Quantitative Aptitude PYQs

Shortcut Trick

For a series of a repeated digit 'd' in the form d + dd + ddd + ... up to n terms (where n ≤ 9):

Sum = d × 123...n

Here, the digit d = 4 and there are 5 terms.

Sum = 4 × 12345 = 49380

∴ The correct answer is 49380.

Alternate Method

Given:

Expression = 4 + 44 + 444 + 4444 + 44444

Calculation:

Factoring out 4 from the expression:

⇒ 4 × (1 + 11 + 111 + 1111 + 11111)

Adding the numbers inside the bracket step-by-step:

⇒ 1 + 11 = 12

⇒ 12 + 111 = 123

⇒ 123 + 1111 = 1234

⇒ 1234 + 11111 = 12345

Now, multiply the sum by 4:

⇒ 4 × 12345

Breaking it down:

⇒ 4 × (10000 + 2000 + 300 + 40 + 5)

⇒ 40000 + 8000 + 1200 + 160 + 20 = 49380

∴ The correct answer is 49380.

Additional Information

Sum of Repeated 1s

The sum of a series 1 + 11 + 111 + ... up to n terms (for n ≤ 9) is always a sequential number 123...n.

Sum of Repeated 9s

For a series like 9 + 99 + 999 + ... up to n terms, rewrite each term as (10 − 1) + (100 − 1) + (1000 − 1) ... to simplify the calculation to (10 + 100 + 1000 + ...) − n.

Column Addition Shortcut for Repeated Digits

To add repeated digit terms vertically, count the digits in each column. The units column has n terms, tens has n−1 terms, and so on. Multiply the digit by the count and sequentially carry over the extra value to the next place.

Shortcut Trick

Convert all mixed fractions directly into decimals for faster calculation.

1 ²/₅ = 1 + 0.4 = 1.4

1 ³/₄ = 1 + 0.75 = 1.75

⇒ (1.4 + 2.6) − 1.75

⇒ 4.0 − 1.75 = 2.25

∴ The correct answer is 2.25.

Alternate Method

Given:

(1 ²/₅ + 2.6) − 1 ³/₄

Method Used:

Convert mixed fractions and decimals into improper fractions and simplify using BODMAS.

Calculations:

⇒ 1 ²/₅ = ^{(1 × 5 + 2)}/₅ = ⁷/₅

⇒ 2.6 = ²⁶/₁₀ = ¹³/₅

⇒ 1 ³/₄ = ^{(1 × 4 + 3)}/₄ = ⁷/₄

⇒ Expression = (⁷/₅ + ¹³/₅) − ⁷/₄

⇒ Expression = (²⁰/₅) − ⁷/₄

⇒ Expression = 4 − ⁷/₄

⇒ Expression = ^{(16 − 7)}/₄

⇒ Expression = ⁹/₄

⇒ Expression = 2.25

∴ The correct answer is 2.25.

Additional Information

BODMAS Rule

Simplification problems always follow the BODMAS order: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Converting mixed fractions directly to decimals (e.g., ²/₅ = 0.4 and ³/₄ = 0.75) avoids LCM calculations and saves significant time in exams.

Mixed to Improper Fraction

A mixed fraction A ^B/_C can be converted to an improper fraction using the standard formula (A × C + B) ÷ C.

Shortcut Trick

Rewrite 3 as √9 and 2 as √4 to see the continuous pattern.

Rationalizing 1/(√a − √b) gives √a + √b when a − b = 1.

The given expression evaluates to: (√9 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + √4)

All intermediate terms cancel out, leaving only the first and last parts: √9 + √4 = 3 + 2 = 5

∴ The correct answer is 5.

Alternate Method

Given:

1/(3 − √8) − 1/(√8 − √7) + 1/(√7 − √6) − 1/(√6 − √5) + 1/(√5 − 2)

Formula Used:

(a − b)(a + b) = a² − b²

Calculations:

⇒ Rationalize the first term:

⇒ 1/(3 − √8) = (3 + √8) / (3² − (√8)²) = (3 + √8) / (9 − 8) = 3 + √8

⇒ Rationalize the second term:

⇒ 1/(√8 − √7) = (√8 + √7) / ((√8)² − (√7)²) = (√8 + √7) / (8 − 7) = √8 + √7

⇒ Rationalize the third term:

⇒ 1/(√7 − √6) = (√7 + √6) / (7 − 6) = √7 + √6

⇒ Rationalize the fourth term:

⇒ 1/(√6 − √5) = (√6 + √5) / (6 − 5) = √6 + √5

⇒ Rationalize the fifth term:

⇒ 1/(√5 − 2) = (√5 + 2) / ((√5)² − 2²) = (√5 + 2) / (5 − 4) = √5 + 2

⇒ Substitute all these simplified values back into the original expression:

⇒ (3 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + 2)

⇒ Expand by applying the minus signs:

⇒ 3 + √8 − √8 − √7 + √7 + √6 − √6 − √5 + √5 + 2

⇒ Cancel out the opposite matching pairs:

⇒ 3 + 0 + 0 + 0 + 0 + 2 = 5

∴ The correct answer is 5.

