Search Previous Year Questions

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S = (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200 = 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Total Purses = 270 + 960 + 90 = 1,320

⇒ Total Suitcases = 450 + 1,200 + 240 = 1,890

Solution

Given:

Company C Purses = 90

Formula:

Ratio = First Quantity : Second Quantity

Calculation:

⇒ New Purses of Company C = 90 × 2 = 180

⇒ New total Purses = 270 + 960 + 180 = 1,410

⇒ Required ratio = 1,410 : 1,890 = 47 : 63

∴ Answer = 47 : 63

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S = (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200

= 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Suitcases manufactured by Companies A and C = 450 + 240 = 690

⇒ Total Purses manufactured by all companies = 270 + 960 + 90 = 1,320

Solution

Given:

Suitcases (A & C) = 690

Total Purses = 1,320

Formula:

Percentage = (Required Value ÷ Total Value) × 100

Calculation:

⇒ Percentage = (690 ÷ 1,320) × 100 = 52.27% ≈ 52.3%

∴ Answer = 52.3%

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S = (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200 = 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Purses manufactured by Companies A and B = 270 + 960 = 1,230

⇒ Suitcases manufactured by Companies B and C = 1,200 + 240 = 1,440

Solution

Given:

Purses (A & B) = 1,230

Suitcases (B & C) = 1,440

Formula:

Ratio = First Quantity : Second Quantity

Calculation:

⇒ Required ratio = 1,230 : 1,440 = 41 : 48

∴ Answer = 41 : 48

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S = (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200 = 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Purses manufactured by Companies A and C = 270 + 90 = 360

⇒ Purses manufactured by Company B = 960

Solution

Given:

Company B = 960

Companies A and C = 360

Formula:

Percentage = (Difference ÷ Original Value) × 100

Calculation:

⇒ Difference = 960 – 360 = 600

⇒ Percentage = (600 ÷ 360) × 100 = 166⅔%

∴ Answer = 166(2/3)%

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S = (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200

= 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Purses manufactured by Company B = 960

⇒ Suitcases manufactured by Company B = 1,200

Solution

Given:

Purses = 960

Increase = 25%

Suitcases = 1,200

Formula:

Increase = (Percentage ÷ 100) × Original Value

Calculation:

⇒ Increase = (25 ÷ 100) × 960 = 240

⇒ New Purses = 960 + 240 = 1,200

⇒ Difference = 1,200 − 1,200 = 0

∴ Answer = 0

Common Solution:

Given:

Company A

Given,

P − S = 180

(P + S) : S = 8 : 5

⇒ P + S = (8/5)S

⇒ P = (8/5 − 1)S

= (3/5)S

Now,

P – S = (3/5)S – S = (−2/5)S

Since the difference between the two quantities is 180,

S − P = 180

⇒ (2/5)S = 180

⇒ S = 180 × (5/2) = 450

P = (3/5) × 450 = 270

Company B

Given,

P − S = 240

(P + S) : S = 9 : 5

⇒ P + S = (9/5)S

⇒ P = (9/5 − 1)S = (4/5)S

Now,

S – P = S − (4/5)S = (1/5)S

Since the difference is 240,

⇒ (1/5)S = 240

⇒ S = 240 × 5 = 1,200

P = (4/5) × 1,200 = 960

Company C

Given,

P − S = 150

(P + S) : S = 11 : 8

⇒ P + S = (11/8)S

⇒ P = (11/8 − 1)S = (3/8)S

Now,

S – P = S − (3/8)S = (5/8)S

Since the difference is 150,

⇒ (5/8)S = 150

⇒ S = 150 × (8/5) = 240

P = (3/8) × 240 = 90

Final Table

 

⇒ Total Purses = 270 + 960 + 90 = 1,320

⇒ Average Purses = 1,320 ÷ 3 = 440

⇒ Total Suitcases = 450 + 1,200 + 240 = 1,890

⇒ Average Suitcases = 1,890 ÷ 3 = 630

Solution

Given:

Average Purses = 440

Average Suitcases = 630

Formula:

Percentage = (Required Value ÷ Total Value) × 100

Calculation:

⇒ Percentage

= (440 ÷ 630) × 100 = 69.84% ≈ 70%

∴ Answer = 70%

Shortcut Trick

Perimeter = 56 ⇒ l + b = 28  |  Area = lb = 180

(l − b)² = (l + b)² − 4lb = 784 − 720 = 64 ⇒ l − b = 8

Side of equilateral triangle = 8 cm

Area = √3/4 × 8² = √3/4 × 64 = 16√3 cm²

∴ The correct answer is 16√3 cm².

