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Shortcut Trick

Let the total capacity of the cistern be the LCM of (10, 20, 12) = 60 units.

Combined Efficiency: (A+B) = 6, (B+C) = 3, (C+A) = 5.

Summing them: 2(A+B+C) = 6 + 3 + 5 = 14 ⇒ A+B+C = 7 units/hr.

Efficiency of Pipe B = (A+B+C) − (C+A) = 7 − 5 = 2 units/hr.

Time taken by B = 60 ÷ 2 = 30 hr.

∴ The correct answer is 30 hr.

Alternate Method

Given:

Time taken by (Pipe A + Pipe B) = 10 hr

Time taken by (Pipe B + Pipe C) = 20 hr

Time taken by (Pipe C + Pipe A) = 12 hr

Formula Used:

Efficiency = Total Work ÷ Time taken

Calculations:

⇒ Adding efficiencies: (A + B) + (B + C) + (C + A) = 6 + 3 + 5

⇒ 2(A + B + C) = 14

⇒ A + B + C = 14 ÷ 2 = 7 units/hr

⇒ We know efficiency of (C + A) = 5 units/hr

⇒ Efficiency of B = (A + B + C) − (C + A) = 7 − 5 = 2 units/hr

⇒ Time taken by Pipe B alone = Total Work ÷ Efficiency of B

⇒ Time = 60 ÷ 2 = 30 hr

∴ The correct answer is 30 hr.

Additional Information

Combined Efficiency Rule

When pipes work in pairs, the sum of their individual efficiencies is half the sum of their pairwise efficiencies: (A+B+C) = [(A+B) + (B+C) + (C+A)] ÷ 2.

Pipes and Cisterns Basics

Inlet pipes perform positive work (filling), while outlet pipes perform negative work (emptying). Total work is the volume of the tank/cistern.

Time and Work Relation

Time taken and efficiency are inversely proportional to each other. Time = 1 ÷ Efficiency (for constant work).

Shortcut Trick

Approximate Effective Discount for each Scheme:

(A) 21 + 8 − (21 × 8 / 100) = 27.32%

(B) Successive 38%, 42%, 30% → clearly much higher than 50% (≈ 74.8%)

(C) Weighted Average = [(8 × 17) + (2 × 20)] ÷ 10 = 17.6%

(D) Buy 10 Get 7 Free = [7 ÷ (10 + 7)] × 100 ≈ 41.18%

Comparing the values, 17.6% is the lowest discount, making it the least beneficial.

∴ The correct answer is C.

Alternate Method

Given:

⇒ Total Toys to buy = 10

⇒ Let Marked Price (MP) of 1 toy = ₹100

⇒ Total Marked Price for 10 toys = ₹1000

Formula Used:

⇒ Successive Discount = a + b − (ab/100)

⇒ Buy X Get Y Free Discount % = [Y ÷ (X + Y)] × 100

Calculations:

Scheme (A): Successive 21% and 8%

⇒ Effective Discount = 21 + 8 − (21 × 8 / 100) = 29 − 1.68 = 27.32%

Scheme (B): Successive 38%, 42%, 30%

⇒ First two: 38 + 42 − (38 × 42 / 100) = 80 − 15.96 = 64.04%

⇒ Final: 64.04 + 30 − (64.04 × 30 / 100) = 94.04 − 19.21 = 74.83%

Scheme (C): 17% on first 8, 20% on next 2

⇒ Discount on 8 toys = 8 × 17 = ₹136

⇒ Discount on 2 toys = 2 × 20 = ₹40

⇒ Total Discount = 136 + 40 = ₹176

⇒ Effective % = (176 ÷ 1000) × 100 = 17.6%

Scheme (D): Buy 10 Get 7 Free

⇒ Effective Discount % = [7 ÷ (10 + 7)] × 100 = (7 ÷ 17) × 100 ≈ 41.18%

⇒ Comparing all schemes, 17.6% is the minimum discount.

∴ The correct answer is C.

Additional Information

Successive Discounts

When multiple discounts are applied one after another, the total discount is always less than the sum of individual percentages. Formula: a + b − (ab/100).

Buy X Get Y Free

The effective discount percentage in such offers is calculated as: [Free Items ÷ Total Items (Bought + Free)] × 100.

