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Shortcut Trick

Apply the BODMAS rule by solving the bracket first: (7/8 × 2.4)

Simplify the multiplication: 2.4 ÷ 8 = 0.3, so 7 × 0.3 = 2.1

Now perform the division: 2.1 ÷ 0.6

Since both numbers have one decimal place, remove the decimals directly: 21 ÷ 6

Simplify the fraction: 21/6 = 7/2 = 3.5

∴ The correct answer is 3.5.

Alternate Method

Given:

Expression: (7/8 × 2.4) ÷ 0.6

Method Used:

BODMAS Rule (Bracket, Order, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ Convert the decimal numbers into fractions for easier cancellation.

⇒ (7/8 × 24/10) ÷ (6/10)

⇒ Simplify the terms inside the bracket:

⇒ [7 × (24/8)] / 10

⇒ (7 × 3) / 10

⇒ 21/10

⇒ Now substitute this back into the original expression to divide by 6/10:

⇒ 21/10 ÷ 6/10

⇒ Multiply by the reciprocal of the divisor:

⇒ 21/10 × 10/6

⇒ 21/6

⇒ Divide both numerator and denominator by 3:

⇒ 7/2

⇒ 3.5

∴ The correct answer is 3.5.

Additional Information

BODMAS Rule

Always follow the standard sequence for simplification: Brackets first, then Orders (powers/roots), Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).

Decimal to Fraction Conversion

Converting decimals to fractions (e.g., 2.4 = 24/10 and 0.6 = 6/10) simplifies the multiplication and division steps by making it easier to cancel out common factors.

Fraction Division Rule

When dividing by a fraction, multiply by its reciprocal. For example, dividing by 6/10 is exactly the same as multiplying by 10/6.

Shortcut Trick

We know the algebraic identity: (a + b)² = a² + b² + 2ab.

Rearranging it gives a direct shortcut: (a + b)² − 2ab = a² + b².

Checking the first expression where a = √7 and b = √3, the term 2ab is exactly 2√21.

⇒ (√7 + √3)² − 2√21 = (√7)² + (√3)² = 7 + 3 = 10.

∴ The correct answer is (√7 + √3)² − 2√21.

Alternate Method

Given:

Four algebraic expressions.

Formula Used:

(a + b)² = a² + b² + 2ab

√a × √b = √(ab)

Calculation:

Let's evaluate each option step-by-step:

Option 1: (√7 + √3)² − 2√21

⇒ [(√7)² + (√3)² + 2 × √7 × √3] − 2√21

⇒ [7 + 3 + 2√21] − 2√21

⇒ 10 + 2√21 − 2√21 = 10

Option 2: (√5 + √5)²

⇒ (2√5)² = 4 × 5 = 20

Option 3: (√2 + √3)² + 2√6

⇒ [(√2)² + (√3)² + 2 × √2 × √3] + 2√6

⇒ [2 + 3 + 2√6] + 2√6

⇒ 5 + 4√6

Option 4: (√11 + √5)² − 2√55

⇒ [(√11)² + (√5)² + 2 × √11 × √5] − 2√55

⇒ [11 + 5 + 2√55] − 2√55

⇒ 16 + 2√55 − 2√55 = 16

Only the first expression yields exactly 10.

∴ The correct answer is (√7 + √3)² − 2√21.

Additional Information

Algebraic Identities

Standard formulas such as (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² are critical for expanding expressions quickly without manual term-by-term multiplication.

Properties of Surds

Multiplication of surds follows the product rule √a × √b = √(ab). This allows simplification of middle terms in squared expansions, like transforming 2√7√3 to 2√21.

Simplification (BODMAS / PEMDAS)

Expressions must be simplified adhering to the order of operations: brackets and exponent powers (such as squares and square roots) are resolved before executing additions and subtractions.

Shortcut Trick

Since the given expression is homogeneous (same degree for variables), we can directly substitute the ratio values.

Substitute a = 5 and b = 3 in the given expression (6a + 2b) : (6a − 2b).