Additional Information

Rationalization of Surds

To remove a square root surd from the denominator, multiply both the numerator and denominator by its conjugate. The conjugate of (√a − √b) is (√a + √b).

Difference of Squares Formula

The standard algebraic identity (x − y)(x + y) = x² − y² is widely applied during rationalization to eliminate square root symbols entirely from denominators.

Continuous Series Simplification

When solving a continuous sequence of surd fractions, locating patterns like differences of 1 will allow quick cancellation of intermediate terms, drastically reducing calculation time.

Shortcut Trick

Total Football players = 25 (Both) + 45 (Only Football) = 70

Total Basketball players = 25 (Both) + 35 (Only Basketball) = 60

Ratio of Football to Basketball players = 70 ÷ 60 = 7 : 6

∴ The correct answer is 7:6.

Alternate Method

Given:

Students playing both Football and Basketball = 25

Students playing Football but not Basketball = 45

Students playing Basketball but not Football = 35

Formula Used:

Total players of a sport = Players of only that sport + Players of both sports

Calculation:

⇒ Number of students who play Football = 45 + 25 = 70

⇒ Number of students who play Basketball = 35 + 25 = 60

⇒ Required Ratio = Total Football players : Total Basketball players

⇒ Required Ratio = 70 : 60

⇒ Required Ratio = 7 : 6

∴ The correct answer is 7:6.

Additional Information

Venn Diagram Fundamentals

In a two-set Venn diagram, the total number of elements in set A is calculated by n(A) = n(Only A) + n(A and B).

Union of Two Sets

The total number of elements in at least one set is found using the formula: n(A or B) = n(A) + n(B) − n(A and B).

Ratio & Proportion

A ratio compares two quantities of the same kind. The ratio a:b is in its simplest form when a and b have no common factor other than 1.

Shortcut Trick

Work with ratios in 10-thousands to simplify the calculation (1 unit = ₹10,000).

Tina's equivalent capital = 10 × 6 + 8 × 6 = 60 + 48 = 108

Meera's equivalent capital = 8 × 6 + 12 × 6 = 48 + 72 = 120

Profit Ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Alternate Method

Given:

Tina's initial investment = ₹1,00,000

Meera's initial investment = ₹80,000

After 6 months, Tina withdrew ₹20,000 and Meera added ₹40,000.

₹100k × 6m

₹80k × 6m Tina

₹80k × 6m

₹120k × 6m Meera

Total = 1080k Total = 1200k Final Ratio = 1080 : 1200 = 9 : 10

Formula Used:

Profit Ratio = (C₁ × T₁ + C₂ × T₂) : (C₃ × T₃ + C₄ × T₄)

Calculations:

⇒ For the first 6 months, Tina's investment = ₹1,00,000

⇒ For the next 6 months, Tina's investment = ₹1,00,000 − ₹20,000 = ₹80,000

⇒ Total equivalent capital for Tina = (1,00,000 × 6) + (80,000 × 6) = 6,00,000 + 4,80,000 = ₹10,80,000

⇒ For the first 6 months, Meera's investment = ₹80,000

⇒ For the next 6 months, Meera's investment = ₹80,000 + ₹40,000 = ₹1,20,000

⇒ Total equivalent capital for Meera = (80,000 × 6) + (1,20,000 × 6) = 4,80,000 + 7,20,000 = ₹12,00,000

⇒ Profit sharing ratio = Tina's equivalent capital : Meera's equivalent capital

⇒ Profit sharing ratio = 10,80,000 : 12,00,000

⇒ Profit sharing ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Partnership Basic Rule

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time. P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Compound Partnership

If a partner's capital changes during the year, calculate the equivalent capital by multiplying each unique capital amount by the respective number of months it remained invested, then sum them up.

Active vs Sleeping Partner

An active partner takes part in running the business and typically gets a salary or a fixed percentage of the total profits before the remaining profit is divided based on their capital ratios.

