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Shortcut Trick

Effective Simple Interest rate for 3 years at 5% = 3 × 5% = 15%.

Total Amount = Principal + Interest = 100% + 15% = 115%.

Given that 115% represents ₹6,900.

Principal (100%) = (6,900 ÷ 115) × 100 = 60 × 100 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Total Amount (A) = ₹6,900

Rate of Interest (R) = 5% per annum

Time Period (T) = 3 years

Formula Used:

Amount (A) = Principal (P) + Simple Interest (SI)

A = P + (P × R × T) ÷ 100 ⇒ A = P × [1 + (R × T) ÷ 100]

Calculations:

⇒ 6900 = P × [1 + (5 × 3) ÷ 100]

⇒ 6900 = P × [1 + 15 ÷ 100]

⇒ 6900 = P × [1 + 0.15]

⇒ 6900 = P × 1.15

⇒ P = 6900 ÷ 1.15

⇒ P = (6900 × 100) ÷ 115

⇒ P = 60 × 100

⇒ P = ₹6,000

∴ The correct answer is ₹6,000.

Additional Information

Simple Interest (SI)

It is the interest calculated only on the original principal amount for every time period. Formula: SI = (P × R × T) ÷ 100.

Total Amount in SI

The total sum returned after the time period is A = P + SI = P(1 + RT/100). The Principal remains constant in SI calculations.

Comparison with Compound Interest

Unlike SI, Compound Interest (CI) is calculated on the principal plus the accumulated interest of previous periods. For the first year, SI and CI are equal if compounded annually.

Shortcut Trick

According to the divisibility rule of 9, the sum of all digits must be divisible by 9.

Sum of digits of 4583 = 4 + 5 + 8 + 3 = 20.

The next multiple of 9 after 20 is 27 (since 9 × 3 = 27).

Required number to be added = 27 − 20 = 7.

∴ The correct answer is 7.

Alternate Method

Given: Number = 4583, Divisor = 9

Method: Division algorithm to find the remainder.

⇒ 4583 ÷ 9

⇒ 4583 = (509 × 9) + 2

⇒ Remainder = 2

⇒ Smallest number to be added = Divisor − Remainder

⇒ 9 − 2 = 7

∴ The correct answer is 7.

Additional Information

Divisibility Rule of 9

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

Divisibility Rule of 3

Similar to the rule of 9, a number is divisible by 3 if the sum of its digits is divisible by 3.

Digital Root

The single-digit sum obtained by repeatedly adding the digits of a number is its digital root, which is equal to the remainder when the number is divided by 9 (except when the sum is 9).

Shortcut Trick

The divisibility rule of 8 states that a number is divisible by 8 if its last three digits are divisible by 8.

For the number 76488, we only need to check the last three digits: 488.

Divide 488 by 8: 488 ÷ 8 = 61.

Since 488 is completely divisible by 8, the remainder is 0.

∴ The correct answer is 0.

Alternate Method

Given:

Dividend = 76488

Divisor = 8

Calculations:

⇒ Perform direct division of 76488 by 8:

⇒ 76 ÷ 8 ⇒ 8 × 9 = 72; Remainder = 4

⇒ Bring down 4 to make it 44: 44 ÷ 8 ⇒ 8 × 5 = 40; Remainder = 4

⇒ Bring down 8 to make it 48: 48 ÷ 8 ⇒ 8 × 6 = 48; Remainder = 0

⇒ Bring down the final 8: 8 ÷ 8 ⇒ 8 × 1 = 8; Remainder = 0

⇒ Total Quotient = 9561 and the final remainder is 0.

∴ The correct answer is 0.

Additional Information

Divisibility by 2ⁿ

A number is divisible by 2ⁿ if its last n digits are divisible by 2ⁿ. For 8 (2³), we check the last 3 digits; for 16 (2⁴), we check the last 4 digits.

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4 or if the last two digits are 00.

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively.

Shortcut Trick

Difference in Profit Percentage = 48% − 16% = 32%

This 32% increase in Profit corresponds to ₹80 extra per article.

Unit Cost Price (CP) = (80 ÷ 32) × 100 = ₹250

Number of Articles = Total CP ÷ Unit CP = 8000 ÷ 250 = 32

∴ The correct answer is 32.

Alternate Method

Given:

Initial Profit = 16%

New Profit = 48%

Increase in Selling Price (SP) = ₹80

Total Cost Price of all articles = ₹8,000

Let the Cost Price of one article be x.

⇒ Initial Selling Price (SP₁) = 116% of x = 1.16x

⇒ New Selling Price (SP₂) = 148% of x = 1.48x

⇒ According to the question: SP₂ − SP₁ = 80

⇒ 1.48x − 1.16x = 80

⇒ 0.32x = 80

⇒ x = 80 ÷ 0.32

⇒ x = ₹250 (CP of 1 article)

⇒ Total Cost Price = Number of articles × CP of 1 article

⇒ 8000 = Number of articles × 250

⇒ Number of articles = 8000 ÷ 250 = 32

∴ The correct answer is 32.