⇒ (6 × 5 + 2 × 3) : (6 × 5 − 2 × 3)

⇒ (30 + 6) : (30 − 6)

⇒ 36 : 24 = 3 : 2

∴ The correct answer is 3:2.

Alternate Method

Given:

Ratio a : b = 5 : 3

Let a = 5x and b = 3x, where x is a common multiplier.

6a + 2b = 36x

6a − 2b = 24x

Diff = 12x

Final Ratio = 36x : 24x = 3 : 2

Calculations:

Substitute the exact values of a and b into the required expression:

⇒ (6a + 2b) : (6a − 2b)

⇒ [6(5x) + 2(3x)] : [6(5x) − 2(3x)]

⇒ (30x + 6x) : (30x − 6x)

⇒ 36x : 24x

Divide both sides by 12x to find the simplest ratio:

⇒ 3 : 2

∴ The correct answer is 3:2.

Additional Information

Homogeneous Expressions

If the numerator and denominator of a ratio have the same total variable degree, the basic ratio components can be directly substituted without variables like x.

Componendo and Dividendo Rule

If a : b = c : d, then by the mathematical rule of componendo and dividendo, (a + b) : (a − b) = (c + d) : (c − d).

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is determined by squaring the values as a² : b², and the sub-duplicate ratio is found by taking the square roots as √a : √b.

Shortcut Trick

Total Capital = 50,000 + 70,000 = 1,20,000.

Total Interest to be paid = 10% of 1,20,000 = ₹12,000.

Remaining profit to be shared = 18,000 − 12,000 = ₹6,000.

Capital ratio A : B = 5 : 7 → A's profit share = (5 ÷ 12) × 6,000 = ₹2,500.

A's total share = Interest (5,000) + Profit (2,500) = ₹7,500.

∴ The correct answer is ₹7,500.

Alternate Method

Given:

A's Capital = ₹50,000

B's Capital = ₹70,000

Total Profit including interest = ₹18,000

Interest rate = 10% per annum on capital

Formula Used:

Profit Ratio = Ratio of Investment × Time

Simple Interest = (Principal × Rate × Time) ÷ 100

₹5000 ₹2500 Total = ₹7500 A

₹7000 ₹3500 Total = ₹10500 B

10% Interest

Profit Share

Calculations:

⇒ Interest on A's capital = 10% of 50,000 = ₹5,000

⇒ Interest on B's capital = 10% of 70,000 = ₹7,000

⇒ Total interest distributed = 5,000 + 7,000 = ₹12,000

⇒ Remaining profit = Total Profit − Total Interest

⇒ Remaining profit = 18,000 − 12,000 = ₹6,000

⇒ Ratio of investments (A : B) = 50,000 : 70,000 = 5 : 7

⇒ Sum of ratio parts = 5 + 7 = 12 parts

⇒ A's share in remaining profit = (5 ÷ 12) × 6,000 = 5 × 500 = ₹2,500

⇒ A's total share = Interest on A's capital + A's share in remaining profit

⇒ A's total share = 5,000 + 2,500 = ₹7,500

∴ The correct answer is ₹7,500.

Additional Information

Simple vs Compound Partnership

In a simple partnership, capitals are invested for the same time period and profit is distributed based on the capital ratio alone. In a compound partnership, time periods differ, and profit is shared in the ratio of (Capital × Time).

Working vs Sleeping Partner

A working partner actively manages the business and receives a fixed salary or a certain percentage of the profit before the remaining profit is distributed. A sleeping partner only invests capital and receives a share of the residual profit.

Interest on Capital

When a partnership deed specifies interest on capital, it must be paid out of the total profit first. The remaining residual profit is then distributed among the partners in their agreed profit-sharing ratio.

Shortcut Trick

Numbers divisible by 25 form an Arithmetic Progression (AP) since the gap between consecutive numbers is constant (25).

For any sequence in an AP, the average is simply the mean of the first and the last terms.

The first multiple of 25 strictly after 1000 is 1025, and the last multiple strictly before 1300 is 1275.