Shortcut Trick

A's management fee = 20% of ₹60,000 = ₹12,000

Remaining profit to be shared = 60,000 − 12,000 = ₹48,000

Profit sharing ratio of A and B = 20,000 : 30,000 = 2 : 3

A's profit share = (2/5) × 48,000 = ₹19,200

A's total share = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Alternate Method

Given:

A's investment = ₹20,000

B's investment = ₹30,000

Total profit = ₹60,000

A's management commission = 20% of total profit

Total Profit: ₹60,000

A's Mgmt: ₹12,000

Remaining: ₹48,000

A's Share: ₹19,200

B's: ₹28,800

A's Total: ₹31,200

Calculations:

⇒ Profit distribution ratio = Ratio of investments (since time period is equal)

⇒ Ratio of A : B = 20,000 : 30,000 = 2 : 3

⇒ A's management fee = (20 / 100) × 60,000 = ₹12,000

⇒ Remaining profit = 60,000 − 12,000 = ₹48,000

⇒ A's share from remaining profit = 48,000 × [2 ÷ (2 + 3)]

⇒ A's share = 48,000 × (2 / 5) = ₹19,200

⇒ Total share of A = Management fee + Profit share

⇒ Total share of A = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Additional Information

Active vs Sleeping Partner

An active partner manages the business and receives a salary or a fixed percentage of the profit before the remainder is split. A sleeping partner only invests capital and takes a share of the remaining profit based on investment and time.

Partnership Profit Formula

Profit is directly proportional to the product of capital invested and the time period for which it is invested. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Shortcut Trick

Net increase in average = 48 − 44 = 4 runs

Total extra runs scored above the old average = 132 − 44 = 88 runs

Total number of innings played (new) = Total extra runs ÷ Net increase = 88 ÷ 4 = 22

Earlier innings played = 22 − 1 = 21

∴ The correct answer is 21 innings.

Alternate Method

Given:

Initial average = 44

Score in the next inning = 132

New average = 48

Formula Used:

Average = Total Runs ÷ Number of Innings

Avg = 44x inningsDeficit = 4 × xAvg = 48Next inningScore = 132Surplus = 84New Avg = 48132 runs

Calculation:

Let the number of innings played earlier be x.

⇒ Total runs scored in earlier innings = 44 × x = 44x

⇒ Total runs after the next inning = 44x + 132

⇒ New average = (44x + 132) ÷ (x + 1)

⇒ 48 = (44x + 132) ÷ (x + 1)

⇒ 48 × (x + 1) = 44x + 132

⇒ 48x + 48 = 44x + 132

⇒ 48x − 44x = 132 − 48

⇒ 4x = 84

⇒ x = 21

∴ The correct answer is 21 innings.

Additional Information

Batting Average

Batting average is calculated as the total number of runs scored divided by the total number of times the batsman was dismissed. Formula: Total Runs = Average × Innings.

Bowling Average

Bowling average is the number of runs conceded per wicket taken. Formula: Bowling Average = Total Runs Conceded ÷ Total Wickets Taken.

Deviation Method for Averages

When a new observation is added, the new average changes based on the deviation of the new value from the old average. Change in Average = (New Value − Old Average) ÷ New Total Number of Observations.

Shortcut Trick

Net deviation of the first 10 numbers from the overall average 70 = 5 × (60 − 70) + 5 × (75 − 70) = −50 + 25 = −25.

To balance the overall average, the sum of the last 3 numbers must be = 3 × 70 − (−25) = 235.

Let the 13^th number be x. Then 11^th = x − 6, and 12^th = x + 4.

Sum = (x − 6) + (x + 4) + x = 3x − 2 = 235 ⇒ x = 79.

The 11^th number = 73, and the 12^th number = 83. Their average = (73 + 83) ÷ 2 = 78.

∴ The correct answer is 78.

Alternate Method

Given:

Average of 13 numbers = 70

Average of first 5 numbers = 60

Average of next 5 numbers = 60 + (25% of 60) = 75

11^th number = 13^th number − 6

12^th number = 13^th number + 4

1^st 5 Nums Avg: 60 Sum: 300

Next 5 Nums Avg: 75 Sum: 375

11^th, 12^th, 13^th Avg: ? Sum: 235

Total Sum of 13 Nums = 910 (Avg = 70)

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total sum of 13 numbers = 13 × 70 = 910

⇒ Sum of first 5 numbers = 5 × 60 = 300

⇒ Sum of next 5 numbers = 5 × 75 = 375

⇒ Sum of the remaining 3 numbers (11^th, 12^th, 13^th) = 910 − (300 + 375) = 235

⇒ Let the 13^th number be x.

⇒ 11^th number = x − 6

⇒ 12^th number = x + 4

⇒ Sum of these three numbers = (x − 6) + (x + 4) + x = 3x − 2

⇒ 3x − 2 = 235

⇒ 3x = 237

⇒ x = 79

⇒ 11^th number = 79 − 6 = 73

⇒ 12^th number = 79 + 4 = 83

⇒ Average of 11^th and 12^th numbers = (73 + 83) ÷ 2 = 156 ÷ 2 = 78

∴ The correct answer is 78.

Additional Information

Weighted Average

When different groups have different averages, the combined average is calculated as (n₁A₁ + n₂A₂ + ...) ÷ (n₁ + n₂ + ...).

Net Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. This property is highly useful for finding missing observations in a data set without calculating heavy sum values.

