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Shortcut Trick

Effective Discount % = 15 + 10 − (15 × 10) ÷ 100 = 23.5%.

Net Selling Price = 100% − 23.5% = 76.5% of Marked Price.

76.5% of MP = ₹3,672 ⇒ MP = 3,672 ÷ 0.765 = ₹4,800.

∴ The correct answer is ₹4,800.

Alternate Method

Given:

First Successive Discount (d₁) = 15%

Second Successive Discount (d₂) = 10%

Final Selling Price (SP) = ₹3,672

Formula Used:

SP = MP × (100 − d₁)/100 × (100 − d₂)/100

⇒ 3,672 = MP × (100 − 15)/100 × (100 − 10)/100

⇒ 3,672 = MP × 85/100 × 90/100

⇒ 3,672 = MP × 17/20 × 9/10

⇒ 3,672 = MP × 153/200

⇒ MP = (3,672 × 200) ÷ 153

⇒ MP = 24 × 200

⇒ MP = 4,800

∴ The correct answer is ₹4,800.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Selling Price Calculation

SP = MP − Discount. If multiple discounts are applied, the base for the second discount is the price after the first discount.

Marked Price (List Price)

The price indicated on the product before any discount is applied. Discounts are always calculated on the Marked Price.

Shortcut Trick

Net Profit = Sum of individual profits and losses

Net Profit = 300 − 50 + 200 = ₹450

Overall profit percentage = (Net Profit ÷ Total CP) × 100

Profit % = (450 ÷ 3,951) × 100 ≈ 11.39%

∴ The correct answer is 11.39% profit.

Alternate Method

Given:

Total Cost Price (CP) of 3 bed sheets = ₹3,951

Profit on 1^st sheet = ₹300

Loss on 2^nd sheet = ₹50

Profit on 3^rd sheet = ₹200

Formula Used:

Overall Profit/Loss = Sum of individual Profit/Loss

Profit Percentage = (Overall Profit ÷ Total CP) × 100

Calculations:

⇒ Total Overall Profit = 300 − 50 + 200

⇒ Total Overall Profit = ₹450

⇒ Overall Profit Percentage = (450 ÷ 3,951) × 100

⇒ Profit % = 45000 ÷ 3,951

⇒ Profit % ≈ 11.3895...

⇒ Rounding off to two decimal places, we get 11.39%.

∴ The correct answer is 11.39% profit.

Additional Information

Profit and Loss Percentage

Profit % is always calculated on the Cost Price (CP) unless stated otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Successive Profit/Loss

When multiple items are sold, the net profit is the algebraic sum of individual profits (positive) and losses (negative).

Rounding Rule

To round to two decimal places, look at the third decimal digit; if it is 5 or more, increase the second decimal digit by 1.

Shortcut Trick

Efficiency of A = 3 and B = 2 (taking Total Work as 36 units).

Combined work of A + B for 2 days = (3 + 2) × 2 = 10 units.

Remaining work to be done = 36 − 10 = 26 units.

Time taken by B alone = 26 ÷ 2 = 13 days.

∴ The correct answer is 13 days.

Alternate Method

Given: Time taken by A = 12 days, Time taken by B = 18 days, Duration of joint work = 2 days.

Formula Used: Efficiency = Total Work ÷ Time taken; Work Done = Efficiency × Time.

⇒ Efficiency of A = 3 units/day

⇒ Efficiency of B = 2 units/day

⇒ Total work done by A and B in 2 days = (3 + 2) × 2 = 10 units

⇒ Remaining work = Total Work − Work Done = 36 − 10 = 26 units

⇒ Time taken by B to complete the remaining work = 26 ÷ 2 = 13 days

∴ The correct answer is 13 days.

Additional Information

Relationship between Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of time taken by two people is x : y, the ratio of their efficiencies will be y : x.

Work and Wages

Wages are generally distributed in the ratio of the work done by each individual, which is the product of their efficiency and the time for which they worked.

Combined Work Formula

If A can do a work in x days and B in y days, together they can do the work in (x × y) ÷ (x + y) days.

Shortcut Trick

Net percentage Increase after 15% and 20% = 15 + 20 + (15 × 20 ÷ 100) = 38%.

Let the initial Price be 100, then the current Price = 138.

Required Price for 5% Overall Effective Increment = 105.

Percentage Decrease needed = (138 − 105) ÷ 138 × 100 ≈ 23.91%.

∴ The correct answer is 23.91%.

Alternate Method

Given: First Increase = 15%, Second Increase = 20%, Target Effective Increment = 5%.

⇒ Let the initial Price of the Item = 100.

