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Shortcut Trick

Divide the parts into ratios based on percentage: 50% : 20% : 30% → 5 : 2 : 3 (Total = 10 units).

Apply net percentage change formula: Net % = [5 × 21 + 2 × 36 + 3 × (−29)] / 10.

Net % = (105 + 72 − 87) / 10 = 90 / 10 = 9%.

Since the overall profit is x%, the value of x is 9.

∴ The correct answer is 9.

Alternate Method

Let the total quantity of goods be 100 units and the cost price per unit be ₹1.

Given:

First part: 50% of 100 = 50 units sold at 21% profit.

Second part: 40% of remaining (100 − 50 = 50) = 20 units sold at 36% profit.

Third part: Remaining (50 − 20) = 30 units sold at 29% loss.

Calculations:

⇒ Profit on 1^st part = 50 × 21/100 = ₹10.5

⇒ Profit on 2^nd part = 20 × 36/100 = ₹7.2

⇒ Loss on 3^rd part = 30 × 29/100 = ₹8.7

⇒ Total Net Profit = 10.5 + 7.2 − 8.7 = ₹9

⇒ Overall Profit % = (Total Profit / Total CP) × 100

⇒ x = (9 / 100) × 100 = 9

∴ The correct answer is 9.

Additional Information

Weighted Average Profit %

When goods are sold in parts with different profit/loss rates, the overall profit % = (w₁p₁ + w₂p₂ + ... + w_np_n) / (w₁ + w₂ + ... + w_n), where w represents weight/quantity and p represents profit/loss rate.

Successive Percentage Change

For two successive percentage changes a% and b%, the effective change is [a + b + (ab/100)]%. This is useful when the price is changed multiple times on the same base.

Basic Profit and Loss

Profit % = (Profit / CP) × 100 and Loss % = (Loss / CP) × 100. Always ensure the denominator is the Cost Price (CP) unless specified otherwise.

Shortcut Trick

Sum of Length and Breadth = Half of Perimeter = 56 ÷ 2 = 28 cm.

Let Breadth be x, then Length = 2x + 4.

⇒ (2x + 4) + x = 28

⇒ 3x = 24 ⇒ x = 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

Given:

⇒ Perimeter of Rectangle = 56 cm

⇒ Length (L) = 4 + 2 × Breadth (B)

Formula Used:

⇒ Perimeter = 2 × (Length + Breadth)

Calculations:

⇒ 2 × (L + B) = 56

⇒ L + B = 56 ÷ 2

⇒ L + B = 28

Substitute L = 2B + 4 in the equation:

⇒ (2B + 4) + B = 28

⇒ 3B + 4 = 28

⇒ 3B = 28 − 4

⇒ 3B = 24

⇒ B = 24 ÷ 3

⇒ B = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Area of Rectangle

The total space occupied by the rectangle is given by the product of its dimensions: Area = Length × Breadth.

Diagonal of Rectangle

The distance between opposite corners can be found using the Pythagorean theorem: Diagonal = √(Length² + Breadth²).

Perimeter of a Square

A square is a special rectangle where Length = Breadth = s. Its perimeter is 4 × s.

Shortcut Trick

Simplify using BODMAS in segments:

Segment 1: (14/24 ÷ 14/9) = (14/24 × 9/14) = 3/8

Segment 2: (3/7 × 14/9 + 5/8) = (2/3 + 5/8) = 31/24

Segment 3: (9/10) ÷ (31/24 of 24/10) = (9/10) ÷ (31/10) = 9/31

Calculation: (3/8 ÷ 31/24) + 9/31 = (3/8 × 24/31) + 9/31 = 9/31 + 9/31 = 18/31

∴ The correct answer is 18/31.