Successive Percentage Increase

If a base value is increased by x%, the new value is obtained by multiplying the base by (1 + x/100). For example, a 25% increase on a base of 60 makes it 60 × 1.25 = 75.

Shortcut Trick

Let the total eligible voters be V.

V × 80% × 95% × 60% = 11400

V × (4/5) × (19/20) × (3/5) = 11400

V × (228/500) = 11400 ⇒ V = (11400 × 500) ÷ 228 = 25000

∴ The correct answer is 25000.

Alternate Method

Given:

Votes cast = 80% of total eligible voters

Invalid votes = 5% of cast votes

Winner's votes = 60% of valid votes = 11400

Total Eligible Voters = 100x

Votes Cast = 80x

Valid Votes = 76x

Winner's Votes = 45.6x = 11400

80% cast

95% valid

60% of valid

Calculations:

⇒ Let the total eligible voters be 100x.

⇒ Votes cast = 80% of 100x = 80x

⇒ Valid votes = 80x − (5% of 80x) = 80x − 4x = 76x

⇒ Winner's votes = 60% of 76x = 45.6x

⇒ We are given that Winner's votes = 11400

⇒ 45.6x = 11400

⇒ x = 11400 ÷ 45.6 = 250

⇒ Total eligible voters = 100x = 100 × 250 = 25000

∴ The correct answer is 25000.

Additional Information

Election Percentage Concept

Always maintain the chain: Total Voters → Votes Cast → Valid Votes → Winner's Votes. Subtract invalid votes from cast votes, not from total eligible voters.

Successive Percentage

When a value is subjected to multiple percentage changes sequentially, Final Value = Initial Value × (1 ± p₁/100) × (1 ± p₂/100).

Base Change in Percentage

The base for the winner's 60% share is 'Valid Votes' and not 'Total Voters'. Always correctly identify the base "of what" the percentage is calculated.

Shortcut Trick

Effective rate for the 1^st year = 18%

Effective rate for the next 1/2 year = 18% ÷ 2 = 9%

Total effective Compound Interest % = 18 + 9 + (18 × 9) / 100 = 28.62%

According to the question, 28.62% of Principal = ₹2862

Principal = (2862 ÷ 28.62) × 100 = 10000

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Compound Interest (CI) = ₹2862

Rate of Interest (R) = 18% p.a.

Time (T) = 1.5 years

Formula Used:

For time = n + f years (where n is the complete year and f is the fraction), compounded yearly:

Amount (A) = P × (1 + R/100)ⁿ × (1 + fR/100)

Compound Interest (CI) = A − P

100% (P)

+18% (1 Year)

118%

+9% (1/2 Year)

128.62%

Total CI = 28.62% = ₹2862

Calculations:

⇒ Amount (A) = P × (1 + 18/100) × (1 + (1/2 × 18)/100)

⇒ A = P × (118/100) × (109/100)

⇒ A = P × 1.18 × 1.09

⇒ A = 1.2862 P

⇒ CI = A − P = 1.2862 P − P = 0.2862 P

⇒ 0.2862 P = 2862

⇒ P = 2862 ÷ 0.2862

⇒ P = 10000

∴ The correct answer is ₹10,000.

Additional Information

Fractional Year Compounding

When time is given in fractions (e.g., n full years and f fraction of a year), interest for the fractional year is applied as simple interest on the compounded amount: A = P × (1 + R/100)ⁿ × (1 + fR/100).

Effective Interest Rate

For two successive periods with different rates R₁ and R₂, the net effective rate is given by the formula: R₁ + R₂ + (R₁ × R₂) / 100.

Half-Yearly & Quarterly Compounding

If compounded half-yearly, the rate becomes R/2 and the time periods become 2T. If quarterly, the rate is R/4 and the time is 4T.

Shortcut Trick

Rate = 16 2/3% = 1/6.

Let Principal = 6³ = 216 units.

Amount after 3 years = 7³ = 343 units ⇒ CI = 343 − 216 = 127 units.

Given CI = ₹1270 ⇒ 127 units = 1270 ⇒ 1 unit = ₹10.

SI for 3 years = 216 × (1/6) × 3 = 108 units = 108 × 10 = ₹1080.

∴ The correct answer is ₹1080.

Alternate Method

Given:

Rate (R) = 16 2/3% = 50/3% per annum

Time (T) = 3 years

Compound Interest (CI) = ₹1270

Formula Used:

Amount (A) = P × (1 + R/100)^T

CI = A − P

SI = (P × R × T) ÷ 100

Principal = ₹2160

Year 1 SI = ₹360

Year 2 SI = ₹360

Year 3 SI = ₹360

Total SI = ₹1080

Calculations:

⇒ Let the Principal be P.