Additional Information

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Profit and Loss are always calculated on the Cost Price unless stated otherwise.

Selling Price Formula

SP = CP × (100 + Profit %) ÷ 100 or SP = CP × (100 − Loss %) ÷ 100. This is the base for solving change-in-price problems.

Successive Percentage Change

When profit changes from P₁% to P₂%, the effective change in Selling Price is (P₂ − P₁)% of the Cost Price.

Shortcut Trick

When three numbers a, b, and c are in continued proportion, the relationship is a : b :: b : c.

This means the third proportional (c) is calculated as c = b² ÷ a.

Given: a = 4 cm and b = 6 cm. Substituting these values: c = (6 × 6) ÷ 4 = 36 ÷ 4 = 9.

∴ The correct answer is 9 cm.

Alternate Method

Given: Length of the first rod (a) = 4 cm, Length of the second rod (b) = 6 cm.

Formula Used: For three rods of lengths a, b, and c to be in continued proportion, the ratio a : b must be equal to b : c.

Let the length of the third rod be x cm.

⇒ 4 ÷ 6 = 6 ÷ x

⇒ 4 × x = 6 × 6

⇒ 4x = 36

⇒ x = 36 ÷ 4

⇒ x = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Continued Proportion

Three quantities a, b, and c are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third (a/b = b/c).

Mean Proportional

In the proportion a : b :: b : c, 'b' is called the mean proportional between a and c, where b = √(a × c).

Third Proportional

In the proportion a : b :: b : c, 'c' is known as the third proportional to a and b, calculated as c = b²/a.

Fourth Proportional

For four numbers a, b, c, and d in proportion (a : b :: c : d), 'd' is the fourth proportional and d = (b × c) ÷ a.

Shortcut Trick

When two points are on opposite sides of a vertical object, the height h = D ÷ (cot α + cot β).

Given horizontal distance D = 94 m, α = 30° and β = 60°.

h = 94 ÷ (cot 30° + cot 60°) = 94 ÷ (√3 + 1/√3) = 94 ÷ (4/√3) = (94 × √3) ÷ 4.

h = 23.5 × 1.73 = 40.655 m.

∴ The correct answer is 40.655 m.

Alternate Method

Given:

Horizontal distance between two points A and B = 94 m.

Angles of elevation of the delivery drone (P) from A and B are 30° and 60° respectively.

Let the vertical height of the drone be h (PD), where D is the point on the ground directly below the drone.

Formula Used:

Tangent of an angle (θ) = Perpendicular ÷ Base

Calculations:

In right-angled triangle PAD:

⇒ tan 30° = PD ÷ AD

⇒ 1 ÷ √3 = h ÷ AD

⇒ AD = h × √3

In right-angled triangle PBD:

⇒ tan 60° = PD ÷ BD

⇒ √3 = h ÷ BD

⇒ BD = h ÷ √3

Since A and B are on opposite sides:

⇒ AB = AD + BD = 94

⇒ h√3 + h/√3 = 94

⇒ (3h + h) ÷ √3 = 94

⇒ 4h ÷ √3 = 94

⇒ h = (94 × √3) ÷ 4

⇒ h = 23.5 × 1.73

⇒ h = 40.655 m

∴ The correct answer is 40.655 m.

Additional Information

Trigonometric Ratios for 30° and 60°

Values for calculations: tan 30° = 1/√3, tan 60° = √3. Cot ratios are reciprocals: cot 30° = √3, cot 60° = 1/√3.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when the object is above the eye level.

Complementary Angles Rule

If angles of elevation from two points are complementary (α + β = 90°), then height h = √(AD × BD).

Heights and Distances (Same Side)

If points are on the same side of the drone, the horizontal distance D = h × (cot α − cot β).

Shortcut Trick

Assume Total Capacity is the LCM of 24 and 32 = 96 units.

Efficiency of M = 4 units/min and N = 3 units/min.

Combined efficiency (M + N + O) = 96 ÷ 32 = 3 units/min.

Efficiency of Outlet O = 3 − (4 + 3) = −4 units/min (negative sign denotes drainage).

Time for O to drain = 96 ÷ 4 = 24 min.

∴ The correct answer is 24 min.

