Given:

HCF = 8

LCM = 40

Formula:

The product of the HCF and LCM of two numbers is equal to the product of the two numbers.

Solution:

Product of the two numbers = HCF × LCM = 8 × 40 = 320

Therefore, the product of the numbers whose HCF and LCM are 8 and 40 respectively is 320.

Given

Akhil's work time: 26 days

Akhil & Sakshi's work time: 20 days

Sakshi's work time: 6 days

Concept:

Work and Time calculation.

Calculation:

⇒ Akhil's work rate = 1/26 work/day

⇒ Akhil & Sakshi's work rate = 1/20 work/day

⇒ Work done by Sakshi and Akhil in 6 days =  1/20 ×  6 = 3/10 of the total work

⇒ Remaining work = 1 - work done by Sakshi = 1 - 3/10 = 7/10 of the total work

⇒ Time taken by Akhil to complete remaining work = Remaining work/Akhil's work rate = (7/10)/(1/26) = 91/5 days

Hence, Akhil will complete the remaining work in 91/5 days.

Given

Cost of goods = ₹1,280

Loss on selling one-third of goods = 15%

Required overall gain = 15%

Formula Used:

Gain = SP - CP

Calculation:

⇒ Cost of 1/3 of goods = ₹1,280/3 = ₹426.67

⇒ Loss on this part = 15% of ₹426.67 = ₹64

⇒ So, remaining cost = ₹1,280 - ₹426.67 = ₹853.33

⇒ Overall gain required = 15% of ₹1,280 = ₹192

⇒ Therefore, the gain required on the remaining part

₹192 (overall required gain) + ₹64 (loss on first part) = ₹256

⇒ Gain percentage on the remaining part

(₹256/₹853.33) x 100% = 30 percent

Therefore, the remaining goods should be sold at a gain of 30 percent

Given

Rate of interest = 10% per annum

Time = 2 years

Difference = Rs. 464

Concept:

For 2 years,

D = P × (r/100)²,

where P is the principal amount and r is the rate of interest.

Calculation:

P = Difference/((Rate of interest / 100)²)

⇒ P = 464/((10/100)²) = Rs. 46,400

Therefore, the sum invested is Rs. 46,400.

Given data:

Initial  624 litres

Initial ratio of milk to water: 5 ∶ 3

Desired ratio of milk to water: 5 ∶ 4

Concept:

To keep the amount of milk constant, calculate the amount of water that needs to be added to achieve the desired ratio.

Calculation:

⇒ Volume of milk = 5 / (5 + 3) x 624 litres = 390 litres

⇒ Volume of water in new mixture = 4 / 5 x 390 litres = 312 litres

⇒ Volume of water to be added = (3 / (5 + 3) x 624 litres)  - 312 = 78 litres

Therefore, 78 litres of water needs to be added to achieve the desired ratio.

Given:

Side of cube: 5 cm

Concept Used:

Total surface area of the resulting cuboid is 2lw + 2lh + 2wh, where l is length, w is width, and h is height.

Solution:

⇒ Here, length l = 5 x 3 = 15 cm (since three cubes are joined end to end), width w = 5 cm, height h = 5 cm

⇒ 2lw + 2lh + 2wh = 2(15 x 5) + 2(15 x 5) + 2(5 x 5)

⇒ 150 + 150 + 50 = 350 cm²

Therefore, the total surface area of the resulting cuboid is 350 cm².

Given:

The cost price of the item is Rs.950.

The selling price of the item is Rs.1,150.

Concept:

The percentage profit is calculated by the formula:

((Selling price - Cost price)/Cost price) × 100%.

Solution:

⇒ The percentage profit = ((1,150 - 950)/950) × 100% = 21%.

Therefore, her percentage profit is 21%

Given:

A can paint 7 chairs in an hour.

B can paint 4 chairs in an hour.

Formula:

Total work = (A's rate)(B's rate)

Time = Total work/Combined rate

Solution:

The total work is 55 chairs. The combined rate is 7 + 4 = 11 chairs per hour. Therefore, the time required is:

Time = 55/11 = 5 hours

Therefore, A and B will require 5 hours to paint 55 chairs when they work together.

Given values:

Average salary of all workers = Rs. 3,000

Number of higher officers = 12

Average salary of higher officers = Rs. 12,000

Average salary of remaining workers = Rs. 2,500

Concept:

Total salary = Average salary × total number of individuals.

Also, total salary = salary of group A + salary of group B.

Calculation:

⇒ Total salary of higher officers = number of higher officers × average salary of higher officers = 12 × 12,000

⇒ Let x be the total number of workers. Therefore, total salary of all workers = x × 3,000

⇒ Total salary of remaining workers = total salary of all workers - total salary of higher officers = x × 3,000 - 12 × 12,000

⇒ As total salary of remaining workers = (x - 12) × 2,500

⇒ (x - 12) × 2,500 = x × 3,000 - 12 × 12,000

⇒ 2500x - 30000 = x × 3,000 - 144,000

⇒ 500x = 114000

⇒ x = 228

Hence, the total number of workers is 228.

Given data:

Selling price of each pen: Rs. 12

Gain on first pen: 20%

Loss on second pen: 20%

Concept:

Cost Price = 100 × SP/(100 ± Profit%/Loss%)

Loss = CP - SP

Calculation:

SP of each pen = Rs. 12

Gain on first pen: 20%

⇒ CP₁ = 100 × 12/(100 + 20)

⇒ CP1 = Rs. 10

Loss on second pen: 20%

⇒ CP2 = 100 × 12/(100 - 20)

⇒ CP2 = Rs. 15

⇒ Total CP = 10 + 15 = Rs. 25

⇒ Total SP = 12 + 12 = Rs.24

Loss = CP - SP 

⇒ 25 - 24 = Rs. 1

Therefore, On the whole, he lost Rs. 1.