⇒ Amount (A) = P × (1 + 1/6)³

⇒ A = P × (7/6)³ = P × (343/216)

⇒ CI = A − P = P × (343/216) − P = P × (127/216)

⇒ P × (127/216) = 1270

⇒ P = 1270 × (216/127) = 10 × 216 = 2160

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

⇒ SI = (2160 × 50/3 × 3) ÷ 100

⇒ SI = (2160 × 50) ÷ 100 = 1080

∴ The correct answer is ₹1080.

Additional Information

Difference between CI and SI for 2 years

The difference between compound interest and simple interest for 2 years on a certain sum is directly calculated by: D = P(R/100)².

Difference between CI and SI for 3 years

For a 3-year period, the difference formula extends to: D = P(R/100)² × (3 + R/100).

Multiplying Factor Method

When the rate of interest fraction is 1/x, the principal is usually assumed as x³ for 3 years to simplify compound interest calculations and avoid decimals.

Shortcut Trick Calculate the net profit or loss per unit directly using individual deviations from the final Selling Price: Loss on Darjeeling tea = (300 − 260) = ₹40/kg ⇒ Total = 15 × 40 = ₹600 (Loss) Profit on Assam tea = (260 − 240) = ₹20/kg ⇒ Total = 25 × 20 = ₹500 (Profit) Net overall effect = 500 (Profit) − 600 (Loss) = −100 (Net Loss) ∴ The correct answer is Loss of ₹100.

Alternate Method Given: Quantity of Darjeeling tea = 15 kg at ₹300/kg Quantity of Assam tea = 25 kg at ₹240/kg Selling price of the entire mixture = ₹260/kg

Formula Used: Total Cost Price (CP) = Sum of (Quantity × Rate per kg) Total Selling Price (SP) = Total Quantity × SP per kg Profit or Loss = Total SP − Total CP

Darjeeling Tea 15 kg × ₹300 = ₹4500

Assam Tea 25 kg × ₹240 = ₹6000

Total CP = ₹10500

Loss = ₹100

40 kg × ₹260 Total SP = ₹10400

Calculations: ⇒ Total CP of Darjeeling tea = 15 × 300 = ₹4500 ⇒ Total CP of Assam tea = 25 × 240 = ₹6000 ⇒ Total Cost Price (CP) of the mixture = 4500 + 6000 = ₹10500 ⇒ Total quantity of the mixture = 15 + 25 = 40 kg ⇒ Total Selling Price (SP) of the mixture = 40 × 260 = ₹10400 ⇒ Since Total CP > Total SP, the seller incurs a loss. ⇒ Loss = Total CP − Total SP = 10500 − 10400 = ₹100 ∴ The correct answer is Loss of ₹100.

Additional Information Alligation Method Used to find the ratio in which two ingredients are mixed to produce a mixture of a desired mean price. The ratio of quantities is inversely proportional to the differences in their prices from the mean price. Weighted Average To find the mean price directly without calculating overall totals, use the formula: Mean Price = (Q1 × P1 + Q2 × P2) ÷ (Q1 + Q2). Profit and Loss Percentage Profit % = (Profit ÷ Total CP) × 100. Loss % = (Loss ÷ Total CP) × 100. Both are always calculated on the total Cost Price unless specifically mentioned otherwise.

Shortcut Trick

To avoid decimal calculations, multiply the SP ratio 4 : 5 : 6 by 9 to get assumed SP values: 36, 45, and 54.

Total SP = 36 + 45 + 54 = 135 units.

CP of A = 36 ÷ 0.90 = 40, CP of B = 45 ÷ 1.50 = 30, and CP of C = 54 (no profit/loss).

Total CP = 40 + 30 + 54 = 124 units.

Profit = 135 − 124 = 11 units. ⇒ Profit % = (11 ÷ 124) × 100 ≈ 8.87% (approx. 8.88%).

∴ The correct answer is profit of 8.88%.

Alternate Method

Given:

Selling price (SP) ratio of A, B, and C = 4 : 5 : 6

Loss on item A = 10%

Profit on item B = 50%

No profit or loss on item C

Formula Used:

Cost Price (CP) = (SP × 100) ÷ (100 ± Profit/Loss %)

Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Total Cost Price (CP) = 124 unitsTotal Selling Price (SP) = 135 unitsProfit = 11Overall Profit % = (11 ÷ 124) × 100 ≈ 8.87%

Calculations:

⇒ Let the SP of items A, B, and C be 36x, 45x, and 54x respectively. (Multiplying the ratio by 9x to avoid fractional CP)

⇒ Total SP = 36x + 45x + 54x = 135x

⇒ CP of A = (36x × 100) ÷ 90 = 40x

⇒ CP of B = (45x × 100) ÷ 150 = 30x

⇒ CP of C = 54x (Since he broke even on item C, CP = SP)

⇒ Total CP = 40x + 30x + 54x = 124x

⇒ Overall Profit = Total SP − Total CP

⇒ Overall Profit = 135x − 124x = 11x

⇒ Profit % = (11x ÷ 124x) × 100

⇒ Profit % = 1100 ÷ 124 ≈ 8.87%

⇒ The closest calculated value matches option 2.