Alternate Method

Given: { (14/24) ÷ (14/9) } ÷ (3/7 × 14/9 + 5/8) + 9/10 ÷ 31/24 of 24/10

Simplify using the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction):

⇒ Solve the first set of brackets: (14/24) × (9/14) = 9/24 = 3/8

⇒ Solve the inner multiplication in the second bracket: (3/7 × 14/9) = (1 × 2) / (1 × 3) = 2/3

⇒ Complete the second bracket by addition: 2/3 + 5/8 = (16 + 15) / 24 = 31/24

⇒ Solve the 'of' operator: 31/24 of 24/10 = 31/24 × 24/10 = 31/10

⇒ Rewrite the expression: (3/8 ÷ 31/24) + (9/10 ÷ 31/10)

⇒ Apply division by multiplying by the reciprocal: (3/8 × 24/31) + (9/10 × 10/31)

⇒ 9/31 + 9/31 = 18/31

∴ The correct answer is 18/31.

Additional Information

BODMAS Rule

It defines the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

'Of' Operator Precedence

The 'of' operator is treated as a multiplication but must be performed before any division or standard multiplication in the expression.

Division of Fractions

To divide one fraction by another, multiply the first fraction by the reciprocal of the second (a/b ÷ c/d = a/b × d/c).

Shortcut Trick

Scheme A: Net Discount = 5 + 30 − (5 × 30)/100 = 33.5%

Scheme B: Net Discount ≈ 18 + 15 + 5 = 38% (approx, but effectively around 33.8%)

Scheme C: For exactly 5 toys, only the 11% slab applies ⇒ 11%

Scheme D: Buy 5 Get 2 Free ⇒ Discount = (2 / 7) × 100 ≈ 28.57%

Comparing all, Scheme C (11%) provides the minimum discount, making it the least beneficial.

∴ The correct answer is Scheme C.

Alternate Method

Let the Marked Price (MP) of each toy be ₹100. For 5 toys, Total MP = ₹500.

Scheme A (Successive 5% and 30%):

⇒ Price after 5% = 500 − 25 = ₹475

⇒ Price after 30% = 475 − 142.5 = ₹332.5

⇒ Total Discount = 500 − 332.5 = ₹167.5 (33.5%)

Scheme B (Successive 18%, 15%, 5%):

⇒ Net Discount % = 100 − [100 × (0.82 × 0.85 × 0.95)]

⇒ Net Discount % = 100 − 66.215 = 33.785%

Scheme C (11% on first 5 toys):

⇒ Since the customer buys exactly 5 toys, the discount is strictly 11%.

⇒ Total Discount = 11% of 500 = ₹55

Scheme D (2 toys free on buying 5):

⇒ Customer pays for 5 toys and receives 7 toys.

⇒ Effective Discount % = (Free Items ÷ Total Items) × 100

⇒ Discount % = (2 ÷ 7) × 100 ≈ 28.57%

Comparing the effective discount percentages, Scheme C offers the lowest value (11%), which means the customer pays the highest price under this scheme.

∴ The correct answer is Scheme C.

Additional Information

Successive Discount Formula

For two successive discounts d₁% and d₂%, the single equivalent discount is (d₁ + d₂ − d₁d₂/100)%.

Buy X Get Y Free

The effective discount percentage in such schemes is calculated as [Y ÷ (X + Y)] × 100.

Least Beneficial Scheme

For a customer, the least beneficial scheme is the one that offers the lowest net discount or results in the highest Selling Price (SP).

Shortcut Trick

Given Remainder = 15.

Divisor = 4 × 15 = 60.

Quotient = 60 ÷ 5 = 12.

Dividend = (60 × 12) + 15 = 720 + 15 = 735.

∴ The correct answer is 735.

Alternate Method

Given:

In a division sum:

Remainder (R) = 15

Divisor (D) = 4 × R

Divisor (D) = 5 × Quotient (Q)

Formula Used:

Dividend = (Divisor × Quotient) + Remainder

Calculations:

⇒ Divisor = 4 × 15 = 60

⇒ 60 = 5 × Quotient

⇒ Quotient = 60 ÷ 5 = 12

⇒ Dividend = (60 × 12) + 15

⇒ Dividend = 720 + 15

⇒ Dividend = 735

∴ The correct answer is 735.

