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Shortcut Trick

Convert dimensions to cm: 456 cm and 768 cm.

The largest tile size required to pave the floor is the HCF of its dimensions.

Successive subtraction: 768 − 456 = 312; 456 − 312 = 144; 312 − (144 × 2) = 24.

Since 24 divides 144 exactly, the HCF is 24.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Length of the courtyard = 4.56 m = 456 cm

Width of the courtyard = 7.68 m = 768 cm

Formula Used:

The side of the largest possible square tile = HCF (Length, Width)

Calculations:

⇒ To find the HCF of 456 and 768 using the division method:

⇒ 768 = 456 × 1 + 312

⇒ 456 = 312 × 1 + 144

⇒ 312 = 144 × 2 + 24

⇒ 144 = 24 × 6 + 0

⇒ The remainder is zero, so the HCF is the last divisor.

⇒ HCF = 24

∴ The correct answer is 24 cm.

Additional Information

Total Number of Tiles

The number of tiles required to cover the area is given by: (Area of floor) ÷ (Area of one tile) = (Length × Width) ÷ (Side)².

Prime Factorization Method

HCF can also be found by writing prime factors: 456 = 2³ × 3 × 19 and 768 = 2⁸ × 3. The common factors are 2³ × 3 = 24.

Fundamental Relation

For any two numbers, the product of their HCF and LCM is equal to the product of the two numbers itself (HCF × LCM = a × b).

Shortcut Trick

Assume B = 100.

A is 5 times more than B ⇒ A = 100 + (5 × 100) = 600.

Percentage more = (Difference ÷ Base) × 100.

Percentage more = (500 ÷ 100) × 100 = 500%.

∴ The correct answer is 500 percent.

Alternate Method

Given: A is 5 times more than B.

Formula: Percentage Increase = [(Final Value − Initial Value) ÷ Initial Value] × 100

⇒ Let the value of B be x.

⇒ A is 5 times more than B, so A = x + 5x = 6x.

⇒ The excess value of A over B = A − B = 6x − x = 5x.

⇒ Required percentage more = (Excess Value ÷ B) × 100.

⇒ Percentage more = (5x ÷ x) × 100.

⇒ Percentage more = 5 × 100 = 500%.

∴ The correct answer is 500 percent.

Additional Information

Times vs. Times More

If a value is n times of x, the value is nx. If it is n times more than x, the value is x + nx.

Percentage Comparison

When calculating X is what percent more than Y, the value Y is always the base (denominator). Formula: [(X − Y) ÷ Y] × 100.

Relationship Shortcut

If A = n × B, then A is (n − 1) × 100 percent more than B. Here A = 6B, so (6 − 1) × 100 = 500%.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) is inversely proportional to the number of articles.

CP of 120 = SP of y ⇒ CP ÷ SP = y ÷ 120

Given loss of 4%, SP = 96% of CP ⇒ CP ÷ SP = 100 ÷ 96

Equating the ratios: y ÷ 120 = 100 ÷ 96

y = (100 × 120) ÷ 96 = 125

∴ The correct answer is 125.

Alternate Method

Given:

Cost price of 120 articles = Selling price of y articles

Loss percentage = 4%

Formula Used:

SP = CP × (100 − Loss%) ÷ 100

Calculations:

⇒ Let the cost price of one article be ₹1.

⇒ Total CP of 120 articles = 120 × 1 = ₹120.

⇒ According to the question, selling price of y articles = CP of 120 articles = ₹120.

⇒ SP of one article = 120 ÷ y.

⇒ Loss% = [(CP − SP) ÷ CP] × 100

⇒ 4 = [(1 − 120/y) ÷ 1] × 100

⇒ 4 ÷ 100 = 1 − 120/y

⇒ 0.04 = 1 − 120/y

⇒ 120 ÷ y = 1 − 0.04

⇒ 120 ÷ y = 0.96

⇒ y = 120 ÷ 0.96

⇒ y = 12000 ÷ 96

⇒ y = 125

∴ The correct answer is 125.

Additional Information

Profit and Loss on Article Quantity

If the cost price of 'x' articles is equal to the selling price of 'y' articles, then the Profit or Loss percentage is given by [(x − y) ÷ y] × 100. A positive result indicates profit, while a negative result indicates loss.

Relation between CP, SP and Profit/Loss

The Selling Price can be directly calculated from Cost Price using SP = CP × (100 ± P/L%) ÷ 100, where P/L is the profit or loss percentage.