Least Beneficial vs Most Beneficial

For a customer, the least beneficial scheme is the one offering the lowest effective discount percentage, while the most beneficial offers the highest.

Shortcut Trick

Convert decimals to fractions for faster calculation: 0.25 = 1/4 and 0.5 = 1/2.

Numerator: (1/4) × 1.6 = 0.4

Denominator: 0.04 × (1/2) = 0.02

Result = 0.4 ÷ 0.02 = 40 ÷ 2 = 20

∴ The correct answer is 20.

Alternate Method

Given: (0.25 × 1.6) ÷ (0.04 × 0.5)

Method Used: Apply the BODMAS rule (perform operations inside Brackets first).

⇒ Step 1 (First bracket): 0.25 × 1.6 = 0.400 = 0.4

⇒ Step 2 (Second bracket): 0.04 × 0.5 = 0.020 = 0.02

⇒ Step 3 (Division): 0.4 ÷ 0.02

⇒ To remove the decimal from the divisor, multiply both numerator and denominator by 100.

⇒ (0.4 × 100) ÷ (0.02 × 100) = 40 ÷ 2

⇒ Result = 20

∴ The correct answer is 20.

Additional Information

BODMAS Rule

It stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction. Operations must be performed in this specific order to get the correct result.

Multiplying Decimals

Multiply the numbers as if they were whole numbers, then place the decimal point in the product so that it has the same number of decimal places as the sum of the decimal places in the factors.

Dividing by Decimals

Move the decimal point in the divisor to the right until it is a whole number, and move the decimal point in the dividend the same number of places to the right before dividing.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 100 + 150 + 50 = 300 km

Total Time = (100/50) + (150/60) + (50/40) = 2 + 2.5 + 1.25 = 5.75 hours

Average Speed = 300 ÷ 5.75 ≈ 52.17 km/h

∴ The correct answer is 52.17 km/h.

Alternate Method

Given:

Distance₁ = 100 km, Speed₁ = 50 km/h

Distance₂ = 150 km, Speed₂ = 60 km/h

Distance₃ = 50 km, Speed₃ = 40 km/h

Formula Used:

Time = Distance ÷ Speed

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Time for first part (t₁) = 100 ÷ 50 = 2 hours

⇒ Time for second part (t₂) = 150 ÷ 60 = 2.5 hours

⇒ Time for third part (t₃) = 50 ÷ 40 = 1.25 hours

⇒ Total Time (T) = 2 + 2.5 + 1.25 = 5.75 hours

⇒ Average Speed = 300 ÷ 5.75

⇒ Average Speed = 30000 ÷ 575 = 1200 ÷ 23

⇒ Average Speed ≈ 52.1739... km/h

∴ The correct answer is 52.17 km/h.

Additional Information

Average Speed for Equal Distances

When a journey is divided into two equal distances with speeds x and y, the Average Speed is the harmonic mean: (2 × x × y) ÷ (x + y).

Average Speed for Equal Time Intervals

If an object travels at speed x for time T and then at speed y for the same time T, the Average Speed is the arithmetic mean: (x + y) ÷ 2.

Unit Conversion

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

Shortcut Trick

Assume the total quantity is 12 units (LCM of 3 and 4).

Net profit/loss units = (1/3 × 12 × 20) + (1/4 × 12 × (−10)) + (5/12 × 12 × 0)

Net effect = 80 − 30 + 0 = 50 units.

Overall percentage = 50 ÷ 12 = 25/6 = 4 ¹/₆%.

∴ The correct answer is 4 ¹/₆% profit.

Alternate Method

Given: Total Cost Price (CP) = ₹2400

Formula Used: Profit % = (Total Profit ÷ Total CP) × 100

⇒ Profit on first part = (1/3) × 2400 × 20% = 800 × 0.20 = ₹160

⇒ Loss on second part = (1/4) × 2400 × 10% = 600 × 0.10 = ₹60

⇒ Remaining goods = 1 − (1/3 + 1/4) = 1 − 7/12 = 5/12

⇒ Profit/Loss on remaining part = (5/12) × 2400 × 0 = ₹0

⇒ Total Net Profit = 160 − 60 + 0 = ₹100

⇒ Overall Profit Percentage = (100 ÷ 2400) × 100 = 100/24 = 25/6 = 4 ¹/₆%

∴ The correct answer is 4 ¹/₆% profit.

