculation without finding the actual distance.

Percentage Increase in Speed

A 100% increase means the value doubles (New = Old × 2). Similarly, a 200% increase means the value triples.

Average Speed

For two equal distances covered at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

From the data:

Shop C: 64 umbrellas

Shop D: 80 umbrellas

Total umbrellas sold by C and D:

Total C + D = 64 + 80 = 144

From the data:

Shop B: 160 umbrellas

Now, let's calculate the percentage.

We use the formula:

Percentage = Part / Whole × 100

Here:

Part = Total umbrellas sold by C and D = 144

Whole = Umbrellas sold by B = 160

Substitute the values:

Percentage = 144 / 160× 100

Simplify:

Percentage = 0.9 × 100 = 90%

Thus, the total number of umbrellas sold by shops C and D together is 90% of the number of umbrellas sold by shop B.

Total umbrellas sold by Shop A.

From the data:

Total umbrellas sold by Shop A = 96

The ratio of folded to unfolded umbrellas is 5:3.

Let:

Folded umbrellas = 5x 

Unfolded umbrellas = 3x 

Total umbrellas:

5x + 3x = 8x

Given that the total umbrellas sold by Shop A is 96:

8x = 96

x = 12

Now:

Folded umbrellas = 5x = 5 × 12 = 60 

Unfolded umbrellas = 3x = 3 × 12 = 36 

3/4^th of folded umbrellas are black.

Black folded umbrellas:

3/4 × 60 = 45

2/3^th of unfolded umbrellas are black.

Black unfolded umbrellas:

2/3 × 36 = 24

Total black umbrellas sold by Shop A.

Total black umbrellas = Black folded umbrellas + Black unfolded umbrellas

Total black umbrellas = 45 + 24 = 69

Thus, the number of black umbrellas sold by Shop A is 69.

From the data, the total number of umbrellas sold by Shop D is 80.

It is given that the number of unfolded umbrellas sold by Shop D is 55% of the total umbrellas sold by Shop D. 

Unfolded umbrellas sold by D = 55% × 80 = 0.55 × 80 = 44

The total umbrellas sold by Shop D consist of folded and unfolded umbrellas. Therefore:

Folded umbrellas sold by D = Total umbrellas sold by D - Unfolded umbrellas sold by D

Folded umbrellas sold by D = 80 - 44 = 36

It is given that the total number of unsold folded umbrellas in Shop D is 5/9 of the total number of folded umbrellas sold by Shop D. 

Unsold folded umbrellas in D = 5 / 9 × Folded umbrellas sold by D

Unsold folded umbrellas in D = 5 / 9 × 36 = 20

The total number of unsold folded umbrellas in Shop D is 20.

From the data:

Shop A: 96 umbrellas

Shop C: 64 umbrellas

Shop E: 116 umbrellas

Total umbrellas A + C + E = 96 + 64 + 116

Total umbrellas A + C + E = 276

The average is calculated by dividing the total number of umbrellas by the number of shops 3 in this case:

Average = (Total umbrellas A + C + E) / Number of shops

Average = 276 / 3 = 92

Thus, the average of the total number of umbrellas sold by shops A, C, and E is 92.

Total umbrellas sold by Shop B = 160

Ratio of folded to unfolded umbrellas = 3:5

Let the number of folded umbrellas = 3x 

Let the number of unfolded umbrellas = 5x 

Total umbrellas = 3x + 5x = 8x 

So, 8x = 160 

Solving for x:

x = 160 / 8 = 20

Folded umbrellas sold by Shop B:

3x = 3 × 20 = 60

Total umbrellas sold by Shop D = 80

Ratio of folded to unfolded umbrellas = 1:3

Let the number of folded umbrellas = y 

Let the number of unfolded umbrellas = 3y 

Total umbrellas = y + 3y = 4y 

So, 4y = 80 

Solving for y:

y = 80 / 4 = 20

Folded umbrellas sold by Shop D:

y = 20

Folded umbrellas Shop B = 60

Folded umbrellas Shop D = 20

Total folded umbrellas:

60 + 20 = 80

Thus, the correct answer is 80.

Shortcut Trick

Sum of m and n = 36 × 2 = 72. So, n = 72 − m.

