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Shortcut Trick

Total Volume = Volume of Cylinder + Volume of Hemisphere.

Total Volume = πr²h + (2/3)πr³ = πr²(h + 2r/3)

Substituting r = 7 and h = 3: Volume = (22/7) × 49 × (3 + 14/3) = 154 × (23/3)

Using 23/3 ≈ 7.67 ⇒ 154 × 7.67 = 1181.18 cm³.

∴ The correct answer is 1181.18 cubic. cm.

Alternate Method

Given:

Radius of Hemisphere (r) = 7 cm

Radius of Cylindrical base (r) = 7 cm

Height of Cylindrical base (h) = 3 cm

Formula Used:

Volume of Cylinder = πr²h

Volume of Hemisphere = (2/3)πr³

Calculations:

⇒ Volume of Cylindrical base = (22/7) × 7 × 7 × 3

⇒ Volume of Cylindrical base = 22 × 7 × 3 = 154 × 3 = 462 cm³

⇒ Volume of Hemispherical top = (2/3) × (22/7) × 7 × 7 × 7

⇒ Volume of Hemispherical top = (2/3) × 22 × 49 = 2156 ÷ 3 ≈ 719.18 cm³

⇒ Total Volume of the toy = 462 + 719.18

⇒ Total Volume = 1181.18 cm³

∴ The correct answer is 1181.18 cubic. cm.

Additional Information

Total Surface Area of Toy

The surface area consists of the Curved Surface Area of the hemisphere (2πr²) and the Curved Surface Area of the cylinder (2πrh) plus the base area (πr²).

Volume of a Sphere

A hemisphere is exactly half of a sphere. Volume of sphere = (4/3)πr³.

Units Conversion

1 cubic cm (cm³) is equivalent to 1 millilitre (ml). 1000 cm³ = 1 Litre.

Given:

A sum of money on compound interest amount to Rs. 9680 in 2 years and Rs.10648 in 3 years.

Concept used:

In case of compound interest,

Rate of interest per annum = {Amount of (n + 1)^th year/Amount of (n)th year} × 100 - 100

Calculations:

Rate of interest per annum = {Amount of (n + 1)th year/Amount of (n)th year} × 100 - 100

⇒ (10648/9680) × 100 - 100

⇒ (11/10) × 100 - 100

⇒ 10

∴ The rate of interest per annum is 10%.

Given:

The income ratio of A and B is 5:7

Expenditure ratio is 1:1

A saves Rs. 2000

B saves Rs. 4000

Calculation:

Let's assume the common factor of their incomes is x:

A's income = 5x

B's income = 7x

Since the expenditure ratio is 1:1, and

A and B save Rs. 2000 and Rs. 4000 respectively:

A's expenditure = 5x - 2000

B's expenditure = 7x - 4000

Since the expenditures are equal:

5x - 2000 = 7x - 4000

Now, we'll solve for x:

7x - 5x = 4000 - 2000

2x = 2000

x = 2000 / 2 = 1000

Now, A's income:

A's income = 5 multiplied by 1000 = 5000

So, the income of A is Rs. 5000.

Given:

A person bought a shirt at 10% discount and sold it to his friend at a loss of 10%

His friend paid him Rs. 729 for the shirt

Concept used:

We are calculated discount on marked price and

profit or loss on cost price.

Calculation:

Let undiscounted price = 100x

Cost price(discounted price) = 90x

Selling price = 90x × (9/10) = 81x

according to the question,

Selling price = 81x → 729

⇒ x = 9

∴ The undiscounted price of the shirt is 100x is 900.

Shortcut Trick

Side (a) = 15 cm

Diagonal d₁ = 15 + 60% of 15 = 15 + 9 = 24 cm

Using the property 4a² = d₁² + d₂²

4 × 15² = 24² + d₂² ⇒ 900 − 576 = d₂² ⇒ d₂ = √324 = 18 cm

∴ The correct answer is 18 cm.

Alternate Method

Given: Side of rhombus = 15 cm; Diagonal₁ (d₁) = 160% of side.

Formula Used: In a rhombus, diagonals bisect each other at 90°. (Side)² = (d₁/2)² + (d₂/2)².

