Given:

Profit at one-third of the goods = 15%

Profit at 25% of the goods = 20%

Loss at rest of the goods = 10%

Profit in entire transaction = Rs.350

Concept used:

Profit = SP - CP

Profit % = Profit/CP × 100

Loss = CP - SP

Loss % = Loss/CP × 100

Where CP = Cost price and SP = Selling Price

Profit % and Loss % are always calculated on the Cost price

Calculations:

Let the total number of goods be 120 and the price of each good is 1 unit

One-third of the goods means = 1/3 × 120 = 40 

Selling price of 40 goods = 40 × 115/100 = 46 unit (∵ 15% profit)

25% of the goods means = 25/100 × 120 = 30

Selling price of 30 goods = 30 × 120/100 = 36 unit (∵ 20% profit)

Remaining goods = 120 - 40 - 30 = 50

Selling price of 50 goods = 50 × 90/100 = 45 unit (∵ 10% loss)

Total selling price = 46 + 36 + 45 = 127 units

Total cost price = 120 units 

Profit = 127 - 120 = 7 unit

According to the question, the total  profit is Rs.350

∴ 7 unit → Rs.350 ⇒ 1 unit → Rs.50

⇒ 120 → 50 × 120 = Rs.6000

∴ The total value of the goods is Rs.6,000.

Alternate Method

Let total goods be 120

One-third of the goods = 40

Profit on 40 goods = 40 × 15/100 = 6 (∵ 15% profit)

25% of the goods = 25/100 × 120 = 30

Profit on 30 goods = 30 × 20/100 = 6 (∵ 20% profit)

Remaining goods = 120 - 70 = 50

Loss on 50 goods = 50 × 10/100 = 5 (∵ 10% loss)

Total profit on all goods = 6 + 6 - 5 = 7

According to the question, 7 unit → Rs.350

⇒ 120 units → 350/7 × 120 = Rs.6000

∴ The total value of the goods is Rs.6,000

Given:

The sum of three number s 98

The ratio between the first and second be 2 : 3 and that between the second and third be 5 : 8

Concept:

LCM : The least common multiple or lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.

Calculation:

Let the numbers be x, y and z

⇒ x + y + z = 98

x : y = 2 : 3 and y : z = 5 : 8

As both ratios have y, write x : y : z as one ratio around y’s LCM

⇒ LCM(3, 5) = 15

x : y = 10 : 15 and y : z = 15 : 24

⇒ x : y : z = 10 : 15 : 24

Let x = 10a, y = 15a and z = 24a, where 'a' is a positive integer

⇒ 10a + 15a + 24a = 98

⇒ 49a = 98 

⇒ a = 2

⇒ x = 20, y = 30, z = 48

∴ The second number is 30

Calculation:

The HCF of 12 and 16 is 4 and the LCM is 48.

Therefore, the sum of HCF and LCM is 4 + 48 = 52.

As we know 23 × 31 = 713

⇒ 713/31 = 23

⇒ 0.0713 ÷ 3.1 = (713 × 10) / (31 × 10000)

⇒ 0.0713 ÷ 3.1 = 713/31000 = 0.023

Given:

The quantity of the mixture = 28 litres

Calculation:

When 2 litres of water is added

The new quantity of the water = 8 + 2 = 10 litres

The new ratio of the milk and water = 20 : 10 = 2 : 1

∴ The required result will be 2 : 1.

Given:

Principal = Rs 17500

Amount = Rs 29575

Rate = 30%

Formula used:

A = P(1 + R/100)ⁿ        (n = Time in years) 

Calculation:

29575 = 17500(1 + 30/100)ⁿ

⇒ 29575/17500 = (13/10)ⁿ

⇒ 169/100 = (13/10)ⁿ

⇒ (13/10)² = (13/10)ⁿ

On comparing we get: n = 2 years

Hence, the correct answer is "2".

Given:

Distance upstream = 33 km

Distance downstream = 21 km

Speed of motorboat = 25 km/h

Speed of stream = 3 km/h

Concept used:

Let the Speed of the motorboat = v

And Speed of stream = u

Time taken by boat upstream and downstream = t

Distance covered by boat downstream = (v + u)t

Distance covered by boat upstream = (v - u)t

Calculation:

let the time taken be t

⇒ v - u = 25 - 3 = 22 km/h

⇒ v + u = 25 + 3 = 28 km/h

According to the question,

⇒ t = 33/(v - u) + 21/(v + u)

⇒ 33/22 + 21/28

⇒ 3/2 + 3/4

⇒ 2.25 h

After converting 0.25 h in minutes,

⇒ t = 2 h 15 min

∴ Total time taken by boat is 2 h 15 min.

Given:

Increase % in the price of sugar = 25%

Formula used:

Expense = Price × Quantity

Calculation:

Let the initial expenses on sugar was Rs.100

Now, the price rises by 25%

⇒ New price = (100 + 100 × 25%) = 125

In order to keep the expense unaltered, Rs.25 has to be cut from Rs.125.

⇒ 125 - 25 = 100

∴ The decrease in the consumption = 25/125 × 100 = 20%

Alternate Method

Reduction % = (Old quantity - New quantity)/Old quantity ×100

Let the price of 1 kg sugar = Rs.20

Let the the quantity of sugar bought = 5 kg

⇒ Expense = Price × Consumption = 20 × 5 = Rs.100

After increase of 25% in the price = Rs.20 × 125/100 = Rs.25

To keep the expense unchanged, that is Rs.100

⇒ New consumption = Rs.100/25 = 4 kg

Reduce in the consumption = 5 kg - 4 kg = 1 kg

∴ Percentage reduction in consumption = 1/5 × 100 = 20%

Shortcut Trick

100 →  + 25% → 125 _____ - Y% → 100 

Now, Y = 25/125 × 100 = 20%

Shortcut Trick

Put x = 7, y = 1

x³ - 8y³ = 7³ - 8(1)³

⇒ 343 - 8 = 335

Alternate Method

Given:

x² + 4y² = 53 and x - 2y = 5

Formula:

(a - b)² = a² + b² - 2ab

a³ - b³ = (a - b)(a² + ab + b²)

Calculation:

⇒ (x - 2y)² = x² - 4xy + 4y²

⇒ 25 = 53 - 4xy

⇒ xy = 7

Then,

⇒ x³ - 8y³ = (x)³ - (2y)³

⇒ x³ - (2y)³ = (x - 2y)(x² + 2xy + 4y²)

= 5 × (53 + 2 × 7)

∴ x³ - 8y³ = 335