Successive Discounts

When two successive discounts of a% and b% are given, the equivalent single discount is (a + b − ab ÷ 100)%.

Shortcut Trick

Combined efficiency of R and S = (1/40 + 1/50) = 9/200 work/day.

Work completed in 10 days = 10 × (9/200) = 9/20 part.

∴ The correct answer is 9/20 part.

Alternate Method

Given: R alone completes work in 40 days and S alone in 50 days.

⇒ Combined efficiency of R and S = 5 + 4 = 9 units/day

⇒ Work done in 10 days = 9 × 10 = 90 units

⇒ Part of total work completed = 90 / 200 = 9/20

∴ The correct answer is 9/20 part.

Additional Information

Efficiency and Time Relation

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

Remaining Work Calculation

If a fraction 'x' of work is completed, the remaining work is calculated as (1 − x). For this question, it would be 1 − 9/20 = 11/20.

Combined Work Formula

If two people finish work in A and B days, they finish (A + B) / (A × B) part of the work in one day.

Shortcut Trick

Total Work = LCM(8, 6, 12) = 24 units.

Efficiencies: (M + R) = 3, (R + P) = 4, (P + M) = 2.

Sum of efficiencies: 2(M + R + P) = 3 + 4 + 2 = 9 ⇒ (M + R + P) = 4.5.

Efficiency of P = (M + R + P) − (M + R) = 4.5 − 3 = 1.5 units/day.

Time for P alone = 24 ÷ 1.5 = 16 days.

∴ The correct answer is 16 days.

Alternate Method

Given: Time taken by (M + R) = 8 days, (R + P) = 6 days, (P + M) = 12 days.

Formula Used: Efficiency = Total Work ÷ Time taken.

Calculations:

⇒ Total efficiency of the pairs = (M + R) + (R + P) + (P + M) = 3 + 4 + 2 = 9

⇒ 2(M + R + P) = 9

⇒ Efficiency of (M + R + P) = 9 ÷ 2 = 4.5

⇒ Efficiency of P = Efficiency of (M + R + P) − Efficiency of (M + R)

⇒ Efficiency of P = 4.5 − 3 = 1.5

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

⇒ Time = 24 ÷ 1.5 = 16 days.

∴ The correct answer is 16 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Combined Work Rule for Pairs

When work is done in pairs (A+B, B+C, C+A), the sum of their efficiencies is always twice the combined efficiency of A, B, and C together.

Work and Wages

Wages are distributed in the direct ratio of the work done or the ratio of efficiencies when the duration of work is the same for all.

Shortcut Trick

Total distance to be covered = Train Length + Bridge Length

Total Distance = 210 + 590 = 800 m

Time = Total Distance ÷ Speed = 800 ÷ 25

Time = 32 seconds

∴ The correct answer is 32 sec.

Alternate Method

Given:

Length of the Train (L₁) = 210 m

Length of the Bridge (L₂) = 590 m

Speed of the train (S) = 25 m/s

Formula Used:

Time (T) = Total Distance ÷ Speed

Calculations:

⇒ Total Distance = L₁ + L₂

⇒ Total Distance = 210 + 590 = 800 m

⇒ Time = 800 ÷ 25

⇒ Time = 32 sec

∴ The correct answer is 32 sec.

Additional Information

Crossing Objects with Length

When a train crosses a platform, bridge, or another train, the distance covered is the sum of their lengths: Distance = L_train + L_object.

Unit Conversion

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

Relative Speed

If two trains move in the same direction, relative speed is S₁ − S₂. In opposite directions, it is S₁ + S₂.

Shortcut Trick

Since the Speed of the train is constant, Time is directly proportional to the Total Distance (Train + Bridge).

Total Distance 1 = 250 + 350 = 600 m

Total Distance 2 = 250 + 900 = 1150 m

Time Ratio = Distance Ratio = 600 : 1150 = 12 : 23

As 12 units = 12 sec, then 23 units = 23 sec.

∴ The correct answer is 23 sec.

Alternate Method

Given: Train Length = 250 m, Bridge 1 = 350 m, Time 1 = 12 sec, Bridge 2 = 900 m.

Formula Used: Speed = Total Distance ÷ Total Time

⇒ Case 1: Total Distance 1 = 250 + 350 = 600 m

⇒ Speed of the train = 600 ÷ 12 = 50 m/sec

⇒ Case 2: Total Distance 2 = 250 + 900 = 1150 m

⇒ Time to cross the second bridge = Total Distance 2 ÷ Speed

⇒ Time = 1150 ÷ 50 = 23 sec

∴ The correct answer is 23 sec.

