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Shortcut Trick

When x% of x = k, the value of x is directly calculated as 10 × √k.

Here, k = 225.

⇒ x = 10 × √225

⇒ x = 10 × 15 = 150.

∴ The correct answer is 150.

Alternate Method

Given:

The expression is x percent of x is equal to 225.

Calculations:

⇒ (x ÷ 100) × x = 225

⇒ x² ÷ 100 = 225

⇒ x² = 225 × 100

⇒ x² = 22500

⇒ x = √22500

⇒ x = 150

∴ The correct answer is 150.

Additional Information

Basic Percentage Formula

The percentage of a number is calculated as: Value = (Percentage × Total) ÷ 100.

Square Root Calculation

To find the value from its square, take the square root. For example, √(a × b) = √a × √b. Here, √22500 = √225 × √100 = 15 × 10 = 150.

Percentage and Fractions

Any percentage x% can be written as a fraction x/100 or a decimal 0.01x.

Shortcut Trick

25% = 1/4 (fractional value).

Loss is calculated on Selling Price (SP), so let SP = 4 units and Loss = 1 unit.

Cost Price (CP) = SP + Loss = 4 + 1 = 5 units.

Loss percentage on Cost Price = (Loss ÷ CP) × 100 = (1 ÷ 5) × 100.

∴ The correct answer is 20%.

Alternate Method

Given:

Loss percentage on Selling Price = 25%

Formula Used:

Loss = Selling Price × (Loss% on SP ÷ 100)

Cost Price = Selling Price + Loss

Loss% on Cost Price = (Loss ÷ Cost Price) × 100

Calculations:

⇒ Let the Selling Price (SP) be ₹100.

⇒ Loss = 25% of ₹100 = (25 ÷ 100) × 100 = ₹25.

⇒ Cost Price (CP) = SP + Loss = 100 + 25 = ₹125.

⇒ Loss% on CP = (Loss ÷ CP) × 100.

⇒ Loss% = (25 ÷ 125) × 100.

⇒ Loss% = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Base for Profit and Loss

Profit or loss is always calculated on the Cost Price (CP) unless the question explicitly mentions it is calculated on the selling price.

Profit Percentage Formula

Profit % = (Profit ÷ CP) × 100, where Profit = Selling Price − Cost Price.

Cost Price Conversion

If loss is x% on SP, then loss % on CP = [x ÷ (100 + x)] × 100. If profit is x% on SP, then profit % on CP = [x ÷ (100 − x)] × 100.

Shortcut Trick

Ratio of P : Q : R = 75 : 100 : 60 = 15 : 20 : 12

Total sum = 15 + 20 + 12 = 47 units

Given: 47 units = 1410 → 1 unit = 1410 ÷ 47 = 30

Difference (Q − R) = 20 − 12 = 8 units

Required difference = 8 × 30 = 240

∴ The correct answer is 240 marks.

Alternate Method

Given:

P = Q − 25% of Q

R = Q − 40% of Q

Sum of P, Q, and R = 1410

Formula Used:

Total Marks = P + Q + R

Difference = Q − R

⇒ Let the marks of Q be 100x.

⇒ Marks of P = 100x − 25% of 100x = 75x

⇒ Marks of R = 100x − 40% of 100x = 60x

⇒ Total Marks = 75x + 100x + 60x = 235x

⇒ 235x = 1410

⇒ x = 1410 ÷ 235 = 6

⇒ Difference between Q and R = 100x − 60x = 40x

⇒ Value of difference = 40 × 6 = 240

∴ The correct answer is 240 marks.

Additional Information

Percentage Base Rule

Always identify the base value (denominator) which typically follows keywords like 'than' or 'of' to set it as 100%.

Ratio Method for Percentages

Converting percentages to fractions (e.g., 25% = 1/4, 40% = 2/5) helps in establishing quick ratios between multiple entities.

Relative Percentage Change

If A is x% less than B, then the difference in their values is calculated as (x ÷ 100) × B.

Shortcut Trick

Efficiency of C = Efficiency (A + B + C) − Efficiency (A + B)

Total Work = LCM of (18, 24) = 72 units

Efficiency of (A + B + C) = 72 ÷ 18 = 4 units/day

Efficiency of (A + B) = 72 ÷ 24 = 3 units/day

Efficiency of C = 4 − 3 = 1 unit/day

∴ The correct answer is 72 days.

Alternate Method

Given:

Time taken by A, B, and C together = 18 days

Time taken by A and B together = 24 days

Formula Used:

Total Work = Time taken × Efficiency

Calculations:

⇒ Let the total work be the LCM of 18 and 24 = 72 units.

⇒ Efficiency of (A + B + C) = 72 ÷ 18 = 4 units/day.

⇒ Efficiency of (A + B) = 72 ÷ 24 = 3 units/day.