Alternate Method

Given:

Perimeter of rectangle = 56 cm ⇒ 2(l + b) = 56 ⇒ l + b = 28

Area of rectangle = lb = 180 cm²

Side of equilateral triangle = l − b

Formula used:

(l − b)² = (l + b)² − 4lb

Area of equilateral triangle = √3/4 × a² where a = side

Calculations:

Step 1: Find l + b and lb

⇒ 2(l + b) = 56 ⇒ l + b = 28

⇒ lb = 180

Step 2: Find l − b using identity

⇒ (l − b)² = (l + b)² − 4lb

⇒ (l − b)² = 28² − 4 × 180

⇒ (l − b)² = 784 − 720 = 64

⇒ l − b = √64 = 8 cm

Step 3: Find l and b (for verification)

⇒ l + b = 28 and l − b = 8

⇒ 2l = 36 ⇒ l = 18 cm

⇒ b = 28 − 18 = 10 cm

⇒ Verification: lb = 18 × 10 = 180 

Step 4: Find area of equilateral triangle

⇒ Side a = l − b = 8 cm

⇒ Area = √3/4 × a²

⇒ Area = √3/4 × 8²

⇒ Area = √3/4 × 64

⇒ Area = 16√3 cm²

⇒ Area ≈ 16 × 1.732 = 27.71 cm²

∴ The correct answer is 16√3 cm².

Additional Information

Key Identity: (l − b)² = (l + b)² − 4lb

This identity is the fastest way to find l − b when sum and product are known — no need to solve a quadratic. Similarly (l + b)² = (l − b)² + 4lb. In rectangle problems: always extract l+b from perimeter and lb from area first, then use this identity to find l−b directly.

Equilateral Triangle — All Formulas

For equilateral triangle with side a: Area = √3/4 × a² | Height = √3/2 × a | Perimeter = 3a | Inradius = a/(2√3) | Circumradius = a/√3. For a = 8: Area = 16√3 ≈ 27.71 cm² | Height = 4√3 ≈ 6.93 cm.

Rectangle — Diagonal Formula

Diagonal of rectangle d = √(l² + b²).

Note:

l² + b² =

Shortcut Trick

Let total voters = 100

Votes cast = 80 | Valid votes = 80 × 90% = 72

Winner = 60% of 72 = 43.2 | Loser = 40% of 72 = 28.8

Margin = 43.2 − 28.8 = 14.4 units ≡ 11520 votes

Total voters = (11520 ÷ 14.4) × 100 = 80000

∴ The correct answer is 80,000 voters.

Alternate Method

Given:

Total voters on list = N (to find)

Voters who did NOT vote = 20% of N

Votes cast = 80% of N

Invalid votes = 10% of votes cast

Valid votes = 90% of votes cast = 90% of 80% of N = 72% of N

Winner got = 60% of valid votes

Loser got = 40% of valid votes

Winning margin = 11,520 votes

Formula used:

Valid votes = 80% × 90% × N = 0.72N

Winning margin = (60% − 40%) of valid votes = 20% of valid votes

N = Margin ÷ (20% of 72% of N) × N

Total Voters = N (100%) Voting List Votes Cast = 80% of N = 0.80N Not Voted 20% Valid Votes = 72% of N = 0.72N Invalid 8% Winner 60% = 43.2% of N Loser 40% = 28.8% of N Margin = 20% of 72% of N = 14.4% of N = 11520

Calculations:

Let total voters = N

⇒ Votes cast = 80% of N = 0.80N

⇒ Valid votes = 90% of votes cast = 90% of 0.80N = 0.72N

⇒ Winner votes = 60% of valid = 60% of 0.72N = 0.432N

⇒ Loser votes = 40% of valid = 40% of 0.72N = 0.288N

⇒ Winning margin = Winner − Loser = 0.432N − 0.288N = 0.144N

⇒ 0.144N = 11520

⇒ N = 11520 ÷ 0.144

⇒ N = 11520 × 1000 ÷ 144

⇒ N = 11520000 ÷ 144

⇒ N = 80,000

Verification:

⇒ Votes cast = 80% of 80000 = 64000

⇒ Valid votes = 90% of 64000 = 57600

⇒ Winner = 60% of 57600 = 34560

⇒ Loser = 40% of 57600 = 23040

⇒ Margin = 34560 − 23040 = 11520 

∴ The correct answer is 80,000 voters.