∴ The correct answer is profit of 8.88%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Dishonest Dealer

When a dealer sells goods at CP but uses false weights, the profit percentage is calculated as: Profit % = [(True Weight − False Weight) ÷ False Weight] × 100.

Successive Discounts

The equivalent single discount for two successive discounts of a% and b% is given by: Equivalent Discount % = (a + b − ab ÷ 100)%.

Shortcut Trick

Remaining mixture volume = 75 − 15 = 60 liters.

Since the ratio is 3:2, Milk = 60 × (3 / 5) = 36 liters and Water = 60 × (2 / 5) = 24 liters.

After adding 10 liters of pure milk, the new volume of Milk = 36 + 10 = 46 liters.

The volume of Water remains the same at 24 liters.

New ratio of Milk to Water = 46 : 24 = 23 : 12.

∴ The correct answer is 23:12.

Alternate Method

Given:

Total initial volume of mixture = 75 liters

Initial ratio of Milk to Water = 3 : 2

Volume of mixture removed = 15 liters

Volume of pure milk added = 10 liters

60 L Mix Milk: 36 L Water: 24 L

10 L Pure Milk

=

70 L Final Milk: 46 L Water: 24 L

Mixture Replacement Visualization Final Ratio = 46 : 24 = 23 : 12

Calculations:

⇒ Quantity of mixture remaining after removal = 75 − 15 = 60 liters

⇒ In the remaining 60 liters, the ratio of Milk to Water remains 3 : 2

⇒ Quantity of milk in the remaining mixture = 60 × (3 / 5) = 36 liters

⇒ Quantity of water in the remaining mixture = 60 × (2 / 5) = 24 liters

⇒ Now, 10 liters of pure milk is added to the mixture

⇒ Total milk in the final mixture = 36 + 10 = 46 liters

⇒ Total water in the final mixture remains strictly unchanged = 24 liters

⇒ Final ratio of Milk to Water = 46 : 24

⇒ Simplifying the ratio by dividing by 2 = 23 : 12

∴ The correct answer is 23:12.

Additional Information

Ratio Stability Principle

When any portion of a uniform mixture is drawn out, the ratio of its constituents remains exactly the same in both the drawn-off part and the remaining mixture.

Single Component Addition

If pure milk is explicitly added to a milk-water mixture, the absolute quantity of water naturally remains fully unchanged from its previous state, altering only the total total volume and the milk fraction.

Successive Replacement Formula

When a volume 'v' of a mixture of total volume 'V' is repeatedly drawn out and completely replaced by a pure component 'n' times, the remaining quantity of the other untouched component is given by Initial Quantity × (1 − v/V)ⁿ.

Shortcut Trick

Applying successive percentage multipliers on the base cost C yields the final selling price:

Final SP = C × 1.5 (markup) × 0.8 (discount) × 1.4 (markup) × 0.9 (discount)

⇒ 18144 = C × 1.2 × 1.26

⇒ 18144 = 1.512C

⇒ C = 18144 ÷ 1.512 = 12000

∴ The correct answer is ₹12,000.

Alternate Method

Given:

Factory base cost = ₹ C

Wholesaler markup = 50%, Trade discount = 20%

Retailer markup = 40%, Customer discount = 10%

Final amount paid by customer = ₹18,144

Base Cost₹ CRetailer CP1.2 CFinal SP1.512 CMarkup 50%Discount 20%Markup 40%Discount 10%1.512 C = ₹18,144

Calculations:

⇒ Wholesaler Marked Price (MP) = C × (1 + 50 ÷ 100) = 1.5C

⇒ Retailer Cost Price (CP) = 1.5C × (1 − 20 ÷ 100) = 1.5C × 0.8 = 1.2C

⇒ Retailer Marked Price (MP) = 1.2C × (1 + 40 ÷ 100) = 1.2C × 1.4 = 1.68C

⇒ Final Selling Price (SP) = 1.68C × (1 − 10 ÷ 100) = 1.68C × 0.9 = 1.512C

Equating this to the final amount paid:

⇒ 1.512C = 18144

⇒ C = 18144 ÷ 1.512

⇒ C = 12000

∴ The correct answer is ₹12,000.

Additional Information

Successive Percentage Change

When a value undergoes multiple sequential percentage changes (like markup followed by discount), the final value is found by multiplying the initial base value by the combined multiplying factors (1 + R ÷ 100) or (1 − R ÷ 100).

Marked Price & Discount

Discount is always calculated on the Marked Price (also known as Printed Price or Label Price) and never on the Cost Price. Formula: Selling Price = Marked Price × (1 − Discount%).