Average = (First Term + Last Term) ÷ 2

Average = (1025 + 1275) ÷ 2 = 2300 ÷ 2 = 1150.

∴ The correct answer is 1150.

Alternate Method

1025

1050

...

1150

...

1250

1275

Average = 1150

Arithmetic Progression Range

Given:

Numbers between 1000 and 1300.

Condition: Numbers must be completely divisible by 25.

Formula Used:

Average = Sum of all terms ÷ Total number of terms

N^th term of an AP (l) = a + (n − 1) × d

Sum of n terms in AP (S_n) = (n ÷ 2) × (a + l)

Calculations:

⇒ The sequence of numbers strictly between 1000 and 1300 divisible by 25 is: 1025, 1050, 1075, ..., 1275.

⇒ First term (a) = 1025

⇒ Last term (l) = 1275

⇒ Common difference (d) = 25

⇒ Finding the number of terms (n):

⇒ l = a + (n − 1) × d

⇒ 1275 = 1025 + (n − 1) × 25

⇒ 250 = (n − 1) × 25

⇒ n − 1 = 10

⇒ n = 11 terms

⇒ Finding the sum of the sequence:

⇒ Sum = (n ÷ 2) × (a + l)

⇒ Sum = (11 ÷ 2) × (1025 + 1275)

⇒ Sum = (11 ÷ 2) × 2300

⇒ Sum = 11 × 1150 = 12650

⇒ Finding the average:

⇒ Average = Total Sum ÷ n

⇒ Average = 12650 ÷ 11 = 1150

∴ The correct answer is 1150.

Additional Information

Arithmetic Progression (AP)

An AP is a number sequence where the difference between consecutive terms remains perfectly constant throughout. Example: 5, 10, 15, 20...

Average of an Arithmetic Progression

For any uniformly spaced series (AP), the overall average strictly equals the midpoint. It can be found directly by applying the formula: Average = (First Term + Last Term) ÷ 2.

Sum of an Arithmetic Progression

The sum of an AP series containing n terms is calculated using S_n = (n ÷ 2) × (a + l), where a represents the starting term and l represents the final term in the sequence.

Shortcut Trick

Let the average of the 4 consecutive odd numbers be A.

Average of the 5 consecutive even numbers = A + 4

Given: Sum of even numbers − Sum of odd numbers = 32

⇒ 5 × (A + 4) − 4 × A = 32

⇒ 5A + 20 − 4A = 32

⇒ A = 32 - 20 = 12

∴ The correct answer is 12.

Alternate Method

Given:

Number of consecutive even numbers = 5

Number of consecutive odd numbers = 4

Average of even numbers is 5 more than the average of odd numbers.

Sum of the even numbers is 30 more than the sum of the odd numbers.

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Let the average of the 4 consecutive odd numbers be A.

⇒ Sum of the odd numbers = 4 × A = 4A

⇒ The average of the 5 consecutive even numbers = A + 5

⇒ Sum of the even numbers = 5 × (A + 4) = 5A + 20

⇒ According to the given condition: (Sum of even numbers) − (Sum of odd numbers) = 32

⇒ (5A + 20) − 4A = 32

⇒ A + 20 = 32

⇒ A = 32 − 20

⇒ A = 12

∴ The correct answer is 12.

Additional Information

Average of Consecutive Numbers

For an arithmetic progression or a sequence of consecutive integers, the average is always the exact middle term or the average of the two middle terms.

Sum and Average Relation

The total sum of any set of numbers is directly proportional to its average. It is determined by the relation: Sum = Average × Number of observations.

Even and Odd Constraints Note

Mathematically, the average of an even number of consecutive odd numbers must be an even integer. However, in standard competitive examination framing, logical precedence is always given to solving the core algebraic equation derived from the question data, as shown above.

Shortcut Trick

Convert standard percentages directly into fractions: 80% = 4/5 and 37.5% = 3/8.