Additional Information

Crossing a Stationary Point Object

When a train crosses a pole, tree, or a stationary man, the distance covered is equal to the Length of the Train itself.

Crossing an Object with Length

When a train crosses a bridge, platform, or another stationary train, the distance covered is (Length of Train + Length of Object).

Relative Speed

If two objects move in opposite directions, Relative Speed = S₁ + S₂. In the same direction, Relative Speed = S₁ − S₂.

Unit Conversion

To convert Speed from km/hr to m/sec, multiply by 5/18. To convert from m/sec to km/hr, multiply by 18/5.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) at 30% profit is 10 : 13.

Given Discount = (2/5) SP ⇒ Ratio Discount : SP = 2 : 5.

Equating SP (LCM of 13 and 5 = 65) ⇒ CP : SP : Discount = 50 : 65 : 26.

Marked Price (MP) = SP + Discount = 65 + 26 = 91.

∴ The correct answer is (50/91) of marked price.

Alternate Method

Given:

Profit Percentage = 30%

Discount = (2/5) × Selling Price

Formula Used:

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

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

Calculations:

⇒ Let the Cost Price (CP) of the article be 100.

⇒ Selling Price (SP) = 100 + 30% of 100 = 130.

⇒ Discount = (2/5) × 130 = 2 × 26 = 52.

⇒ Marked Price (MP) = SP + Discount = 130 + 52 = 182.

⇒ To find CP in terms of MP: CP/MP = 100/182.

⇒ Dividing numerator and denominator by 2 ⇒ CP/MP = 50/91.

⇒ CP = (50/91) × MP.

∴ The correct answer is (50/91) of marked price.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price can be directly calculated as CP/MP = (100 − Discount%) / (100 + Profit%).

Discount Percentage

Discount% is always calculated on the Marked Price. Discount% = (Discount / MP) × 100.

Successive Discounts

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

Shortcut Trick

To find Selling Price (SP), apply the multiplier (100 − D%) / 100 to the Marked Price (MP).

For Option 4: MP = ₹5000 and Discount = 30%

SP = 5000 × 0.70 = ₹3500.

Check other options: 4000 × 0.85 = 3400; 4500 × 0.90 = 4050; 5500 × 0.60 = 3300.

∴ The correct answer is Marked price = ₹5000 and Discount = 30%.

Alternate Method

Given:

Marked Price (MP) = ₹5000

Discount Percentage = 30%

Formula Used:

Selling Price (SP) = MP − Discount Amount

Discount Amount = (Discount % / 100) × MP

Calculations:

⇒ Discount Amount = (30 / 100) × 5000

⇒ Discount Amount = 30 × 50 = ₹1500

⇒ Selling Price (SP) = 5000 − 1500

⇒ Selling Price (SP) = ₹3500

∴ The correct answer is Marked price = ₹5000 and Discount = 30%.

Additional Information

Relationship between MP, SP, and CP

Marked Price (MP) is the price printed on an item. Selling Price (SP) = MP × (100 − Discount%)/100. Also, SP = Cost Price (CP) × (100 + Profit%)/100.

Successive Discounts

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

Markup Percentage

Markup% = [(MP − CP) / CP] × 100. It represents how much the price is increased above the cost price before offering a discount.

Shortcut Trick

Let Cost Price (CP) = 100%, then Marked Price (MP) = 120% and Selling Price (SP) = 110%.

Given MP = 120% = Rs. 1200 ⇒ 1% = Rs. 10.

∴ SP = 110% = 110 × 10 = Rs. 1100.

∴ The correct answer is Rs. 1100.

Alternate Method

Given: Marked Price (MP) = Rs. 1200, Markup Percentage = 20%, Profit Percentage = 10%.

Formula Used: CP = MP ÷ (1 + Markup%) and SP = CP × (1 + Profit%)

⇒ Cost Price (CP) = 1200 ÷ (1 + 20/100)

⇒ CP = 1200 ÷ 1.2 = Rs. 1000

⇒ Selling Price (SP) = CP × (1 + 10/100)

⇒ SP = 1000 × 1.1 = Rs. 1100

∴ The correct answer is Rs. 1100.

Additional Information

CP, MP, Profit and Discount Relation

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

Markup Percentage

Markup percentage is always calculated on the Cost Price: Markup% = [(MP − CP) / CP] × 100.

Discount Calculation

Discount is the reduction offered on the Marked Price: Discount = MP − SP. Discount% = (Discount / MP) × 100.