Effective Discount Formula

If two successive discounts of a% and b% are given, the equivalent single discount percentage is given by the formula: (a + b − a × b ÷ 100)%.

Shortcut Trick

Total Work = LCM(16, 12, 4) = 48 units

Efficiencies: A = 48 ÷ 16 = 3, B = 48 ÷ 12 = 4, A + B + C = 48 ÷ 4 = 12

Efficiency of C = 12 − (3 + 4) = 5 units/day

Time taken by C alone = 48 ÷ 5 = 9 ³/₅ days

∴ The correct answer is 9 ³/₅ days.

Alternate Method

Given:

⇒ Time taken by A = 16 days

⇒ Time taken by B = 12 days

⇒ Time taken by A, B and C together = 4 days

Formula Used:

⇒ Total Work = Efficiency × Time

Calculations:

⇒ Total efficiency of A, B and C = 12 units/day

⇒ Efficiency of A = 3 units/day

⇒ Efficiency of B = 4 units/day

⇒ Efficiency of C = Total efficiency − (Efficiency of A + Efficiency of B)

⇒ Efficiency of C = 12 − (3 + 4) = 5 units/day

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

⇒ Time taken by C alone = 48 ÷ 5 = 9 ³/₅ days

∴ The correct answer is 9 ³/₅ days.

Additional Information

Work & Wages

Wages are distributed in direct proportion to the work done. If individuals work for the same number of days, their wages are divided exactly in the ratio of their daily efficiencies.

Pipes & Cisterns

This concept operates similarly to time and work, but inlet pipes have a positive efficiency (filling) while outlet pipes or leaks have a negative efficiency (emptying).

Alternate Days Work

When workers operate on alternate days, find the combined work done in a complete cycle (e.g., 2 days for 2 workers) and multiply to approach the total work limit before calculating the remaining days.

Shortcut Trick

Cost Price (CP) of mixture = 55 × (100 ÷ 110) = ₹50/kg.

Using the net deviation method from the average CP (₹50):

Net deviation = 2 × (40 − 50) + 3 × (50 − 50) + 5 × (x − 50) = 0

⇒ −20 + 0 + 5(x − 50) = 0

⇒ 5(x − 50) = 20 ⇒ x − 50 = 4 ⇒ x = 54

∴ The correct answer is ₹54.00.

Alternate Method

Given:

Prices of the first two varieties = ₹40/kg and ₹50/kg

Ratio of the three varieties = 2 : 3 : 5

Selling Price (SP) of the mixture = ₹55/kg at 10% profit

Mixture CP = ₹55 × (100 ÷ 110) = ₹50/kg2 parts₹40/kg3 parts₹50/kg5 parts₹x/kgTotal = 10 partsAvg CP = ₹50/kg

Formula Used:

Cost Price (CP) = (SP × 100) ÷ (100 + Profit %)

Average Cost = (Total Cost) ÷ (Total parts)

Calculations:

⇒ CP of the mixture = (55 × 100) ÷ (100 + 10) = ₹50/kg

⇒ Let the price of the third variety be ₹x/kg.

⇒ Total parts of the mixture = 2 + 3 + 5 = 10 parts

⇒ Total CP for 10 parts = 10 × 50 = ₹500

⇒ Sum of individual costs = (2 × 40) + (3 × 50) + (5 × x)

⇒ 80 + 150 + 5x = 500

⇒ 230 + 5x = 500

⇒ 5x = 500 − 230 = 270

⇒ x = 270 ÷ 5 = 54

∴ The correct answer is ₹54.00.

Additional Information

Rule of Alligation

When two ingredients are mixed, the ratio of their quantities is inversely proportional to the difference in their prices from the average price. Quantity of cheaper / Quantity of dearer = (CP of dearer − Mean price) / (Mean price − CP of cheaper).

Weighted Average Method

For multiple mixtures, the weighted average cost is calculated as A_w = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n), where w represents the weight or ratio and x represents the value.

Cost Price from Selling Price

When profit percentage (P%) is given, Cost Price (CP) = SP × 100 ÷ (100 + P%). All mixture calculations must always be done on Cost Price, never on Selling Price directly.

Shortcut Trick

Let total work be LCM(8, 3) = 24 units.

Efficiency of A = 24 ÷ 8 = +3 units/day, Efficiency of B = 24 ÷ 3 = −8 units/day (negative as B breaks the wall).

Work done in first 4 days by A = 4 × 3 = +12 units.

Work done in next 2 days by A and B together = 2 × (3 − 8) = −10 units.

Total work completed = 12 − 10 = 2 units. Remaining work = 24 − 2 = 22 units.

Time for A to build the remaining work = 22 ÷ 3 = 7¹/₃ days.

∴ The correct answer is 7¹/₃ days.

