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Given data:

Principal (P) = ₹500

Time (T) = 2 years

Difference between compound interest and simple interest (CI - SI) = ₹1.25

Concept Used:

The difference between compound interest and simple interest for two years is

given by the formula CI - SI = P × (r/100)²,

where r is the rate of interest.

Calculation:

⇒ Rate of interest (r) = √((CI - SI) × 100² / P)

⇒ Rate of interest (r) = √((1.25 × 100²)/500) = 5%

Hence, the rate of interest per annum is 5%.

Given:

S.I. for 5 years = 20% of Principal

Formula used:

S.I. = (P × R × T)/100

where P → Principal

R → Rate of interest

T → Time period

Calculation:

Let the principle be Rs. P

According to the question,

⇒ (P × 5 × rate)/100 = 20% of P

⇒ rate = 4%

Now, let in t years the simple interest will be equal to the principal

⇒ Simple interest = (P × 4 × t)/100

⇒ P = (P × t)/25

∴ t = 25 years

Shortcut TrickRatio of Principal to S.I. in 5 years = 100 : 20

Let the Principal be 100, so we are required to make S.I. = 100

⇒ S.I. of 20 = 5  years

⇒ S.I. of 100 = 5/20 × 100 = 25 years

Shortcut Trick

Total Cost = Length × Width × Rate per m²

Cost = 5.5 × 3.75 × 800

Multiply 3.75 by 800 first: 3.75 × 800 = 3000

Cost = 5.5 × 3000 = 16500

∴ The correct answer is Rs. 16500.

Alternate Method

Given:

Length of the room (L) = 5.5 m

Width of the room (W) = 3.75 m

Rate of paving = Rs. 800 per m²

Formula Used:

Area of a rectangle = Length × Width

Total Cost = Area × Rate per unit area

Calculations:

⇒ Area of the floor = 5.5 m × 3.75 m

⇒ Area = 20.625 m²

⇒ Total Cost = 20.625 × 800

⇒ Total Cost = 16500

∴ The correct answer is Rs. 16500.

Additional Information

Area of Rectangle

The total space enclosed within the boundaries of a rectangle. Formula: Area = Length × Breadth.

Perimeter of Rectangle

The total length of the boundary of a rectangle. Formula: Perimeter = 2 × (Length + Breadth). This is used for finding the cost of fencing or placing borders.

Area of Square

A square is a special type of rectangle where all four sides are equal. Formula: Area = Side².

Given :

Principal(P) = Rs.1250

Amount after two yrs.(A) = Rs.1800

Time = 2 yrs.

Formula used :

\(A = P\bigg(1+\dfrac{R}{100}\bigg)^2\)

Calculation :

Let the rate of interest be R % .

\(1800 = 1250\bigg(1+\dfrac{R}{100}\bigg)^2\) , \(\dfrac{1800}{1250} = \bigg(1+\dfrac{R}{100} \bigg)^2\) or \(\dfrac{6}{5} = \bigg(1+\dfrac{R}{100}\bigg)\) 

So, R = 20%

∴ The answer is 20 % .

Shortcut Trick ⇒ Ratio of Principle and Amount = 1250/1800 = 25/36

⇒ In 2 yrs Principle has changed from, 25x to 36x

To find Rate of change per year we've to take square root on both the sides as it has change in 2 yrs.

⇒ Principle → Amount

⇒ √{25} → √{36} 

⇒ 5 → 6

∴ % Rate of change per year = {(6 - 5)/5} × 100 = 20%

Shortcut Trick

If the area of a new circle is equal to the sum of the areas of two circles with radii r₁ and r₂, the radius R of the new circle is simply given by the Pythagorean relation: R = √(r₁² + r₂²).

Given, r₁ = 15 cm and r₂ = 8 cm.

R = √(15² + 8²)

R = √(225 + 64) = √289

R = 17 cm.

∴ The correct answer is 17 cm.

Alternate Method

Given:

Radius of the first circle (r₁) = 15 cm

Radius of the second circle (r₂) = 8 cm

Formula Used:

Area of a circle = πr²

Calculations:

Let the radius of the larger circle be R.

⇒ Area of new circle = Area of first circle + Area of second circle

⇒ πR² = π(r₁)² + π(r₂)²

⇒ πR² = π × (15² + 8²)

⇒ R² = 225 + 64

⇒ R² = 289

⇒ R = √289

⇒ R = 17 cm

∴ The correct answer is 17 cm.

Additional Information

Pythagorean Triplets

Three positive integers a, b, and c that satisfy a² + b² = c² form a Pythagorean triplet. Common examples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Spotting these saves calculation time.

Circumference Sum Relation

If the circumference of a circle equals the sum of the circumferences of two circles with radii r₁ and r₂, then the radius R of the new circle follows a direct linear addition: R = r₁ + r₂.

Area of Semicircle and Quadrant

Area of a Semicircle = ½ πr² and its Perimeter = πr + 2r. Area of a Quadrant = ¼ πr² and its Perimeter = πr/2 + 2r.

Shortcut Trick

Ratio of Volume (V) to Curved Surface Area (CSA) = r / 2

r / 2 = 1336.5 / 594 = 2.25

⇒ Radius (r) = 4.5 cm

Height (h) = CSA ÷ (2 × π × r)

h = 594 ÷ (2 × 22/7 × 4.5) = 21 cm

∴ The correct answer is 21 cm.

