Formula Used:

a³ + b³ = (a + b)(a² + b² - ab)

Calculations:

Now, (0.04)³ + (0.21)³ = [(0.04) + (0.21)][(0.04)² + (0.21)² - (0.04 × 0.21)]

⇒ √[(0.04)² + (0.21)² - (0.04 × 0.21)]/[(0.04) + (0.21)][(0.04)² + (0.21)² - (0.04 × 0.21)]

⇒ √(1/[(0.04) + (0.21)]

⇒ √1/0.25 = √4

⇒ 2

∴ The value of given expression is 4.

Given:

Discount (d1) = 12%

Discount (d2) = 18%

Discount (d3) = 25%

Formula:

Where “d₁” and “d₂” are successive discounts offered.

Calculation:

Given:

The initial investment made by Mr. Shubham is RS 20,000.

Mr. Karan's initial investment is Rs 40,000 but after one year of business starting.

Concept:

The ratio of profit = Amount invested by one partner × time period of investment ⦂ Amount invested by other partner × time period of investment

Calculation:

⇒ The ratio of share of profit = Shubham : Karan

⇒ Amount invested by shubham × time period of investment ⦂ Amount invested by Karan × time period of investment

⇒ 20,000 × 2 : 40,000 × 1

⇒ 40,000 : 40,000 ⇒ 1 : 1

Karan’s share = (1/2) × 45,000

⇒ Rs 22,500

let the fourth proportional be x

⇒ 6 ∶ 24 ∶∶ 83 ∶ x

⇒ 6/24 = 83/x

⇒ x = 83 × (24/6) = 332

Total production of the year is 150000 tonnes

Woolen garments = 15%

So, 150000 × 15/100

⇒ 22500

∴ The production of woolen garments (in tonnes) in 2014 is 22500

Given:

Time taken to complete the whole work by A = 7 days

Time taken to complete the whole work by B = 14 days

Number of days A and B worked together = 4 days

Concept used:

If A and B can do a piece of work in 'x' days and 'y' days, respectively.

They start working together and after 't' days B leaves the work,

then the time taken to finish the whole work will be [(x/y) × (y - t)] days.

Calculation:

Here, x = 7 days, y = 14 days, t = 4 days

As we know that

Required time = [(x/y) × (y - t)] days

⇒ [(7/14) × (14 - 4)]

⇒ (1/2) × 10

⇒ 5 days

∴ Total work would be finished in 5 days.

Alternate Method

Calculation:

One day work of (A + B) = (1/7) + (1/14)

⇒ (3/14) units

Work done by (A + B) in 4 days = 4 × (3/14)

⇒ (6/7) units

Remaining Work = 1 - (6/7)

⇒ (1/7) units

Since B left after 4 days.

Remaining work done by A = (1/7) units

Time taken by A to do remaining work = [(1/7)/(1/7)] days

⇒ 1 day

So, Total days = (4 + 1) days

⇒ 5 days

∴ Total work would be finished in 5 days.

Given:

A and B together complete a job = 12 days

A gets for this job = Rs. 4980

B gets for this job = Rs. 3320

Calculation:

A and B price ratio = 4980 : 3320 = 3 : 2

⇒ A and B efficiency ratio = 3 : 2

⇒ Total work = 12 × (2 + 3) = 60 units

⇒ A one day work efficiency = 3 unit/day

⇒ A alone completed the work = 60/3 = 20 days

∴ A alone completed the job in 20 days.

Given:

The rate of interest (r) = 30% per annum

The time period (t) = 5 years

The interest in 5 years exceeds the amount lent by Rs.3200

Formula Used:

The formula for simple interest is I = PRT/100

Where:

I is the interest, P is the principal amount 

R is the rate of interest, T is the time period

Calculation:

From the given problem,

I = P + 3200

(P × 30 × 5)/100 = P + 3200

1.5P = P + 3200

0.5P = 3200

P = 3200/0.5 

⇒ P = 6400

Hence, the loan amount (Principal) is Rs. 6400.

Given:

A and B contain spirit and water in the ratio 3 : 8 and 6 : 5 respectively

Calculation:

∴ Required ratio = (1/11) : (2/11) = 1 : 2

Alternate Method

Let the total quantity in A be 11x and in B be 11y, which is to be mixed

According to the question:

(3x + 6y)/(8x + 5y) = 5/6

⇒ 18x + 36y = 40x + 25y

⇒ 22x = 11y

⇒ x/y = 1/2

∴ They should be mixed in the ratio 1 : 2

Concept used:

Follow BODMAS rule to solve this question, as per the order given below:

Step-1: Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step-2: Any mathematical 'Of' or 'Exponent' must be solved next,

Step-3: Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step-4: Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

Now,

Calculation:

Considering the given equation