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Shortcut Trick

Total percentage interest = Rate × Time = 8% × 4 = 32%.

According to the question, 32% of the Principal = ₹1,320.

1% of the Principal = 1320 ÷ 32 = 41.25.

100% (Original Sum) = 41.25 × 100 = ₹4,125.

∴ The correct answer is ₹4,125.

Alternate Method

Given:

Simple Interest (SI) = ₹1,320

Rate of Interest (R) = 8% per annum

Time (T) = 4 years

Formula Used:

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

Calculations:

⇒ P = (1320 × 100) ÷ (8 × 4)

⇒ P = 132000 ÷ 32

⇒ P = 4125

∴ The correct answer is ₹4,125.

Additional Information

Amount in Simple Interest

The total Amount (A) is the sum of the Principal and the Simple Interest: A = P + SI.

Compound Interest (CI)

Unlike Simple Interest, Compound Interest is calculated on the Principal plus the interest accumulated over previous periods: A = P(1 + r/100)ⁿ.

Difference between SI and CI

For a period of 1 year at the same rate, Simple Interest and Compound Interest are always equal, provided the interest is compounded annually.

Rule of 72

A quick way to estimate how long it takes to double your money with Compound Interest: Time ≈ 72 ÷ Interest Rate.

Shortcut Trick

Ratio of Profit = Ratio of Investments (since time is the same).

Profit Ratio of P : Q = 3000 : 600 = 5 : 1.

Total units = 5 + 1 = 6 units → ₹54,846.

P's share (5 units) = (54,846 ÷ 6) × 5 = 9,141 × 5 = ₹45,705.

∴ The correct answer is ₹45,705.

Alternate Method

Given:

Investment of P = ₹3,000

Investment of Q = ₹600

Total Profit at the end of 1 year = ₹54,846

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Profit Ratio (P : Q) = 3000 : 600

⇒ Profit Ratio (P : Q) = 5 : 1

⇒ Sum of ratios = 5 + 1 = 6

⇒ P's share in profit = (5 / 6) × 54,846

⇒ P's share = 5 × 9,141

⇒ P's share = ₹45,705

∴ The correct answer is ₹45,705.

Additional Information

Basic Rule of Partnership

Profit is distributed in the ratio of the product of Capital and Time Period (P = C × T).

Variable Time Investment

If P invests for T₁ months and Q for T₂ months, their profit ratio is (C₁ × T₁) : (C₂ × T₂).

Active vs. Sleeping Partner

An active partner may receive an additional salary or commission from the total profit before the remaining profit is shared according to the investment ratio.

Shortcut Trick

Total Area = Area₁ + Area₂ + Area₃

Area₃ = 1/2 × 12 × 38 = 6 × 38 = 228 cm²

Summing all areas ⇒ 24 + 30 + 228 = 282 cm²

∴ The correct answer is 282 cm².

Alternate Method

Given:

Area of Triangle 1 = 24 cm²

Area of Triangle 2 = 30 cm²

Base of Triangle 3 = 12 cm, Height of Triangle 3 = 38 cm

Formula Used:

Area of a Triangle = (1/2) × base × height

Total Area of Pentagon = Area of Triangle 1 + Area of Triangle 2 + Area of Triangle 3

Calculations:

⇒ Area of first triangle = 24 cm²

⇒ Area of second triangle = 30 cm²

⇒ Area of third triangle = (1/2) × 12 × 38

⇒ Area of third triangle = 6 × 38 = 228 cm²

⇒ Total Area of Pentagon = 24 + 30 + 228

⇒ Total Area of Pentagon = 54 + 228 = 282 cm²

∴ The correct answer is 282 cm².

Additional Information

Polygon Triangulation

Any convex polygon with n sides can be divided into (n − 2) triangles by drawing all possible diagonals from a single vertex.

Additive Property of Area

The total area of a geometric figure is the sum of the areas of its non-overlapping constituent parts.

Sum of Interior Angles

The sum of the interior angles of a pentagon (5 sides) is (5 − 2) × 180° = 540°.

Shortcut Trick

To find the net percentage change for successive increases of a% and b%, use the formula: a + b + (a × b)/100

Here, a = 9% and b = 13%

Total percentage increase = 9 + 13 + (9 × 13)/100

Total percentage increase = 22 + 117/100 = 22 + 1.17 = 23.17%

∴ The correct answer is 23.17%.

Alternate Method

Given:

First increase = 9%

Second increase = 13%

Calculations:

⇒ Let the initial price of the article be 100.