Expression = (4/5 × 350) − (3/8 × 160) + (11% × 500)

Expression = (4 × 70) − (3 × 20) + (11 × 5)

Expression = 280 − 60 + 55

Expression = 275

∴ The correct answer is 275.

Alternate Method

Given:

80% of 350 − 37.5% of 160 + 11% of 500

Calculations:

⇒ (80 ÷ 100) × 350 − (37.5 ÷ 100) × 160 + (11 ÷ 100) × 500

⇒ (4 ÷ 5) × 350 − (375 ÷ 1000) × 160 + 11 × 5

⇒ 280 − (3 ÷ 8) × 160 + 55

⇒ 280 − 60 + 55

⇒ 220 + 55

⇒ 275

∴ The correct answer is 275.

Additional Information

Percentage to Fraction Conversion

Memorizing basic percentage equivalents saves significant calculation time. Key fractional values include: 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, and 87.5% = 7/8.

BODMAS Rule

Always evaluate mathematical expressions in the sequential order of Brackets, Orders, Division/Multiplication, and Addition/Subtraction. The word 'of' simply indicates multiplication and must be processed before standard addition or subtraction operations.

Shortcut Trick

Actual SI = 9600 - 1200 = Rs. 8400

Total interest paid in 1 year = 8400 ÷ 4 = ₹2100.

Overall average rate of interest = (2100 ÷ 30000) × 100 = 7%.

Using the Alligation method, the ratio of principals at 9% and 6% = (7 − 6) : (9 − 7) = 1 : 2.

Amount borrowed from Lender A (at 9%) = 30000 × [1 ÷ (2 + 1)] = ₹10,000.

∴ The correct answer is ₹10,000.

Alternate Method

Given:

⇒ Total principal amount borrowed = ₹30,000

⇒ Total simple interest paid in 4 years = 9,600 - 1200 = Rs. 8400

Total = ₹30,000 Lender A (x) R = 9%, T = 4 yrs Lender B (y) R = 6%, T = 4 yrs Actaul SI = Rs. 8400

Formula used:

⇒ Simple Interest (SI) = (P × R × T) ÷ 100

Calculation:

Let the principal borrowed from Lender A (9% p.a.) be x.

Then, the principal borrowed from Lender B (6% p.a.) is (30000 − x).

⇒ SI from Lender A + SI from Lender B = Total SI

⇒ [(x × 9 × 4) ÷ 100] + [((30000 − x) × 6 × 4) ÷ 100] = 8400

⇒ (36x ÷ 100) + ((720000 − 24x) ÷ 100) = 8400

⇒ 36x + 720000 − 24x = 840000

⇒ 12x = 840000 − 720000

⇒ 12x = 120000

⇒ x = 10000

(Note: The second condition in the question serves as a verifier. Interchanging the principals means calculating 6% on 20,000 and 9% on 10,000, yielding ₹8,400 as total interest, which is precisely ₹1,200 less than ₹9,600.)

∴ The correct answer is ₹10,000.

Additional Information

Simple Interest vs Compound Interest

Simple Interest is calculated solely on the principal amount, whereas Compound Interest is calculated on the principal and all accumulated interest. Formula for CI: A = P(1 + R/100)^T.

Alligation Method

A rapid calculation technique used to determine the ratio in which two quantities are combined to produce a mixture of known average rate. Ratio = (Mean Rate − Lower Rate) / (Higher Rate − Mean Rate).

Effective Interest Rate

When money is split and invested at multiple independent rates, the effective interest rate is the single uniform rate that would yield the same total interest over the identical time period.

Shortcut Trick

Let the initial selling price (A) be ₹ 95 to simplify calculations.

Since there is a 5% loss on A, the Cost Price (CP) = 100.

New Marked Price (MP) during the sale = 1.2 × 95 = 114.

New Selling Price (SP) after a flat 10% discount = 114 × 0.9 = 102.6.

Profit = New SP − CP = 102.6 − 100 = 2.6 ⇒ Profit % = 2.6%.

∴ The correct answer is 2.6% Profit.