Ratio equation: (0.8m) / (72 − m + 13) = 9/10.

8m = 9 × (85 − m) ⇒ 8m = 765 − 9m ⇒ 17m = 765.

m = 765 ÷ 17 = 45.

∴ The correct answer is 45.

Alternate Method

Given:

Average of m and n = 36

Ratio [80% of m] : [n + 13] = 9 : 10

Calculations:

⇒ (m + n) / 2 = 36

⇒ m + n = 72

⇒ n = 72 − m

⇒ 80% of m = 0.8m = 4m/5

⇒ (4m/5) / (n + 13) = 9/10

⇒ (4m/5) / (72 − m + 13) = 9/10

⇒ (4m) / [5 × (85 − m)] = 9/10

⇒ 40m = 45 × (85 − m)

⇒ 8m = 9 × (85 − m)

⇒ 8m = 765 − 9m

⇒ 17m = 765

⇒ m = 45

∴ The correct answer is 45.

Additional Information

Average Concept

Average is the sum of observations divided by the number of observations. For two numbers, Average = (m + n) ÷ 2.

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 80% = 80/100 = 4/5 or 0.8.

Ratio and Proportion

If a:b = c:d, then a/b = c/d. This allows for cross-multiplication to solve for unknown variables.

Shortcut Trick

Profit ratio is calculated as (Investment × Time).

Profit Ratio X : Y : Z = (4 × 3) : (6 × 4) : (5 × 6) = 12 : 24 : 30.

Simplified Profit Ratio = 2 : 4 : 5.

X's share (2 units) = Rs 18,000 ⇒ 1 unit = Rs 9,000.

Total profit (2 + 4 + 5 = 11 units) = 11 × 9,000 = Rs 99,000.

∴ The correct answer is Rs 99,000.

Alternate Method

Given → Investment ratio = 4 : 6 : 5 | Time ratio = 3 : 4 : 6 | X's profit = Rs 18,000. Formulas: Profit = Investment × Time | Total Profit = Sum of individual profit shares.

⇒ Calculate profit ratio for X, Y, and Z: (4 × 3) : (6 × 4) : (5 × 6) = 12 : 24 : 30. ⇒ Reducing the ratio by dividing by 6: 2 : 4 : 5. ⇒ Let the profit shares be 2k, 4k, and 5k. ⇒ Given X's share 2k = 18,000 ⇒ k = 9,000. ⇒ Total Profit = 2k + 4k + 5k = 11k. ⇒ Total Profit = 11 × 9,000 = 99,000.

∴ The correct answer is Rs 99,000.

Additional Information

Partnership Basics

In a partnership, the profit is distributed in the ratio of the product of Capital (Investment) and Time. If time is constant, profit is proportional to investment.

Ratio of Investments

If profits (P) and time (T) are known, the investment (I) ratio can be found using I = P ÷ T.

Shortcut Trick

Let the Cost Price (CP) be 100 units.

Marked Price (MP) = 100 + 50% of 100 = 150 units.

Selling Price (SP) = 150 − 20% of 150 = 150 − 30 = 120 units.

Profit = SP − CP = 120 − 100 = 20 units. Given 20 units = ₹660, so 1 unit = ₹33.

Discount = MP − SP = 150 − 120 = 30 units.

Difference between Discount and Profit = 30 − 20 = 10 units = 10 × 33 = ₹330.

∴ The correct answer is ₹330.

Alternate Method

Given: Markup% = 50%, Discount% = 20%, Profit = ₹660. Formulas: Profit = SP − CP | SP = MP × (100 − D%) ÷ 100 | MP = CP × (100 + Markup%) ÷ 100.

⇒ Let CP = x. Then MP = 1.5x and SP = 0.8 × 1.5x = 1.2x. ⇒ Profit = 1.2x − x = 0.2x. Given 0.2x = 660 ⇒ x = 3300. ⇒ Profit = 0.2 × 3300 = ₹660. ⇒ Discount = MP − SP = 1.5x − 1.2x = 0.3x = 0.3 × 3300 = ₹990. ⇒ Difference = 990 − 660 = ₹330.

∴ The correct answer is ₹330.