⇒ Price after first increase (9%) = 100 + (9% of 100) = 109.

⇒ Price after second increase (13%) = 109 + (13% of 109).

⇒ Price after second increase = 109 + (13 × 109)/100 = 109 + 14.17 = 123.17.

⇒ Net increase in price = 123.17 − 100 = 23.17.

⇒ Percentage increase = (23.17 / 100) × 100 = 23.17%.

∴ The correct answer is 23.17%.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the effective change is [x + y + (xy/100)]%.

Successive Discounts

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

Multiplying Factor Method

Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100), where r₁ and r₂ are the percentage increase rates.

Shortcut Trick

The number N must divide (228 − 18) = 210.

According to the remainder rule, the divisor N must be greater than the remainder 18 (N > 18).

List factors of 210 in descending order: 210, 105, 70, 42, 35, 30, 21, 15...

The largest two-digit factor that satisfies N > 18 is 70.

∴ The correct answer is 70.

Alternate Method

Given: Dividend = 228, Remainder = 18.

Formula Used: Dividend = (Divisor × Quotient) + Remainder

⇒ Let the divisor be N and the quotient be q.

⇒ 228 = (N × q) + 18

⇒ N × q = 228 − 18 = 210

⇒ This implies N is a factor of 210.

⇒ Factors of 210 = 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.

⇒ Two-digit factors of 210 are 10, 14, 15, 21, 30, 35, 42, 70.

⇒ Since the remainder is 18, N must be greater than 18.

⇒ Possible values for N = 21, 30, 35, 42, 70.

⇒ The greatest two-digit value of N is 70.

∴ The correct answer is 70.

Additional Information

Euclid's Division Lemma

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

Divisor and Remainder Relationship

The divisor is always strictly greater than the remainder. If a number divides another with a remainder r, then the divisor must be a factor of (Dividend − r).

Finding Factors

To find all factors of a number, check divisibility by integers from 1 up to the square root of that number.

Shortcut Trick

Combine all terms of x and constants in the Angle Sum Property equation:

⇒ (3x + 6x + 4x) + (−7 − 14 + 6) = 180

⇒ 13x − 15 = 180 ⇒ 13x = 195 ⇒ x = 15

Required sum ∠A + ∠C = (3x − 7) + (4x + 6) = 7x − 1

⇒ 7(15) − 1 = 105 − 1 = 104°

∴ The correct answer is 104°.

Alternate Method

Given:

∠A = (3x − 7)°

∠B = 2∠A = 2(3x − 7) = (6x − 14)°

∠C = (4x + 6)°

According to the Angle Sum Property of a triangle:

⇒ ∠A + ∠B + ∠C = 180°

⇒ (3x − 7) + (6x − 14) + (4x + 6) = 180

⇒ 13x − 15 = 180

⇒ 13x = 180 + 15

⇒ 13x = 195

⇒ x = 195 ÷ 13

⇒ x = 15

Now, substitute the value of x to find the angles:

⇒ ∠A = 3(15) − 7 = 45 − 7 = 38°

⇒ ∠C = 4(15) + 6 = 60 + 6 = 66°

Sum of ∠A and ∠C:

⇒ ∠A + ∠C = 38° + 66° = 104°

∴ The correct answer is 104°.

Additional Information

Angle Sum Property

The sum of the interior angles of any triangle is always 180°. This is a fundamental property in Euclidean geometry.

Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. For instance, Exterior ∠ACD = ∠A + ∠B.

Types of Triangles by Angles

An Acute triangle has all angles < 90°, a Right triangle has one angle = 90°, and an Obtuse triangle has one angle > 90°.

Shortcut Trick

In both cases, the relative speed (S_F − S_S) remains the same.

Time to pass total distance (L_F + L_S) = 36 sec.

Time for passenger in faster train to pass length L_S = 12 sec.

⇒ (L_F + L_S) : L_S = 36 : 12 = 3 : 1

⇒ L_F : L_S = (3 − 1) : 1 = 2 : 1

∴ The correct answer is 2 : 1.

Alternate Method

Given:

Time taken to pass slower train completely = 36 sec.

Time taken by passenger to pass slower train = 12 sec.

Let length of faster train be L_F and slower train be L_S.

Let speeds be S_F and S_S. Relative speed V_R = S_F − S_S.

