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Interest for 5 years (13 − 8) = ₹64000 − ₹49000 = ₹15000

Interest for 1 year = ₹15000 ÷ 5 = ₹3000

Interest for 8 years = 8 × ₹3000 = ₹24000

Sum invested = Amount in 8 years − Interest for 8 years = ₹49000 − ₹24000 = ₹25000

∴ The correct answer is ₹25000.

Alternate Method

Given:

Amount after 8 years (A₈) = ₹49000

Amount after 13 years (A₁₃) = ₹64000

Formula Used:

Simple Interest (SI) = (P × R × T) / 100

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

Calculations:

⇒ Interest for 5 years = A₁₃ − A₈

⇒ Interest for 5 years = 64000 − 49000 = 15000

⇒ Simple Interest for 1 year = 15000 ÷ 5 = 3000

⇒ Simple Interest for 8 years = 8 × 3000 = 24000

⇒ Sum invested (P) = A₈ − SI for 8 years

⇒ P = 49000 − 24000 = 25000

∴ The correct answer is ₹25000.

Additional Information

Simple Interest Concept

In simple interest, the interest remains the same for every year because it is always calculated on the original Principal (P).

Rate of Interest Formula

Rate (R) = (SI × 100) / (P × T). In this question, R = (3000 × 100) / (25000 × 1) = 12% per annum.

Compound Interest Difference

Unlike simple interest, Compound Interest (CI) increases every year as interest is calculated on the previous year's amount (Interest on Interest).

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The centroid (G) divides any median (AD) in the ratio 2 : 1 from the vertex.

Distance from vertex to centroid (AG) = 2 units.

Distance from centroid to midpoint (GD) = 1 unit.

Total length of the median (AD) = 2 + 1 = 3 units.

Required Ratio = Median (AD) : Vertex to Centroid (AG) = 3 : 2.

∴ The correct answer is 3 : 2.

Alternate Method

Given: In triangle ABC, D, E, and F are the midpoints of sides BC, CA, and AB. The point G is the centroid where medians AD, BE, and CF intersect.

Formula: In any triangle, the centroid divides the median in a ratio of 2 : 1, where the part towards the vertex is twice the part towards the midpoint of the side.

⇒ Let the length of AG (vertex to centroid) = 2x

⇒ Let the length of GD (centroid to midpoint) = 1x

⇒ Total length of the median AD = AG + GD

⇒ AD = 2x + 1x = 3x

⇒ Ratio = Median : Distance from vertex to centroid

⇒ Ratio = AD : AG = 3x : 2x

⇒ Ratio = 3 : 2

∴ The correct answer is 3 : 2.

Additional Information

Apollonius Theorem

It relates the lengths of sides and the median: AB² + AC² = 2(AD² + BD²), where AD is the median to side BC.

Equal Area Property

The three medians of a triangle divide it into six small triangles of equal area.

Coordinate of Centroid

If vertices are (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid G is ( (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3 ).

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The question explicitly states that the premium amount paid by F is to be shared in proportion to the original capital investments of D and E.

Ratio of D's investment to E's investment = 30,000 : 45,000.

Simplifying the ratio: 30,000 ÷ 15,000 : 45,000 ÷ 15,000 = 2 : 3.

∴ The correct answer is 2:3.

Alternate Method

Given:

Initial investment of D = ₹30,000

Initial investment of E = ₹45,000

Premium amount paid by F = ₹1,50,000

According to the problem, the premium is shared in the ratio of their initial investments.

⇒ Required Ratio = Investment of D : Investment of E

⇒ Required Ratio = 30,000 : 45,000

⇒ Required Ratio = (30,000 ÷ 15,000) : (45,000 ÷ 15,000)

⇒ Required Ratio = 2 : 3

∴ The correct answer is 2:3.

Additional Information

Capital Sharing Ratio

In a standard partnership, profits are shared in the ratio of (Capital × Time). If time is constant, profit sharing depends solely on the capital investment.

Premium for Goodwill

When a new partner joins, they often pay a premium. Normally, this is shared in the sacrificing ratio, but specific contract terms can dictate sharing by capital ratio.

Sacrificing Ratio

Sacrificing Ratio = Old Profit Sharing Ratio − New Profit Sharing Ratio. It represents the portion of profit share given up by old partners for the new partner.

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Ratio of bottles = 8 : 20 = 2 : 5

Since the ratio of oil stays constant, Oil ratio = 2 : 5

2 units = 120 ml ⇒ 1 unit = 60 ml

5 units = 5 × 60 ml = 300 ml

∴ The correct answer is 300 ml.