Additional Information

Weighted Average Profit/Loss

When goods are sold in parts, the net profit/loss percentage is the weighted average: % = (w₁p₁ + w₂p₂ + ...) ÷ (w₁ + w₂ + ...).

Cost Price Selling

Selling an item at its Cost Price (CP) implies there is no profit and no loss (0% change).

Fractional Parts Calculation

Always ensure the sum of all fractional parts equals 1 (e.g., 1/3 + 1/4 + 5/12 = 12/12 = 1).

Shortcut Trick

Let the HCF of the three numbers be x.

The numbers are then 3x, 7x, and 2x.

LCM of (3x, 7x, 2x) = x × LCM(3, 7, 2) = x × 42.

Given, 42x = 8820.

x = 8820 ÷ 42 = 210.

∴ The correct answer is 210.

Alternate Method

Given:

Ratio of three numbers = 3 : 7 : 2

LCM of the numbers = 8820

Formula Used:

If numbers are in the ratio a : b : c, then the numbers can be represented as ax, bx, and cx, where x is their HCF.

The LCM of these numbers = x × LCM(a, b, c).

Calculations:

⇒ Let the numbers be 3x, 7x, and 2x.

⇒ LCM(3, 7, 2) = 3 × 7 × 2 = 42 (as they are co-prime).

⇒ LCM of (3x, 7x, 2x) = 42x.

⇒ 42x = 8820.

⇒ x = 8820 ÷ 42.

⇒ x = 210.

∴ The correct answer is 210.

Additional Information

HCF (Highest Common Factor)

The greatest number that divides two or more numbers without leaving a remainder.

LCM (Least Common Multiple)

The smallest non-zero number that is a multiple of two or more numbers.

Product Rule for Two Numbers

For any two numbers A and B: A × B = HCF(A, B) × LCM(A, B).

Co-prime Numbers

Two or more numbers are co-prime if their HCF is 1. Their LCM is simply the product of the numbers.

Shortcut Trick

Effective Interest Rate for 2 years at 4% = 4 + 4 + (4 × 4)/100 = 8.16%.

Given that 8.16% of the Principal (y) = ₹765.

The Total Amount (A) percentage = 100% + 8.16% = 108.16%.

⇒ Amount = (765 ÷ 8.16) × 108.16

⇒ Amount = 93.75 × 108.16 = ₹10,140.

∴ The correct answer is ₹10,140.

Alternate Method

Given: Principal (P) = ₹y, Rate (R) = 4% p.a., Time (n) = 2 years, Compound Interest (CI) = ₹765.

Formula Used: CI = P[(1 + R/100)ⁿ − 1] and Amount (A) = P + CI

⇒ 765 = y[(1 + 4/100)² − 1]

⇒ 765 = y[(1 + 1/25)² − 1]

⇒ 765 = y[(26/25)² − 1]

⇒ 765 = y[676/625 − 1]

⇒ 765 = y[(676 − 625)/625]

⇒ 765 = y[51/625]

⇒ y = (765 × 625) ÷ 51

⇒ y = 15 × 625 = ₹9,375

⇒ Amount (A) = Principal (y) + CI

⇒ A = 9,375 + 765 = ₹10,140

∴ The correct answer is ₹10,140.

Additional Information

Compound Interest Formula

The amount A is calculated as A = P(1 + r/100)ⁿ, where P is principal, r is rate, and n is time.

Effective Rate for 2 Years

The net percentage interest for two years is given by x + y + (xy/100), where x and y are annual rates.

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is P(R/100)².

Shortcut Trick

Net percentage change = a + b + (ab ÷ 100)

Here, a = +12% and b = −10%

Net change = 12 − 10 + (12 × (−10) ÷ 100) = 2 − 1.2 = 0.8%

Given, 0.8% increase = ₹1,584

Original Salary (100%) = (1,584 ÷ 0.8) × 100 = 1,980 × 100 = ₹1,98,000

∴ The correct answer is ₹1,98,000.

Alternate Method

Given:

First year revision = increased by 12%

Second year revision = decreased by 10%

Increase in original monthly salary = ₹1,584

Formula Used:

Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100)

Calculations:

Let Sangeeta's salary be ₹x.