Alternate Method

Given:

Time taken by A to build the wall = 8 days

Time taken by B to break the wall = 3 days

Calculations:

⇒ Work completed by A alone in the first 4 days = 4 × 3 = 12 units

⇒ Combined efficiency of A and B when working together = 3 − 8 = −5 units/day

⇒ Work done by A and B together in 2 days = 2 × (−5) = −10 units

⇒ Total work completed by the end of 6 days = 12 − 10 = 2 units

⇒ Remaining work to be completed = 24 − 2 = 22 units

⇒ Time required by A alone to build the remaining wall = Remaining work ÷ Efficiency of A

⇒ Time required = 22 ÷ 3 = 7¹/₃ days

∴ The correct answer is 7¹/₃ days.

Additional Information

Pipes & Cisterns Analogy (Negative Work)

When a person breaks a wall or a pipe empties a tank, their work is considered negative. Combined work = Positive Efficiency − Negative Efficiency.

Alternate Day Work

When people work on alternate days, calculate the work done in a complete cycle of days, then find the remaining work and assign it sequentially to the person whose turn it is next.

Work & Wages

Wages are strictly distributed in the ratio of the total work done by each individual. If the time worked by all is the same, the wages are simply distributed in the ratio of their efficiencies.

Shortcut Trick

When distance is constant, speed is inversely proportional to time.

Time ratio = 5 : 1

Speed ratio = 1 : 5

Given 1 unit of speed = 240 km/h

Required speed = 5 units = 5 × 240 = 1200 km/h.

∴ The correct answer is 1200 km/h.

Alternate Method

Case 1: 5 Hours

Distance = D Speed = 240 km/h | Time = 5 h

Case 2: 1 Hour

Distance = D Speed = ? | Time = 1 h

Distance is constant in both cases

Given:

Initial speed (S₁) = 240 km/h

Initial time (T₁) = 5 hours

Required time (T₂) = 1 hour

Formula Used:

Distance = Speed × Time

Speed = Distance ÷ Time

Calculations:

⇒ Total distance covered = S₁ × T₁

⇒ Distance = 240 × 5 = 1200 km

⇒ To cover the same distance in 1 hour, required speed (S₂) = Distance ÷ T₂

⇒ S₂ = 1200 ÷ 1 = 1200 km/h

∴ The correct answer is 1200 km/h.

Additional Information

Unit Conversion

To convert speed from km/h to m/s, multiply the speed by 5 ÷ 18. To convert speed from m/s to km/h, multiply by 18 ÷ 5.

Average Speed

Average Speed = Total Distance ÷ Total Time. If two equal distances are covered at speeds a and b respectively, then the average speed is given by (2 × a × b) ÷ (a + b).

Relative Speed

When two objects move in the same direction, their relative speed is S₁ − S₂. When they move in opposite directions towards each other, their relative speed is S₁ + S₂.

Shortcut Trick

Distance = [ (S₁ × S₂) / |S₁ − S₂| ] × Δt

Here, S₁ = 50 km/h, S₂ = 30 km/h.

Time difference Δt = 1 hour early + 1 hour late = 2 hours.

⇒ Distance = [ (50 × 30) / (50 − 30) ] × 2

⇒ Distance = [ 1500 / 20 ] × 2 = 150 km.

∴ The correct answer is 150 km.

Alternate Method

City A City B

Distance = d km

50 km/h → Arrives 1 hr early

30 km/h → Arrives 1 hr late

Given:

Speed 1 (S₁) = 50 km/h, Time 1 (t₁) = t − 1 hours

Speed 2 (S₂) = 30 km/h, Time 2 (t₂) = t + 1 hours

Formula Used:

Distance = Speed × Time

Calculations:

Let the usual time taken to cover distance d be t hours.

⇒ d = 50 × (t − 1) --- (Equation 1)

⇒ d = 30 × (t + 1) --- (Equation 2)

Equating both the distances since the actual distance is constant:

⇒ 50 × (t − 1) = 30 × (t + 1)

⇒ 5t − 5 = 3t + 3

⇒ 2t = 8

⇒ t = 4 hours

Substituting the value of t in Equation 1:

⇒ d = 50 × (4 − 1)

⇒ d = 50 × 3 = 150 km

∴ The correct answer is 150 km.

Additional Information

Average Speed

When a person travels equal distances at two different speeds S₁ and S₂, the average speed for the entire journey is given by (2 × S₁ × S₂) / (S₁ + S₂).

Relative Speed

If two bodies are moving in the same direction, relative speed = |S₁ − S₂|. If they are moving in opposite directions, relative speed = S₁ + S₂.

Time Difference Formula

When travelling the same distance with varying speeds resulting in early or late arrivals, the direct distance is always (S₁ × S₂) / |S₁ − S₂| × Total Time Difference.