Additional Information

Election Problem — Quick Margin Formula

Winning margin = (Winner% − Loser%) of valid votes = (60% − 40%) = 20% of valid votes. Since valid votes = 72% of N: margin = 20% of 72% of N = 14.4% of N. Setting 14.4% of N = 11520 gives N directly. This chain percentage method is the fastest approach for election problems.

Chain Percentage — Successive Application

When percentages are applied one after another: 80% of 90% of N = 0.80 × 0.90 × N = 0.72N. Always multiply the decimal factors together first before applying to N. Chain: 100% → 80% (voted) → 72% (valid) → 43.2% (winner) → 28.8% (loser) → 14.4% (margin).

Working with 100 as Base — Assumption Method

Assume total = 100. Compute all values numerically: valid = 72, winner = 43.2, loser = 28.8, margin = 14.4. Then scale: actual margin/assumed margin = actual total/assumed total ⇒ N = 11520/14.4 × 100 = 80000. This avoids algebra entirely and works for all percentage-chain problems.

Vote Share vs Winning Margin

If winner gets p% of valid votes: loser gets (100−p)%. Margin = (2p − 100)% of valid votes. Here p = 60%: margin = (120−100)% = 20% of valid votes. If winner gets 55%: margin = 10% of valid votes. If winner gets 75%: margin = 50% of valid votes — these ratios are useful for quick mental calculation.

Shortcut Trick

Net effect = (1 + 25/100) × (1 − 12/100) × 100 − 100

= 1.25 × 0.88 × 100 − 100

= 110 − 100 = 10%

∴ The correct answer is 10% profit.

Alternate Method

Given:

Mark up = 25% above CP

Discount = 12% on Marked Price (MP)

Find: Overall profit %

Formula used:

MP = CP × (100 + markup%) ÷ 100

SP = MP × (100 − discount%) ÷ 100

Profit % = (SP − CP) ÷ CP × 100

Direct formula: Profit % = (1 + m/100)(1 − d/100) × 100 − 100

where m = markup% and d = discount%

CP ₹100 +25% markup MP ₹125 −12% discount SP ₹110 Overall Profit = 110 − 100 = ₹10 = 10% base final SP disc=₹15 Profit = 10%

Calculations:

Let CP = ₹100

⇒ MP = CP + 25% of CP = 100 + 25 = ₹125

⇒ Discount = 12% of MP = 12% of 125 = ₹15

⇒ SP = MP − Discount = 125 − 15 = ₹110

⇒ Profit = SP − CP = 110 − 100 = ₹10

⇒ Profit % = (10 ÷ 100) × 100 = 10%

Using direct formula:

⇒ Profit % = (1 + 25/100) × (1 − 12/100) × 100 − 100

⇒ = (125/100) × (88/100) × 100 − 100

⇒ = (125 × 88) ÷ 100 − 100

⇒ = 11000 ÷ 100 − 100

⇒ = 110 − 100 = 10%

∴ The correct answer is 10% profit.

Additional Information

Direct Formula — Profit% with Markup and Discount

Profit % = (1 + m/100)(1 − d/100) × 100 − 100 where m = markup% and d = discount%. If result is positive ⇒ profit | negative ⇒ loss. Breakeven condition: m/100 × d/100 = (m − d)/100 ⇒ md = 100(m−d). For m=25, d=12: 25×12=300 ≠ 100(13)=1300 ⇒ profit exists.

When Does Profit = Loss = 0 (Breakeven)?

SP = CP when (1+m/100)(1−d/100) = 1. This gives d = 100m/(100+m). For m=25%: breakeven discount = 100×25/125 = 20%. So if discount is exactly 20% on 25% markup ⇒ no profit no loss. Any discount below 20% gives profit | above 20% gives loss.