⇒ Diagonal₁ (d₁) = 15 × (160/100) = 24 cm

⇒ Half of d₁ = 24 ÷ 2 = 12 cm

⇒ Let half of d₂ be x. In the right-angled triangle formed by the intersection:

⇒ 15² = 12² + x²

⇒ 225 = 144 + x²

⇒ x² = 225 − 144 = 81

⇒ x = √81 = 9 cm

⇒ Full Diagonal₂ (d₂) = 2 × 9 = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Area of Rhombus

The area of a rhombus is calculated as Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals.

Perimeter of Rhombus

Since all sides of a rhombus are equal, the perimeter is P = 4 × side.

Diagonal Property

The sum of the squares of the diagonals is equal to four times the square of the side: d₁² + d₂² = 4a².

Given

A and B can do job in 5 days.

B and C can do job in 6 days.

C and A can do job in 7.5 days.

Formula used

Total work = Efficiency × Time taken

Calculation

Let total work be 30 units (LCM of 5, 6 and 7.5).

Now, efficiency of A + B = 30/5 = 6 unit/day      ---- (1)

And, efficiency of B + C = 30/6 = 5 units/day      ---- (2)

And, efficiency of A + C = 30/7.5 = 4 units/day      ---- (3)

Adding 1, 2 and 3, we get

2(A + B + C) = 15 units/day

⇒ A + B + C = 15/2 = 7.5 units/day

∴ Time taken by A, B and C to complete the work = 30/7.5 = 4 days

For a number to be divisible by 6, The last digit must be even and sum of the digits of the number must be divisible by 3

Sum of the digits of 35432 = 3 + 5 + 4 + 3 + 2 = 17 it is not divisible by 3

Sum of the digits of 53322 = 5 + 3 + 3+ 2 + 2 =15 it is divisible by 3 and its last digit is even

Sum of the digits of 93242 = 9 + 3 + 2 + 4 + 2 = 20 it is not divisible by 3

Sum of the digits of 85232 = 8 + 5 + 2 + 3 + 2 = 20 it is not divisible by 3

∴ 53322 is divisible by 6

Shortcut Trick

Difference in profit percentage = 60% − 30% = 30%

This 30% corresponds to the price increase of ₹1800.

Actual Selling Price (at 30% profit) = 130% of CP

⇒ Actual SP = (1800 ÷ 30) × 130 = 60 × 130 = 7800

∴ The correct answer is ₹7800.

Alternate Method

Given:

Initial Profit (P₁) = 30%

Final Profit (P₂) = 60%

Increase in SP = ₹1800

Formula Used:

Profit % = [(SP − CP) ÷ CP] × 100

SP = CP × (100 + Profit%) ÷ 100

CP (100%)

Actual SP (130%)

New SP (160%)

+30% Profit

+60% Profit

Diff = ₹1800 (30%)

Calculations:

Let the Cost Price (CP) of the article be 100 units.

⇒ Actual SP = 100 units + 30% of 100 units = 130 units

⇒ New SP = 100 units + 60% of 100 units = 160 units

⇒ Difference in SP = 160 units − 130 units = 30 units

⇒ According to the question: 30 units = 1800

⇒ 1 unit = 1800 ÷ 30 = 60

⇒ Actual SP = 130 units = 130 × 60

⇒ Actual SP = 7800

∴ The correct answer is ₹7800.

Additional Information

Marked Price and Discount

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100. Profit is always calculated on CP.

Successive Percentage Change

If there are two successive changes of a% and b%, the effective change is [a + b + (ab/100)]%.

Dishonest Dealer Concept

Profit% = [Error ÷ (True Value − Error)] × 100, where error is the weight difference.

Volume = a × a × a

Let the side of the cube be ‘a’.

Give, each side of a cube is increased by 10%.

New side = a + 10% of a

⇒ New side = 1.1a

New volume = 1.1a × 1.1a × 1.1a

⇒ New volume = 1.331a

% increase in volume \(= \frac{{1.331a - a}}{a} \times 100%\)

⇒ % increase in volume = 33.1%

30 persons can finish a job in 20 days,

⇒ Total work = Days × men = 20 × 30 = 600 units

Work done in 6 days by 30 men = 6 × 30 = 180 units

⇒ Remaining work = 420 units

As the work is completed in a total of 26 days,

6 days of work is already done 

Remaining days = 26 - 6 = 20 days 

⇒ Remaining 420 units work to be completed in 20 days,

⇒ No of men = 420/20 = 21 men

∴ 9 men should leave.