Alternate Method

Given: Curved Surface Area (CSA) = 594 cm², Volume (V) = 1336.5 cm³

Formula Used: CSA = 2πrh; Volume = πr²h

r

h CSA = 2πrh = 594 Vol = πr²h = 1336.5

⇒ Volume / CSA = (πr²h) / (2πrh) = 1336.5 / 594

⇒ r / 2 = 2.25

⇒ r = 4.5 cm

⇒ 2 × (22 / 7) × 4.5 × h = 594

⇒ (198 / 7) × h = 594

⇒ h = (594 × 7) / 198

⇒ h = 3 × 7 = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Total Surface Area (TSA) of Cylinder

The total surface area includes the curved surface and the areas of the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Hollow Cylinder

If R is the outer radius and r is the inner radius, the volume is V = πh(R² − r²).

Diagonal of a Cylinder

The longest segment within a cylinder connects opposite points on the top and bottom circles, calculated as d = √(h² + (2r)²).

Let the radius, height and slant height of cone be ‘r’, ‘h’ and ‘l’ units respectively

Given, curved surface area = 90% of Total surface area of cone

⇒ πrl = (9/10) × πr(l + r)

⇒ 10l = 9l + 9r

⇒ l = 9r

Also, l² = h² + r²

⇒ 81r² = h² + r²

⇒ h² = 80r²

⇒ r² = h²/80

Now,

Volume of cone = (1/3)πr²h

⇒ Volume of cone = π/3 × h²/80 × h

⇒ Volume of cone = (π/240) × h³

∴ The volume of cone is (π/240) times the cube of the height of the cone

Given 

Area of square = 72 cm²

Formula Used 

Area of square = side²

Area of circle = πR²

Calculation 

Area of square = side²

⇒ side = √72 = 6√2 cm 

Diagonal of square = side√2

⇒ 6√2 × √2 = 12 cm 

Radius of circle = diagonal/2

⇒ 12/2 = 6 cm 

Area of circle = π(6)² 

⇒ 36π cm²

∴ The area of circle is 36π cm²

Short Trick:

U = upstream and D = downstream

3.6(U) + 5.4(D) = 54 and 5.4(U) + 3.6(D) = 58.5

Divide by 18

2U + 3D = 30 and 3U + 2D = 32.5

Now, Multiply by 3 in (1) and by 2 in (2), and then subtract (2) from (1), then

5D = 25 min

∴10D = 50 min

Detailed Method:

Let the speed of boat be x km/hr and speed of current be y km/hr

Upstream speed = (x – y) km/hr

Downstream speed = (x + y) km/hr

According to the question

3.6/(x – y) + 5.4/(x + y) = 54/60

5.4/(x – y) + 3.6/(x + y) = 58.5/60

Let 1/(x – y) = a and 1/(x + y) = b, then

3.6a + 5.4b = 54/60   ---(1)

5.4a + 3.6b = 58.5/60   ---(2)

Add equation (1) and equation (2)

9 (a + b) = 112.5/60

⇒a + b = 12.5/60

⇒a + b = 5/24   ---(3)

Subtract equation (2) from equation (1)

1.8 (b – a) = -4.5/60

⇒b – a = -2.5/60

⇒b – a = -1/24   ---(4)

Add equation (3) and equation (4)

2b = 4/24

⇒b = 1/12

⇒(x + y) = 12 km/hr

Downstream speed = 12 km/hr

∴Time taken to cover 10 km distance with downstream = 10/12 hr or (10/12) × 60 = 50 min

Shortcut Trick

Using the M₁D₁/W₁ = M₂D₂/W₂ chain rule.

Total work (W₁) = 1 unit.

Work part (W₂) = (M₂ × D₂) ÷ (M₁ × D₁)

W₂ = (207 × 20) ÷ (138 × 50) = 4140 ÷ 6900 = 3/5

∴ The correct answer is 3/5.

Alternate Method

Given:

Initial Men (M₁) = 138, Days (D₁) = 50, Work (W₁) = 1

Final Men (M₂) = 207, Days (D₂) = 20, Work (W₂) = ?

Formula Used:

M₁ × D₁ × W₂ = M₂ × D₂ × W₁

Calculations:

⇒ 138 × 50 × W₂ = 207 × 20 × 1

⇒ 6900 × W₂ = 4140

⇒ W₂ = 4140 ÷ 6900

⇒ W₂ = 414 ÷ 690

⇒ W₂ = (69 × 6) ÷ (69 × 10)

⇒ W₂ = 6 ÷ 10 = 3/5

∴ The correct answer is 3/5.

Additional Information

The MDH Formula

Total work is proportional to Men × Days × Hours. The general formula is (M₁D₁H₁)/W₁ = (M₂D₂H₂)/W₂.

Efficiency and Work

If efficiencies (E) are different, the formula becomes M₁D₁E₁/W₁ = M₂D₂E₂/W₂.

Unitary Method

1 man 1 day work = 1 / (M₁ × D₁). Therefore, work by M₂ in D₂ = (M₂ × D₂) / (M₁ × D₁).