Successive Percentage Change Shortcut

Net % change for two successive changes a% and b%: net = a + b + ab/100. Here a = +25 (markup) and b = −12 (discount): net = 25 − 12 + (25 × −12)/100 = 13 − 3 = 10%. This is the fastest mental math shortcut — no need to compute MP and SP separately.

Markup vs Profit — Key Difference

Markup is always calculated on CP | Discount is always calculated on MP | Profit/Loss is always calculated on CP. Markup % ≠ Profit % when discount is given. Here markup = 25% but profit = only 10% after 12% discount. Always trace CP → MP → SP to avoid confusion between markup and profit.

Shortcut Trick

Mixture A (500 ml): Milk = 7/10 × 500 = 350 ml | Water = 150 ml

Mixture B: Milk = 100 ml | Water = w ml

Final mixture: Milk = 350 + 100 = 450 ml | Water = 150 + w ml

Ratio 5:2 ⇒ 450/(150 + w) = 5/2 ⇒ 900 = 750 + 5w ⇒ w = 30 ml

Mixture B ratio = 100 : 30 = 10 : 3

∴ The correct answer is Milk : Water in B = 10 : 3.

Alternate Method

Given:

Mixture A = 500 ml | Milk : Water = 7 : 3

Mixture B = 100 ml milk + w ml water (unknown)

Final mixture ratio = Milk : Water = 5 : 2

Formula used:

Milk in A = 7/(7+3) × 500 = 350 ml

Water in A = 3/(7+3) × 500 = 150 ml

Final Milk / Final Water = 5/2

Mixture A 500 ml | 7:3 Milk = 350 ml Water = 150 ml + Mixture B (100 + w) ml Milk = 100 ml Water = w ml = Final Mix 5 : 2 M=450 W=180 450 / (150 + w) = 5/2 ⇒ 900 = 750 + 5w ⇒ w = 30 ml Total water in final = 150 + 30 = 180 ml ✓  |  Check: 450/180 = 5/2 ✓ Mixture B: Milk = 100 ml | Water = 30 ml Milk : Water in B = 100 : 30 = 10 : 3

Calculations:

Step 1: Find milk and water in Mixture A

⇒ Milk in A = 7/(7+3) × 500 = 7/10 × 500 = 350 ml

⇒ Water in A = 3/10 × 500 = 150 ml

Step 2: Set up final mixture

⇒ Total Milk = 350 + 100 = 450 ml

⇒ Total Water = 150 + w ml

Step 3: Apply final ratio condition

⇒ Milk / Water = 5/2

⇒ 450 / (150 + w) = 5/2

⇒ 2 × 450 = 5 × (150 + w)

⇒ 900 = 750 + 5w

⇒ 5w = 900 − 750 = 150

⇒ w = 30 ml

Step 4: Find ratio in Mixture B

⇒ Milk in B = 100 ml | Water in B = 30 ml

⇒ Milk : Water = 100 : 30 = 10 : 3

Verification:

⇒ Final milk = 450 | Final water = 150 + 30 = 180

⇒ 450 : 180 = 5 : 2 

∴ The correct answer is Milk : Water in Mixture B = 10 : 3.

Additional Information

Alligation Method — When to Use

Alligation is used when two mixtures of known concentrations are mixed to get a target concentration. Cross-multiply: (C_final − C_B) : (C_A − C_final) gives ratio of quantities. Here since quantity of B is unknown but its milk is fixed, the direct equation method is faster than alligation.

Replacement Method in Mixtures

When x litres of a mixture is replaced with pure milk repeatedly n times: Final milk concentration = (1 − x/V)ⁿ where V = total volume. After 1 replacement: milk fraction changes. After n replacements use the power formula. This is different from mixing two separate mixtures.

Mixing Three Mixtures

For three mixtures A, B, C: Final milk = milk_A + milk_B + milk_C | Final water = water_A + water_B + water_C. Always find individual components first then add. Never directly add the ratios — always convert ratio to actual quantities using total volume.

Common Mistake — Ratio vs Quantity

Ratios cannot be added directly: 7:3 + 5:2 ≠ 12:5. Always convert to actual quantities first. Milk in A = 350 (not 7) | Water in A = 150 (not 3). Only after finding actual ml of each component can you add and find the final ratio. This is the most common error in mixture problems.