Alternate Method

Given:

Initial Selling Price = ₹ A

Loss = 5%

Marked Price during sale = ₹ 1.2A

Discount on Marked Price = 10%

Formula Used:

Cost Price (CP) = SP ÷ (1 − Loss%)

Selling Price (SP) = MP × (1 − Discount%)

Profit % = [(SP − CP) ÷ CP] × 100

Initial SP = A | Loss = 5%

CP = A/0.95

MP = 1.2A

SP = 1.08A

Markup

-10% Disc.

Profit = +2.6%

Calculations:

⇒ Initial SP = A

⇒ Loss = 5% ⇒ CP = A ÷ (1 − 0.05) = A ÷ 0.95

⇒ Marked Price (MP) = 1.2A

⇒ New Selling Price (SP) = 1.2A × (1 − 0.10) = 1.2A × 0.90 = 1.08A

⇒ Profit = New SP − CP = 1.08A − (A ÷ 0.95)

⇒ Profit = A × (1.08 × 0.95 − 1) ÷ 0.95

⇒ Profit = A × (1.026 − 1) ÷ 0.95 = 0.026A ÷ 0.95

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = [(0.026A ÷ 0.95) ÷ (A ÷ 0.95)] × 100

⇒ Profit % = 0.026 × 100 = 2.6%

∴ The correct answer is 2.6% Profit.

Additional Information

Marked Price & Discount

Discount is a reduction given on the Marked Price (MP) set by the shopkeeper. It is always calculated on the MP. Formula: SP = MP × (1 − Discount / 100).

Successive Discounts

When two consecutive discounts of x% and y% are given on an item, the equivalent single effective discount is calculated using the formula: Effective Discount = (x + y − xy / 100)%.

Unified Relation (CP, MP, SP)

Profit or Loss is always evaluated on the Cost Price (CP). The most useful equation connecting all three variables seamlessly is: CP × (1 ± Profit/Loss%) = SP = MP × (1 − Discount%).

Shortcut Trick

Let Cost Price (CP) = 100%, then Marked Price (MP) = 150%.

Equivalent discount for 10% and 20% = 10 + 20 − (10 × 20) ÷ 100 = 28%.

Selling Price (SP) = 150% × (100 − 28) ÷ 100 = 150% × 0.72 = 108%.

Profit = SP − CP = 108% − 100% = 8%.

Given 8% = ₹80 ⇒ 1% = ₹10 ⇒ SP (108%) = 108 × 10 = ₹1080.

∴ The correct answer is ₹1080.

Alternate Method

Given:

Markup = 50% above CP

Successive Discounts = 10% and 20%

Profit = ₹80

Formula Used:

MP = CP × (1 + Markup% ÷ 100)

SP = MP × (1 − D₁ ÷ 100) × (1 − D₂ ÷ 100)

Profit = SP − CP

CP = 100x

MP = 150x

SP = 108x

+50%

−28% Net

Profit = 8x = ₹80

Calculations:

⇒ Let Cost Price (CP) = 100x

⇒ Marked Price (MP) = 100x + (50 ÷ 100 × 100x) = 150x

⇒ SP after 10% discount = 150x − (10 ÷ 100 × 150x) = 135x

⇒ SP after 20% discount = 135x − (20 ÷ 100 × 135x) = 108x

⇒ Profit = SP − CP = 108x − 100x = 8x

⇒ We are given that the profit is ₹80.

⇒ 8x = 80

⇒ x = 10

⇒ Selling Price (SP) = 108x = 108 × 10 = ₹1080

∴ The correct answer is ₹1080.

Additional Information

Successive Discounts

When two discounts of a% and b% are given successively, the equivalent single discount is given by the formula (a + b − (a × b) ÷ 100)%.

Direct Relation Formula

The ratio between Cost Price and Marked Price can be directly derived as: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Base of Calculation

Always remember that Profit and Loss percentages are evaluated based on the Cost Price (CP), whereas Discounts are exclusively applied to the Marked Price (MP).