Alternate Method

Given:

Essential oil for 8 bottles = 120 ml

Target production = 20 bottles

Formula Used:

Total oil = (Oil per bottle) × (Number of bottles)

Calculations:

⇒ Essential oil needed for 1 bottle = 120 ÷ 8

⇒ Essential oil needed for 1 bottle = 15 ml

⇒ Essential oil needed for 20 bottles = 20 × 15 ml

⇒ Total essential oil = 300 ml

∴ The correct answer is 300 ml.

Additional Information

Direct Proportion

When two quantities increase or decrease together such that their ratio remains constant (x/y = k).

Inverse Proportion

When an increase in one quantity leads to a decrease in another such that their product remains constant (x × y = k).

Unitary Method

A technique where you first find the value of a single unit and then multiply by the required number of units.

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

Pairwise efficiencies: (A+B) = 4, (B+C) = 3, (A+C) = 2.

Combined efficiency 2(A+B+C) = 4 + 3 + 2 = 9 ⇒ A+B+C = 4.5.

Individual efficiencies: A = 4.5 − 3 = 1.5, B = 4.5 − 2 = 2.5, C = 4.5 − 4 = 0.5.

Efficiency Ratio A : B : C = 1.5 : 2.5 : 0.5 = 3 : 5 : 1.

∴ The correct answer is 3 : 5 : 1.

Alternate Method

Given:

Time taken by (Team A + Team B) = 6 days.

Time taken by (Team B + Team C) = 8 days.

Time taken by (Team A + Team C) = 12 days.

Efficiency Table:

Calculations:

⇒ Let efficiencies of Teams A, B, and C be a, b, and c respectively.

⇒ a + b = 4 — (i)

⇒ b + c = 3 — (ii)

⇒ a + c = 2 — (iii)

⇒ Adding (i), (ii), and (iii): 2(a + b + c) = 4 + 3 + 2 = 9

⇒ a + b + c = 4.5 — (iv)

⇒ a = (a + b + c) − (b + c) = 4.5 − 3 = 1.5

⇒ b = (a + b + c) − (a + c) = 4.5 − 2 = 2.5

⇒ c = (a + b + c) − (a + b) = 4.5 − 4 = 0.5

⇒ Work rate ratio A : B : C = 1.5 : 2.5 : 0.5

⇒ Multiply by 2 to get whole numbers: (1.5 × 2) : (2.5 × 2) : (0.5 × 2) = 3 : 5 : 1

∴ The correct answer is 3 : 5 : 1.

Additional Information

Efficiency and Time Relationship

The work rate (efficiency) of a person is inversely proportional to the time taken to complete the work. Efficiency = Total Work ÷ Time.

Combined Work Formula

If three pairs (A+B), (B+C), and (C+A) finish work in x, y, and z days respectively, then A+B+C together finish work in [2xyz ÷ (xy + yz + zx)] days.

Work and Wages

Wages are generally distributed in the ratio of work done, which is equivalent to the ratio of their efficiencies if they all work for the same duration.

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Slope (m) is the ratio of the vertical change to the horizontal change between two points.

Point 1: (2, 3) | Point 2 (x-intercept): (5, 0)

m = (y₂ − y₁) ÷ (x₂ − x₁)

m = (0 − 3) ÷ (5 − 2) = −3 ÷ 3 = −1

∴ The correct answer is -1.

Alternate Method

Given:

The line passes through point P₁ (2, 3).

The line intersects the x-axis at x = 5, which gives the point P₂ (5, 0).

Formula Used:

Slope (m) = (y₂ − y₁) ÷ (x₂ − x₁)

Calculations:

⇒ Let (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 0).

⇒ m = (0 − 3) ÷ (5 − 2)

⇒ m = −3 ÷ 3

⇒ m = −1

∴ The correct answer is -1.

Additional Information

Slope-Intercept Form

The equation of a line is y = mx + c, where m is the slope and c is the y-intercept.

Parallel and Perpendicular Lines

Two lines are parallel if their slopes are equal (m₁ = m₂) and perpendicular if the product of their slopes is −1 (m₁ × m₂ = −1).

X and Y Intercepts

The x-intercept is the point where y = 0, and the y-intercept is the point where x = 0.

Shortcut Trick

Break 18 into factors where one is the largest possible perfect square.

√18 = √(9 × 2)

Since √9 = 3, the expression simplifies to 3√2.

Comparing 3√2 with a√2, we directly see that a = 3.

∴ The correct answer is 3.

Alternate Method

Given: √18 = a√2

To find the value of a, we can square both sides of the equation to eliminate the radical sign.