Additional Information

Division Algorithm

For any two integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

Properties of Remainder

The remainder is always a non-negative integer and is strictly less than the divisor. If the remainder is 0, the dividend is perfectly divisible by the divisor.

Verification Rule

To verify a division result, subtract the remainder from the dividend; the result must be the product of the divisor and the quotient.

Shortcut Trick

Effective Compound Interest rate for 2 years = 5 + 5 + (5 × 5)/100 = 10.25%

Given CI = 10.25% of Principal = ₹102.50

Principal = (102.50 ÷ 10.25) × 100

Principal = 10 × 100 = ₹1,000

∴ The correct answer is ₹1,000.

Alternate Method

Given:

Time (n) = 2 years, Rate (R) = 5% per annum, Compound Interest (CI) = ₹102.50

Formula Used:

CI = P[(1 + R/100)ⁿ − 1]

Calculations:

⇒ 102.50 = P[(1 + 5/100)² − 1] ⇒ 102.50 = P[(21/20)² − 1] ⇒ 102.50 = P[441/400 − 1] ⇒ 102.50 = P[(441 − 400)/400] ⇒ 102.50 = P × (41/400) ⇒ P = (102.50 × 400) ÷ 41 ⇒ P = 2.5 × 400 ⇒ P = ₹1,000

∴ The correct answer is ₹1,000.

Additional Information

Effective Rate for Compound Interest

For 2 years at rate R%, the net effective interest rate is calculated using the successive percentage formula: (R + R + R²/100)%.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is given by the formula: D = P(R/100)².

Amount Formula

The total Amount (A) accumulated after n years is A = P(1 + R/100)ⁿ, where P is the Principal.

Shortcut Trick

The ratio of the first three terms is 2 : 5 : 7. Let the terms be 2x, 5x, and 7x.

According to the proportion rule: 1^st / 2^nd = 3^rd / 4^th.

⇒ 2 / 5 = 7x / 140

⇒ 7x = (2 × 140) ÷ 5 = 56

⇒ x = 56 ÷ 7 = 8

Middle number (2^nd term) = 5 × 8 = 40.

∴ The correct answer is 40.

Alternate Method

Given: Ratio of three numbers = 2 : 5 : 7 and fourth proportional = 140.

Formula: If a, b, c, and d are in proportion (a : b :: c : d), then a × d = b × c.

⇒ Let the numbers be 2x, 5x, and 7x.

⇒ By proportion property: (2x) × 140 = (5x) × (7x)

⇒ 280x = 35x²

⇒ 280 = 35x (Dividing both sides by x)

⇒ x = 280 ÷ 35

⇒ x = 8

⇒ Middle number = 2^nd term = 5x = 5 × 8 = 40.

∴ The correct answer is 40.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is given by √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a : b :: b : c), then the third proportional c = b² ÷ a.

Fourth Proportional

If four quantities a, b, c, and d are in proportion (a : b :: c : d), then d = (b × c) ÷ a.

Product Rule

For any proportion, the product of the extremes (1^st and 4^th terms) is always equal to the product of the means (2^nd and 3^rd terms).

Shortcut Trick

For divisibility by 9, the sum of digits must be a multiple of 9.

Sum of digits in 6y512 = 6 + y + 5 + 1 + 2 = 14 + y.

The nearest multiple of 9 after 14 is 18.

∴ 14 + y = 18 ⇒ y = 4.

∴ The correct answer is 4.

Alternate Method

Given: The number 6y512 is divisible by both 8 and 9.

Step 1: Check Divisibility by 8

A number is divisible by 8 if its last three digits are divisible by 8.

⇒ Last three digits = 512.

⇒ 512 ÷ 8 = 64 (Remainder = 0).

Since 512 is divisible by 8, the number 6y512 is divisible by 8 for any single digit value of y.