⇒ Price after first 15% Increase = 100 + 15 = 115.

⇒ Price after second 20% Increase = 115 + (20 ÷ 100 × 115) = 115 + 23 = 138.

⇒ Required Final Price with 5% Overall Effective Increment = 100 + 5 = 105.

⇒ Absolute Decrease needed from the current price = 138 − 105 = 33.

⇒ Approximate Percentage Decrease = (33 ÷ 138) × 100.

⇒ Percentage Decrease = 23.913% ≈ 23.91%.

∴ The correct answer is 23.91%.

Additional Information

Successive Percentage Formula

For two successive percentage changes a% and b%, the net effective change is given by the formula: (a + b + ab ÷ 100)%.

Multiplier Method

The final value after multiple changes can be calculated using multipliers: Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100).

Base for Percentage Decrease

When calculating a percentage decrease to return to a target value, the denominator must always be the current (larger) value.

Shortcut Trick

Ratio of Men to Women = 60% : 40% = 3 : 2

Using Weighted Average shortcut:

Average = (Ratio_Men × Avg_Men + Ratio_Women × Avg_Women) ÷ Total Ratio

Average = (3 × 51 + 2 × 55) ÷ (3 + 2) = (153 + 110) ÷ 5 = 263 ÷ 5 = 52.6

∴ The correct answer is 52.6 years.

Alternate Method

Given:

Percentage of Men = 60%

Percentage of Women = 40%

Average age of Men = 51 years

Average age of Women = 55 years

Formula Used:

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂)

Calculations:

Assume total members in the health club = 100

⇒ Number of Men = 60% of 100 = 60

⇒ Number of Women = 40% of 100 = 40

⇒ Total age of Men = 60 × 51 = 3060 years

⇒ Total age of Women = 40 × 55 = 2200 years

⇒ Total age of all members = 3060 + 2200 = 5260 years

⇒ Average age of all members = 5260 ÷ 100

⇒ Average age = 52.6 years

∴ The correct answer is 52.6 years.

Additional Information

Weighted Average Formula

It is used when two groups with different averages are combined. Formula: A_avg = (w₁a₁ + w₂a₂) ÷ (w₁ + w₂), where w represents weight/number and a represents average.

Net Deviation Method

Assume a baseline average and calculate the net increase or decrease relative to that baseline, then distribute the net change over the total number of items.

Alligation Rule

(Quantity 1) ÷ (Quantity 2) = (Average 2 − Combined Average) ÷ (Combined Average − Average 1). This is useful for finding ratios or combined means.

Shortcut Trick

Assume Total Capacity of the cistern = LCM(15, 20, 30) = 60 units.

Efficiencies: Pipe A = 4, Pipe B = 3, Pipe C = −2 (outlet).

Net efficiency for first 5 hours = 4 + 3 − 2 = 5 units/hr. Work done = 5 × 5 = 25 units.

Remaining work = 60 − 25 = 35 units. Time taken by (A + B) = 35 ÷ 7 = 5 hours.

Total time = 5 hr (initial) + 5 hr (remaining) = 10 hours.

∴ The correct answer is 10 hr.

Alternate Method

Given: Filling time for Pipe A = 15 hr, Pipe B = 20 hr; Emptying time for Pipe C = 30 hr.

⇒ Work done by A, B, and C together in 5 hours = 5 × (4 + 3 − 2)

⇒ Work done = 5 × 5 = 25 units

⇒ Remaining Work to be filled = Total Work − Completed Work

⇒ Remaining Work = 60 − 25 = 35 units

⇒ Since Pipe C is closed, only A and B will fill the remaining cistern.

⇒ Combined efficiency of (A + B) = 4 + 3 = 7 units/hr

⇒ Time taken to fill remaining part = 35 ÷ 7 = 5 hours

⇒ Total Time = Initial 5 hours + Remaining 5 hours = 10 hours

∴ The correct answer is 10 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency), while outlet pipes remove water (negative efficiency). Net work is the algebraic sum of their efficiencies.

Time and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work. Efficiency = Total Work ÷ Time.

Pipes and Cisterns Concept

If an inlet pipe fills a tank in x hours and an outlet pipe empties it in y hours, the net work done in 1 hour when both are open is (1/x − 1/y).