⇒ (√18)² = (a√2)²

⇒ 18 = a² × (√2)²

⇒ 18 = a² × 2

⇒ a² = 18 ÷ 2

⇒ a² = 9

⇒ a = √9

⇒ a = 3

∴ The correct answer is 3.

Additional Information

Simplifying Radicals

A radical expression is in simplest form when the radicand (number inside) has no perfect square factors other than 1. Rule: √(x² × y) = x√y.

Squaring Radicals

The square of a square root of a non-negative number is the number itself: (√x)² = x.

Product Rule for Square Roots

The square root of a product is equal to the product of the square roots: √(a × b) = √a × √b.

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Use the Complementary Angle identity: sin θ = cos(90° − θ).

Given sin 75°, compare it with cos θ.

⇒ cos θ = sin 75° = cos(90° − 75°)

⇒ θ = 15°

∴ The correct answer is 15°.

Alternate Method

Given:

The trigonometric expression is cos θ = sin 75°.

Formula Used:

In a right-angled triangle, for any acute angle A, the Co-function Identity states:

sin A = cos(90° − A)

cos A = sin(90° − A)

Calculations:

⇒ Let the unknown angle be θ.

⇒ According to the question: cos θ = sin 75°

⇒ We know that sin A = cos(90° − A)

⇒ Substituting A = 75° in the identity:

⇒ sin 75° = cos(90° − 75°)

⇒ sin 75° = cos 15°

⇒ Comparing cos θ = cos 15°

⇒ θ = 15°

∴ The correct answer is 15°.

Additional Information

Complementary Angle Identities

These identities relate trigonometric functions of complementary angles: tan θ = cot(90° − θ) and sec θ = cosec(90° − θ).

Sum of Angles in a Triangle

In any triangle, the sum of all internal angles is 180°. In a right-angled triangle, the two non-right angles must sum to 90°.

Specific Trigonometric Values

The value of sin 75° (or cos 15°) can be calculated using the addition formula sin(45° + 30°), resulting in (√6 + √2) ÷ 4.

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The Area of a regular octagon is given by the direct formula: Area ≈ 4.828 × s².

Here, the side length (s) is 10 cm.

Area ≈ 4.828 × (10)²

Area ≈ 4.828 × 100 = 482.8 cm².

∴ The correct answer is 482.8 cm².

Alternate Method

Given:

Side of the regular octagon (s) = 10 cm

Formula Used:

Area of a regular octagon = 2 × (1 + √2) × s²

Calculations:

⇒ Area = 2 × (1 + √2) × (10)²

⇒ Area = 2 × (1 + 1.4142) × 100

⇒ Area = 2 × 2.4142 × 100

⇒ Area = 4.8284 × 100

⇒ Area = 482.84 cm²

∴ The correct answer is 482.8 cm².

Additional Information

Interior and Exterior Angles

For a regular octagon, each interior angle is 135° and each exterior angle is 45°.

Perimeter Formula

The perimeter of a regular octagon is calculated as P = 8 × s, where s is the side length.

Sum of Interior Angles

The sum of all interior angles of an octagon is (8 − 2) × 180° = 1080°.

Shortcut Trick

For any regular polygon, the relationship between Area, Perimeter, and Apothem is:

Area = (1/2) × Perimeter × Apothem

Perimeter = 5 × 10 = 50 cm.

Apothem = (2 × 172) ÷ 50 = 344 ÷ 50 = 6.88 cm.

Rounding to the nearest option, we get 6.8 cm.

∴ The correct answer is 6.8 cm.

Alternate Method

Given:

Side length (s) = 10 cm

Area of pentagon (A) = 172 cm²

Formula Used:

Area of a regular polygon = (Perimeter × Apothem) ÷ 2

Calculations:

⇒ Perimeter (P) = number of sides × side length

⇒ P = 5 × 10 = 50 cm

⇒ Area = (P × a) ÷ 2

⇒ 172 = (50 × a) ÷ 2

⇒ 172 = 25 × a

⇒ a = 172 ÷ 25

⇒ a = 6.88 cm

Rounding to one decimal place as per the given options, the value is 6.8 cm.

∴ The correct answer is 6.8 cm.

Additional Information

General Area Formula

The area of a regular polygon with n sides of length s is A = [n × s²] ÷ [4 × tan(180°/n)].

Apothem and Side Relation

The apothem (a) is related to the side length (s) by the formula: a = s ÷ [2 × tan(180°/n)].

Inradius Property

The apothem of a regular polygon is exactly equal to the radius of the inscribed circle (inradius) of that polygon.