Additional Information

Markup and Profit Relationship

Net Profit% can be calculated using the successive percentage formula: Net% = m − d − (m × d)/100, where m is markup% and d is discount%.

Relation between CP and MP

The ratio of CP to MP is given by: CP/MP = (100 − Discount%)/(100 + Profit%).

Shortcut Trick

Ratio of Milk to Water = 6 : 1

Given Water = 15 litres (1 unit = 15 litres)

Total Milk (final) = 6 × 15 = 90 litres

Initial Milk (x) = Final Milk − Added Milk = 90 − 18 = 72 litres

∴ The correct answer is 72 litres.

Alternate Method

Given:

Initial Milk = x litres, Initial Water = 15 litres

Milk added = 18 litres

Final Ratio (Milk : Water) = 6 : 1

Calculations:

⇒ (Initial Milk + Added Milk) ÷ Water = 6 ÷ 1

⇒ (x + 18) ÷ 15 = 6 ÷ 1

⇒ x + 18 = 15 × 6

⇒ x + 18 = 90

⇒ x = 90 − 18

⇒ x = 72

∴ The correct answer is 72 litres.

Additional Information

Ratio and Proportion

A ratio compares two quantities. If a : b = c : d, then product of extremes (a×d) equals product of means (b×c).

Mixture Problems

When only one component is added to a mixture, the quantity of the other component remains constant in terms of absolute units.

Unitary Method

Find the value of one unit by dividing the given total by its ratio share, then multiply to find other values.

General Solution:

Let:

G_A : Questions given to A

G_B : Questions given to B

G_C : Questions given to C

A_A : Questions attempted by A

A_B : Questions attempted by B

A_C : Questions attempted by C

B attempted 50% of the questions he was given:

A_B = 0.5 × G_B

A was given 20 fewer questions than B:

G_A = G_B - 20

C was given 130 questions:

G_C = 130

Combined questions attempted by A and C is 145:

A_A + A_C = 145

Total questions given to all three is twice the total questions they attempted:

G_A + G_B + G_C = 2 × (A_A + A_B + A_C)

Number of questions attempted by B and C is the same:

A_B = A_C

From Equation 1:

A_B = 0.5 × G_B

From Equation 3:

G_C = 130

From Equation 2:

G_A = G_B - 20

From Equation 4:

A_A + A_C = 145

From Equation 5:

GA + G_B + G_C = 2 × (A_A + A_B + A_C)

Substitute G_A = G_B - 20 and G_C = 130 into Equation 5:

G_B - 20 + G_B + 130 = 2 × (A_A + A_B + A_C)

Simplify:

2G_B + 110 = 2 × (A_A + A_B + A_C)

G_B + 55 = A_A + A_B + A_C

From Equation 4:

A_A + A_C = 145

From Equation 6:

A_B = A_C

Substitute A_B = A_C into G_B + 55 = A_A + A_B + A_C:

G_B + 55 = AA + AB + AC

G_B + 55 = A_A + 2A_C

From Equation 4:

A_A = 145 - A_C

Substitute A_A = 145 - A_C into G_B + 55 = A_A + 2A_C:

G_B + 55 = 145 - A_C + 2A_C

Simplify:

G_B + 55 = 145 + A_C

GB = 90 + A_C

From Equation 1:

A_B = 0.5 × G_B

From Equation 6:

AB = A_C

So:

A_C = 0.5 × GB

Substitute A_C = 0.5 × G_B into G_B = 90 + A_C:

G_B = 90 + 0.5 × G_B

Subtract 0.5 × G_B from both sides:

0.5 × G_B = 90

G_B = 180

Now, find A_C:

A_C = 0.5 × G_B = 0.5 × 180 = 90

From Equation 6:

A_B = A_C = 90

From Equation 4:

A_A = 145 - AC = 145 - 90 = 55

From Equation 2:

G_A = G_B - 20 = 180 - 20 = 160

From Equation 3:

G_C = 130

Thus,

 

Calculations:

Questions attempted by A: 55

Questions given to B: 180

The ratio of A_A to G_B is:

A_A / G_B = 55 / 180

Divide both numerator and denominator by 5 to simplify:

55 / 180 = 11 / 36

Thus, the ratio of the number of questions attempted by A to the number of questions given to B is: 11:36