Given:

Expression: 0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a = 1

Formula used:

1. x^m × xⁿ = x^m+n
2. (x^m)ⁿ = x^m×n
3. x⁻ⁿ = 1/xⁿ

Calculation:

0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a = 1

⇒ a = (1/2)² × 5⁵ × 2⁻⁴ × (2²)⁴ × (2 × 5)⁻³

⇒ a = 2⁻² × 5⁵ × 2⁻⁴ × 2⁸ × 2⁻³ × 5⁻³

⇒ a = 2^{(−2 − 4 + 8 − 3)} × 5^{(5 − 3)}

⇒ a = 2⁻¹ × 5²

⇒ a = (1/2) × 25

⇒ a = 12.5

The simplified value of the variable a is 12.5.

∴ The correct answer is option (4).

Shortcut Trick

Let the HCF of the three numbers be x.

Since the ratio is 11 : 3 : 2, and 11, 3, and 2 are co-prime numbers, their LCM will be 11 × 3 × 2 × x.

⇒ 66x = 2706

⇒ x = 2706 ÷ 66

⇒ x = 41

∴ The correct answer is 41.

Alternate Method

Given:

⇒ Ratio of three numbers = 11 : 3 : 2

⇒ LCM of the numbers = 2706

Formula Used:

⇒ Numbers = (Ratio Part) × HCF

⇒ LCM of (ax, bx, cx) = x × LCM(a, b, c), where x is the HCF.

Calculations:

Let the common multiplier (which is the HCF) be x.

⇒ First number = 11x

⇒ Second number = 3x

⇒ Third number = 2x

⇒ LCM (11x, 3x, 2x) = x × LCM (11, 3, 2)

⇒ Since 11, 3, and 2 have no common factors other than 1, their LCM is 11 × 3 × 2 = 66.

⇒ 66x = 2706

⇒ x = 2706 ÷ 66

⇒ x = 41

∴ The correct answer is 41.

Additional Information

HCF (Highest Common Factor)

The greatest number that divides two or more numbers without leaving a remainder. For numbers in a ratio, the HCF is the common multiplier.

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers. LCM = HCF × product of remaining co-prime factors.

Property of Two Numbers

For any two numbers A and B: A × B = HCF(A, B) × LCM(A, B). This property does not directly apply to more than two numbers.

Shortcut Trick

To find the total cost of 9 items when the cost of 5 items is given, find the price per item and multiply by the required quantity.

Cost per dress = 4955 ÷ 5 = 991

Cost of 9 dresses = 991 × 9 = 8919

∴ The correct answer is ₹8,919.

Alternate Method

Given:

Price of 5 identical dresses = ₹4,955

Number of dresses Meena buys = 9

Formula Used:

Unitary Method: Value of 1 unit = Total Value ÷ Total Units

Total Required Value = Value of 1 unit × Required Units

Calculations:

⇒ Cost of 5 dresses = ₹4,955

⇒ Cost of 1 dress = 4955 ÷ 5

⇒ Cost of 1 dress = ₹991

⇒ Cost of 9 dresses = 991 × 9

⇒ Cost of 9 dresses = ₹8,919

∴ The correct answer is ₹8,919.

Additional Information

Unitary Method

A technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Direct Proportion

When two quantities increase or decrease at the same rate. In this case, more dresses lead to a higher cost: x₁/y₁ = x₂/y₂.

Inverse Proportion

When an increase in one quantity results in a decrease in the other. Product remains constant: x₁ × y₁ = x₂ × y₂.

Shortcut Trick

Relative Speed in opposite directions = S₁ + S₂ = 36 + 54 = 90 km/h.

Convert Relative Speed to m/s: 90 × (5/18) = 25 m/s.

Total distance to be covered = Sum of lengths of trains = 100 + 150 = 250 m.

Required Time = Total Distance ÷ Relative Speed = 250 ÷ 25 = 10 sec.

∴ The correct answer is 10 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 100 m

Length of Train 2 (L₂) = 150 m

Speed of Train 1 (S₁) = 36 km/h

Speed of Train 2 (S₂) = 54 km/h

Formula Used:

Time = Total Distance ÷ Relative Speed

Relative Speed (Opposite Direction) = S₁ + S₂

Calculations:

⇒ Total Distance = L₁ + L₂ = 100 + 150 = 250 m

⇒ Relative Speed = S₁ + S₂ = 36 + 54 = 90 km/h

⇒ Convert km/h to m/s: 90 × (5/18) m/s

⇒ Relative Speed = 5 × 5 = 25 m/s

⇒ Time taken = 250 ÷ 25

⇒ Time taken = 10 sec

∴ The correct answer is 10 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂). If they move in the same direction, the relative speed is the difference between their speeds (|S₁ − S₂|).

Unit Conversion

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

Crossing Different Objects

When a train crosses a stationary point/pole, distance = length of train. When it crosses a platform or bridge, distance = length of train + length of platform.