Alternate Method

Given:

Time taken by Inlet M = 24 min

Time taken by Inlet N = 32 min

Time taken by (M + N + Outlet O) = 32 min

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total capacity = LCM(24, 32) = 96 units

⇒ Efficiency of Inlet M = 96 ÷ 24 = 4 units/min

⇒ Efficiency of Inlet N = 96 ÷ 32 = 3 units/min

⇒ Net efficiency when all three operate = 96 ÷ 32 = 3 units/min

⇒ Efficiency of M + N + O = Efficiency of M + Efficiency of N + Efficiency of O

⇒ 3 = 4 + 3 + Efficiency of O

⇒ Efficiency of O = 3 − 7 = −4 units/min

⇒ Time taken by Outlet O alone = Total Capacity ÷ |Efficiency of O|

⇒ Time = 96 ÷ 4 = 24 min

∴ The correct answer is 24 min.

Additional Information

Pipes and Cisterns Basics

Inlets fill the tank (positive work) and Outlets empty the tank (negative work). If a pipe fills a tank in 'x' hours, its efficiency is 1/x per hour.

Combined Rate Formula

For two inlets A and B and one outlet C, the net part filled in 1 hour = (1/A + 1/B − 1/C).

Emptying vs. Filling

If the net efficiency is negative, the tank will be emptied; if positive, the tank will be filled.

Given:

Expression = (((10)/(18)))/(((10)/(9))) ÷ ((4)/(9) × (18)/(12) + (5)/(6)) + (9)/(9) ÷ (27)/(15) of (15)/(9)

Formula used:

According to the BODMAS rule, operations are performed in the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

Step 1: Simplify the first term

⇒ (((10)/(18)))/(((10)/(9))) = 10/18 × 9/10 = 1/2

Step 2: Simplify the expression inside the bracket

⇒ ((4/9) × (18/12) + 5/6) = (2/3 + 5/6)

⇒ (4/6 + 5/6) = 9/6 = 3/2

Step 3: Simplify the part containing of and division

⇒ (9/9) ÷ (27/15 of 15/9) = 1 ÷ (27/15 × 15/9)

⇒ 1 ÷ (27/9) = 1 ÷ 3 = 1/3

Step 4: Combine all simplified parts into the main expression

⇒ (1/2) ÷ (3/2) + 1/3

⇒ (1/2 × 2/3) + 1/3

⇒ 1/3 + 1/3 = 2/3

∴ The correct answer is option (4).

Shortcut Trick

Ratio of Expenditure to Savings = 70 : 30 = 7 : 3.

Using Weighted Average formula: Overall Income % = (E_ratio × E% + S_ratio × S%) ÷ Total_ratio

20 = (7 × 25 + 3 × x) ÷ 10

200 = 175 + 3x ⇒ 3x = 25 ⇒ x = 8¹/₃%.

∴ The correct answer is 8¹/₃% increase.

Alternate Method

Given: Initial Expenditure = 70% of Income. Income increase = 20%. Expenditure increase = 25%.

Formula Used: Income = Expenditure + Savings

⇒ Let Initial Income = 100

⇒ Initial Expenditure = 70% of 100 = 70

⇒ Initial Savings = 100 − 70 = 30

⇒ New Income = 100 + 20% of 100 = 120

⇒ New Expenditure = 70 + 25% of 70 = 70 + 17.5 = 87.5

⇒ New Savings = 120 − 87.5 = 32.5

⇒ Change in Savings = 32.5 − 30 = 2.5

⇒ Percentage Change = (2.5 ÷ 30) × 100 = 25/3 = 8¹/₃%

∴ The correct answer is 8¹/₃% increase.

Additional Information

Income-Expenditure Identity

Total Income is always the sum of Expenditure and Savings (I = E + S).

Percentage Change Formula

% Change = [(New Value − Old Value) ÷ Old Value] × 100.

Weighted Average in Percentages

If a total is divided into two parts, the overall percentage change is the weighted average of the percentage changes of the individual parts.

Shortcut Trick

Identify the bases: a = 20, b = −16, and c = −4.

Check the sum of the bases: 20 + (−16) + (−4) = 0.

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculation: 3 × 20 × (−16) × (−4) = 60 × 64 = 3840.

∴ The correct answer is 3840.

Alternate Method

Given:

The expression to evaluate is 20³ + (−16)³ + (−4)³.

Formula Used:

Direct calculation of cubes of the given numbers.

Calculations:

⇒ 20³ = 20 × 20 × 20 = 8000

⇒ (−16)³ = (−16) × (−16) × (−16) = −4096

⇒ (−4)³ = (−4) × (−4) × (−4) = −64

⇒ Sum = 8000 + (−4096) + (−64)

⇒ Sum = 8000 − 4160

⇒ Result = 3840

∴ The correct answer is 3840.

Additional Information

Algebraic Cube Identity

The expression a³ + b³ + c³ − 3abc is equal to (a + b + c)(a² + b² + c² − ab − bc − ca).

Special Case

When a + b + c = 0, the identity simplifies to a³ + b³ + c³ = 3abc, which is useful for simplifying complex sums of cubes.

Negative Number Powers

Any negative number raised to an odd power results in a negative value, while an even power results in a positive value.