Step 2: Check Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

⇒ Sum = 6 + y + 5 + 1 + 2

⇒ Sum = 14 + y

For the number to be divisible by 9, (14 + y) must be 18, 27, etc.

Since y is a single digit (0 to 9):

⇒ 14 + y = 18

⇒ y = 18 − 14

⇒ y = 4

∴ The correct answer is 4.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8 or if the last three digits are zero.

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is exactly divisible by 9.

Divisibility Rule of 72

Since 72 = 8 × 9 (co-prime factors), a number is divisible by 72 only if it is divisible by both 8 and 9 simultaneously.

Shortcut Trick

Since the cuboid is made from half the cube's material, the difference (250 cm³) is equal to the volume of the cuboid itself.

Volume of cuboid = 250 cm³ ⇒ Ratio 2 : 1 : 1 ⇒ (2x) × (x) × (x) = 250.

2x³ = 250 ⇒ x³ = 125 ⇒ x = 5 cm.

Dimensions: L = 10 cm, B = 5 cm, H = 5 cm.

TSA = 2 × (10 × 5 + 5 × 5 + 5 × 10) = 2 × (50 + 25 + 50) = 250 cm².

∴ The correct answer is 250 cm².

Alternate Method

Given:

Ratio of cuboid dimensions (L : B : H) = 2 : 1 : 1

Difference (V_cube − V_cuboid) = 250 cm³

Volume of cuboid = 1/2 × Volume of cube

Formula Used:

Volume of cuboid = Length × Breadth × Height

Total Surface Area (TSA) = 2(LB + BH + HL)

Calculations:

⇒ Let Volume of Cube be V_c.

⇒ Volume of Cuboid V_b = 1/2 × V_c.

⇒ V_c − V_b = 250 ⇒ V_c − 0.5V_c = 250 ⇒ 0.5V_c = 250.

⇒ V_b = 250 cm³.

⇒ Let dimensions be 2x, x, and x.

⇒ 2x × x × x = 250 ⇒ 2x³ = 250.

⇒ x³ = 125 ⇒ x = 5 cm.

⇒ Length = 10 cm, Breadth = 5 cm, Height = 5 cm.

⇒ TSA = 2 × (10 × 5 + 5 × 5 + 5 × 10).

⇒ TSA = 2 × (50 + 25 + 50) = 2 × 125.

⇒ TSA = 250 cm².

∴ The correct answer is 250 cm².

Additional Information

Volume and Surface Area of Cube

For a cube of side 'a', Volume = a³ and Total Surface Area = 6a².

Lateral Surface Area of Cuboid

LSA = 2 × Height × (Length + Breadth). This represents the area of the four side walls.

Diagonal of a Cuboid

The longest rod that can be placed in a cuboid has length d = √(L² + B² + H²).

Shortcut Trick

The filling rate of the second tap (10 hr) and the emptying rate of the third tap (10 hr) cancel each other out.

Net efficiency is determined solely by the first tap.

Time to fill the full cistern = 4 hr.

Time to fill 1/4^th of the cistern = 4 ÷ 4 = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given: Filling times are 4 hr and 10 hr; Emptying time is 10 hr.

Formula: Total Work = LCM of individual times; Efficiency = Total Work ÷ Time.

⇒ Net Efficiency of all taps = 5 + 2 − 2 = 5 units/hr.

⇒ Target work (1/4^th of cistern) = 20 ÷ 4 = 5 units.

⇒ Time taken = Work ÷ Net Efficiency

⇒ Time taken = 5 ÷ 5 = 1 hr.

∴ The correct answer is 1 hr.

Additional Information

Negative Efficiency

In Pipes and Cisterns, an outlet pipe or a leak is always considered to have negative efficiency because it reduces the total volume of water.

Work-Efficiency Relation

Total Work = Efficiency × Time. For a constant work, Efficiency is inversely proportional to the Time taken to complete the task.

Combined Rate Formula

If two pipes fill in 'x' and 'y' hours and one empties in 'z' hours, the net part filled in 1 hour is (1/x + 1/y − 1/z).