⇒ Salary after 1^st year = x × (112 ÷ 100) = 1.12x

⇒ Salary after 2^nd year = 1.12x × (90 ÷ 100) = 1.008x

⇒ Total increase in salary = 1.008x − x = 0.008x

⇒ 0.008x = 1,584

⇒ x = 1,584 ÷ 0.008

⇒ x = 1,584,000 ÷ 8

⇒ x = 1,98,000

∴ The correct answer is ₹1,98,000.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the net effective change is (a + b + ab/100)%.

Multiplying Factor Method

For an increase of r%, the factor is (1 + r/100). For a decrease, the factor is (1 − r/100). Final value = Initial value × Factor₁ × Factor₂.

Base Value Property

Percentage changes are always calculated on the preceding value in a series of successive changes, not on the original base value each time.

Shortcut Trick

Convert science-to-total ratios into decimals to compare proportions:

Batch A: 5 ÷ (5 + 8) = 5/13 ≈ 0.3846

Batch B: 7 ÷ (7 + 11) = 7/18 ≈ 0.3888

Batch C: 3 ÷ (3 + 5) = 3/8 = 0.3750

Since 0.3888 > 0.3846 > 0.3750, Batch B has the highest proportion.

∴ The correct answer is Batch B.

Alternate Method

Given:

Ratio of Science to Arts students:

Batch A = 5 : 8 | Batch B = 7 : 11 | Batch C = 3 : 5

Formula Used:

Proportion of Science students = Science ÷ (Science + Arts)

Calculations:

⇒ Batch A proportion = 5 / (5 + 8) = 5/13

⇒ Batch B proportion = 7 / (7 + 11) = 7/18

⇒ Batch C proportion = 3 / (3 + 5) = 3/8

⇒ Comparing 5/13 and 7/18 by cross-multiplication: 5 × 18 = 90 and 7 × 13 = 91. Since 91 > 90, Batch B > Batch A.

⇒ Comparing 7/18 and 3/8 by cross-multiplication: 7 × 8 = 56 and 3 × 18 = 54. Since 56 > 54, Batch B > Batch C.

⇒ Batch B has the highest proportion of science students.

∴ The correct answer is Batch B.

Additional Information

Ratio to Proportion Conversion

If the ratio of two components is a:b, the proportion of the first component in the total is given by a/(a+b).

Comparing Fractions Shortcut

To compare a/b and c/d without division, compare the products (a × d) and (b × c). If ad > bc, then a c/d.

Percentage Share

The percentage share of a component is (Value / Total) × 100, which is useful for comparing proportions across different group sizes.

Shortcut Trick

Let Marked Price (MP) = 100 units.

Cost Price (CP) = 100 − 20% = 80 units.

Selling Price (SP) = 80 + 40% of 80 = 112 units.

Difference (SP − MP) = 112 − 100 = 12 units → ₹10.

Marked Price = (10 ÷ 12) × 100 = 1000 ÷ 12 = 83.33.

∴ The correct answer is ₹83.33.

Alternate Method

Given:

Discount on Marked Price = 20%

Selling Price (SP) = Marked Price (MP) + ₹10

Profit Percentage = 40%

⇒ Let the Marked Price be x.

⇒ Cost Price (CP) = x − 20% of x = 0.8x.

⇒ Selling Price (SP) = x + 10.

⇒ Profit = SP − CP = (x + 10) − 0.8x = 0.2x + 10.

⇒ Profit Percentage = (Profit ÷ CP) × 100

⇒ 40 = [(0.2x + 10) ÷ 0.8x] × 100

⇒ 0.40 = (0.2x + 10) ÷ 0.8x

⇒ 0.32x = 0.2x + 10

⇒ 0.12x = 10

⇒ x = 10 ÷ 0.12 = 83.333...

∴ The correct answer is ₹83.33.

Additional Information

Relationship between CP and MP

The direct ratio between Cost Price and Marked Price is given by: CP/MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

Two successive discounts of a% and b% are equivalent to a single discount of [a + b − (a × b)/100]%.

Mark-up Percentage

Mark-up % = [(MP − CP) ÷ CP] × 100. It represents how much more the Marked Price is compared to the Cost Price.