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

⇒ Efficiency of C = 4 − 3 = 1 unit/day.

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

⇒ Time taken by C alone = 72 ÷ 1 = 72 days.

∴ The correct answer is 72 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work. Efficiency = 1 ÷ Time.

Combined Work Formula

If X takes 'a' days and Y takes 'b' days, then together they take (a × b) ÷ (a + b) days.

Pipes and Cisterns

The concept is identical to Time & Work, where an inlet pipe does positive work and an outlet pipe does negative work.

Work and Wages

Wages are always distributed in the ratio of the Efficiency of the workers (or work done) when they work for the same duration.

Shortcut Trick

Ratio of efficiencies (X : Y : Z) = (1/12) : (1/18) : (1/36) = 3 : 2 : 1.

Since they work together, Wage ∝ Efficiency.

Given X wage = 3 units = ₹4800 ⇒ 1 unit = ₹1600.

Difference = |(Y + Z) − (X + Z)| = |Y − X| = |2 − 3| = 1 unit.

Value of 1 unit = ₹1600.

∴ The correct answer is ₹1600.

Alternate Method

Given:

Time taken by X = 12 days, Y = 18 days, Z = 36 days.

Wages of X = ₹4800.

Formula Used:

Total Work = LCM of individual days.

Efficiency = Total Work ÷ Time taken.

Wage Share = (Individual Efficiency ÷ Total Efficiency) × Total Wage.

Calculations:

⇒ Total wages are distributed in the ratio of efficiencies X : Y : Z = 3 : 2 : 1.

⇒ Wage of X = 3 units = ₹4800.

⇒ Value of 1 unit = 4800 ÷ 3 = ₹1600.

⇒ Wage of Y = 2 × 1600 = ₹3200.

⇒ Wage of Z = 1 × 1600 = ₹1600.

⇒ Total wages of (Y + Z) = 3200 + 1600 = ₹4800.

⇒ Total wages of (X + Z) = 4800 + 1600 = ₹6400.

⇒ Absolute Difference = |4800 − 6400| = ₹1600.

∴ The correct answer is ₹1600.

Additional Information

Work and Wages Rule

Wages are always distributed in the ratio of the actual work done by each person. If they work for the same duration, the wage ratio is identical to the efficiency ratio.

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work. Efficiency = 1 ÷ Time.

Pipes and Cisterns

The same LCM and efficiency logic applies to pipes filling or emptying a tank, where inlet efficiency is positive and outlet efficiency is negative.

Shortcut Trick

Relative Speed in same direction = 75 − 25 = 50 km/hr.

Total Distance to cover = Sum of lengths = 120 + 130 = 250 m.

Time = Distance ÷ Relative Speed

Time = 250 ÷ (50 × 5/18) = 250 × 18 ÷ 250 = 18 sec.

∴ The correct answer is 18 sec.

Alternate Method

Given:

Length of Faster Train (L₁) = 130 m

Length of Slower Train (L₂) = 120 m

Speed of Faster Train (S₁) = 75 km/hr

Speed of Slower Train (S₂) = 25 km/hr

Formula Used:

Relative Speed (Same direction) = S₁ − S₂

Time = (L₁ + L₂) ÷ Relative Speed

Calculations:

⇒ Total Length to be crossed = 130 + 120 = 250 m

⇒ Relative Speed = 75 − 25 = 50 km/hr

⇒ Convert Speed to m/s = 50 × (5 ÷ 18) = 250 ÷ 18 m/s

⇒ Time taken = 250 ÷ (250 ÷ 18)

⇒ Time taken = 250 × (18 ÷ 250) = 18 seconds

∴ The correct answer is 18 sec.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed = S₁ − S₂. When they move in the opposite direction, relative speed = S₁ + S₂.

Unit Conversion

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

Crossing Distance

When a train crosses another train or a bridge, the total distance covered is the sum of their lengths (L₁ + L₂).

Shortcut Trick

By definition, the Upstream speed of a boat is the difference between its speed in still water and the speed of the stream.

The question asks for this exact difference: (Speed of boat in still water − Speed of stream).

Given Upstream speed = 15 km/hr.

Required difference = 15 km/hr.

∴ The correct answer is 15 km/hr.

Alternate Method

Given:

Downstream Speed (D) = 25 km/hr

Upstream Speed (U) = 15 km/hr

Formula Used:

Speed of boat in still water (x) = (D + U) ÷ 2

Speed of stream (y) = (D − U) ÷ 2

Calculations:

⇒ Speed of boat in still water (x) = (25 + 15) ÷ 2

⇒ x = 40 ÷ 2 = 20 km/hr

⇒ Speed of stream (y) = (25 − 15) ÷ 2

⇒ y = 10 ÷ 2 = 5 km/hr

⇒ Required Difference = x − y

⇒ Required Difference = 20 − 5 = 15 km/hr

∴ The correct answer is 15 km/hr.