Formula: Distance = Relative Speed × Time

⇒ Case 1: L_F + L_S = (S_F − S_S) × 36 --- (1)

⇒ Case 2: L_S = (S_F − S_S) × 12 --- (2)

⇒ From (2), (S_F − S_S) = L_S / 12

⇒ Substitute in (1): L_F + L_S = (L_S / 12) × 36

⇒ L_F + L_S = 3 × L_S

⇒ L_F = 3L_S − L_S = 2L_S

⇒ L_F / L_S = 2 / 1

∴ The correct answer is 2 : 1.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is (S₁ − S₂). If they move in opposite directions, relative speed is (S₁ + S₂).

Train Crossing a Point Object

When a train passes a passenger, a pole, or a stationary man, the distance covered is exactly equal to the length of the train (or the length of the train being observed).

Unit Conversion

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

Shortcut Trick

Assume total work as LCM(12, 6) = 12 units.

Efficiency: A = 1 unit/day, B = 2 units/day.

Combined efficiency of A + B = 3 units/day.

Work done in 2 days = 3 × 2 = 6 units.

Remaining work = 12 − 6 = 6 units, completed by B in 6 ÷ 2 = 3 days.

Total time taken = 2 + 3 = 5 days.

∴ The correct answer is 5 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 6 days

Formula Used:

Efficiency = Total Work ÷ Time

⇒ Total Work = 12 units.

⇒ Work done by A and B in 2 days = (1 + 2) × 2 = 6 units.

⇒ Remaining work to be completed by B = 12 − 6 = 6 units.

⇒ Time taken by B to finish remaining work = 6 ÷ 2 = 3 days.

⇒ Total time for the whole work = Work together + Work by B alone = 2 + 3 = 5 days.

∴ The correct answer is 5 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work. If the ratio of time is x:y, the ratio of efficiency is y:x.

Work and Wages

Wages are generally distributed in the ratio of the work done by each individual, which is the product of their efficiency and the time they worked.

Combined Work Formula

If A can do a work in x days and B in y days, then together they can do it in (x × y) ÷ (x + y) days.

Shortcut Trick

Total amount to be divided after deductions = 5278 − (67 + 61 + 68) = ₹5,082.

Sum of ratio parts = 8 + 20 + 5 = 33 units.

Value of 1 unit = 5082 ÷ 33 = 154.

X's share after deduction = 8 units = 8 × 154 = ₹1,232.

∴ The correct answer is ₹1,232.

Alternate Method

Given:

Total amount = ₹5,278

Deductions from X, Y, and Z = ₹67, ₹61, and ₹68

Ratio after deduction = 8 : 20 : 5

Calculations:

⇒ Sum of deductions = 67 + 61 + 68 = ₹196

⇒ Amount remaining after deduction = 5278 − 196 = ₹5,082

⇒ Let the parts be 8k, 20k, and 5k

⇒ 8k + 20k + 5k = 5082

⇒ 33k = 5082

⇒ k = 5082 ÷ 33 = 154

⇒ X's share after deduction = 8k = 8 × 154 = 1232

∴ The correct answer is ₹1,232.

Additional Information

Ratio and Proportion

To divide a quantity S in the ratio a : b : c, the parts are [a/(a+b+c)] × S, [b/(a+b+c)] × S, and [c/(a+b+c)] × S respectively.

Deduction Rule

When fixed amounts are subtracted from the original shares to get a new ratio, always subtract the sum of those amounts from the total capital first.

Unitary Method in Ratios

Finding the value of '1 unit' by dividing the total quantity by the sum of ratio terms is the fastest way to calculate individual components.

Shortcut Trick

The third proportional (x) to two numbers a and b is calculated using the direct formula: x = b² ÷ a.

Given: a = 8 and b = 48.2.

Substituting the values: x = (48.2 × 48.2) ÷ 8

x = 2323.24 ÷ 8 = 290.405.

∴ The correct answer is 290.405.

Alternate Method

Given:

First number (a) = 8

Second number (b) = 48.2

Formula Used:

Three numbers a, b, and x 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 : x.

Calculations:

⇒ 8 ÷ 48.2 = 48.2 ÷ x ⇒ 8 × x = 48.2 × 48.2 ⇒ 8x = 2323.24 ⇒ x = 2323.24 ÷ 8 ⇒ x = 290.405

∴ The correct answer is 290.405.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b). It is the middle term when three numbers are in continued proportion.

Fourth Proportional

For four numbers a, b, c, d to be in proportion (a : b :: c : d), the fourth proportional is given by d = (b × c) ÷ a.

Continued Proportion

Numbers a, b, c, d... are said to be in continued proportion if a/b = b/c = c/d and so on.