Additional Information

Downstream Speed

The speed of the boat when traveling with the current: Speed in Still Water + Speed of Current.

Upstream Speed

The speed of the boat when traveling against the current: Speed in Still Water − Speed of Current.

Speed of Boat Formula

Speed of boat in still water is calculated as (Downstream Speed + Upstream Speed) ÷ 2.

Speed of Current Formula

Speed of stream or current is calculated as (Downstream Speed − Upstream Speed) ÷ 2.

Shortcut Trick

Use the ratio relation: CP / MP = (100 − Discount%) / (100 + Profit%)

CP : MP = (100 − 40) : (100 + 50) = 60 : 150 = 2 : 5

Given CP = 2 units = Rs. 1000 ⇒ 1 unit = Rs. 500

Marked Price (MP) = 5 units = 5 × 500 = Rs. 2500

∴ The correct answer is Rs. 2500.

Alternate Method

Given: Cost Price (CP) = Rs. 1000, Discount = 40%, Profit = 50%.

Formula Used: SP = CP × (1 + Profit/100) and SP = MP × (1 − Discount/100).

⇒ Selling Price (SP) = 1000 + 50% of 1000

⇒ SP = 1000 + 500 = Rs. 1500

⇒ According to discount, SP = MP × (100 − 40)/100

⇒ 1500 = MP × 0.6

⇒ MP = 1500 ÷ 0.6 = Rs. 2500

∴ The correct answer is Rs. 2500.

Additional Information

Relation between MP and CP

A very useful direct relation is MP/CP = (100 + Profit%) / (100 − Discount%). This eliminates the need to calculate SP separately.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − (d₁ × d₂)/100)%.

Markup Percentage

The percentage by which Marked Price is higher than Cost Price is Markup% = [(MP − CP) / CP] × 100.

Shortcut Trick

Let Cost Price (CP) = 4 units. For 25% Loss, Selling Price (SP) = 3 units.

Discount = (5/9) × SP = (5/9) × 3 = 5/3 units.

Marked Price (MP) = SP + Discount = 3 + 5/3 = 14/3 units.

Ratio of MP to CP = (14/3) ÷ 4 = 14/12 = 7/6.

∴ The correct answer is (7/6) of cost price.

Alternate Method

Given:

Loss Percentage = 25%

Discount = (5/9) × Selling Price (SP)

Formula Used:

SP = CP × (100 − Loss%)/100

MP = SP + Discount

⇒ Let Cost Price (CP) be C.

⇒ SP = C × (100 − 25)/100 = (75/100)C = (3/4)C

⇒ Marked Price (MP) = SP + Discount

⇒ MP = SP + (5/9)SP = (14/9)SP

⇒ Substituting SP = (3/4)C:

⇒ MP = (14/9) × (3/4)C

⇒ MP = (42/36)C = (7/6)C

∴ The correct answer is (7/6) of cost price.

Additional Information

Relationship between CP, MP, Profit and Discount

The formula relating CP and MP is CP/MP = (100 − Discount%)/(100 + Profit%). In case of loss, replace +Profit% with −Loss%.

Marked Price (MP)

MP is the maximum retail price (MRP) set by the seller. Discount is always calculated as a percentage of the MP: SP = MP × (100 − D%)/100.

Cost Price (CP) and Selling Price (SP)

Profit or loss is determined by the difference between SP and CP. If SP > CP, there is a Profit. If SP < CP, there is a Loss.

Shortcut Trick

Selling Price (SP) = 70% of Marked Price (MP)

⇒ SP = 0.70 × 1000 = 700

Profit = SP − Cost Price (CP)

⇒ Profit = 700 − 520 = 180

∴ The correct answer is Rs. 180.

Alternate Method

Given:

Marked Price (MP) = Rs. 1000

Discount = 30%

Cost Price (CP) = Rs. 520

Formula Used:

Selling Price (SP) = Marked Price (MP) − Discount

Profit = Selling Price (SP) − Cost Price (CP)

Calculations:

⇒ Discount Amount = (30/100) × 1000 = Rs. 300

⇒ Selling Price (SP) = 1000 − 300 = Rs. 700

⇒ Profit = 700 − 520

⇒ Profit = Rs. 180

∴ The correct answer is Rs. 180.

Additional Information

Profit Percentage

Profit % = (Profit ÷ Cost Price) × 100. It measures the gain relative to the investment.

Discount Percentage

Discount % = (Discount ÷ Marked Price) × 100. Discounts are always calculated on the Marked Price.

Relation between CP and MP

MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%). This direct ratio is useful for finding mark-up percentages.