Formula:

After meeting each other two trains of speed x km/hr and y km/hr respectively,

complete his journey in t1 and t2, then

(x/y)² = t₂/t₁

Calculation:

Let t₁ = 2 hrs and t₂ = 8 hrs

Let speed of two trains be x km/hr and y km/hr respectively,

According to the question

(x/y)² = (8/2)

⇒ (x/y)² = 4

⇒ x/y = √4 = 2

∴ x : y = 2 : 1

Given:

a : b = 2 : 3 and b : c = 2 : 3

Formula Used:

If a : b = x : y and b : c = y : z

Then, a : b : c = x : y : z

Calculation:

a : b = 2 : 3     ---(1)

b : c = 2 : 3     ---(2)

⇒ a : b : c = 4 : 6 : 9

Now, (3a² + b² + c²) : (a² + 2b² + c²)

⇒ (3 × 4² + 6² + 9²) : (4² + 2 × 6² + 9²)

⇒ (3 × 16 + 36 + 81) : (16 + 2 × 36 + 81)

⇒ (48 + 36 + 81) : (16 + 72 + 81)

⇒ 165 : 169

Given:

A can do work in days = 15

The efficiency of B = Efficiency of A - 10%

The efficiency of C = Efficiency of A + 20%

Work done by A, B and C = 3 days

Formula Used:

Total work = Efficiency × Days

Calculation:

Let's assume that the efficiency of A is 100 units per day

So, 

Efficiency of B = 100 -10% of 100 = 90 units per day

Efficiency of C = 100 + 20% of 100 = 120 units per day

⇒ Work done by A + B + C in a day = 100 + 90 + 120 = 310 units

⇒ Total work = Work done by A in a day × Days

⇒ Total work = 100 × 15 = 1500 units

⇒ Work done by A+ B + C = 310 × 3 = 930 units

⇒ Remaining work = 1500 - 930 = 570 units

⇒ 570 units of work done by D in days = 19/2

⇒ 1500 units of work will be done by D in days

∴ D will complete work in 25 days.

GIVEN:

Riya  runs 3/2 times as fast as Prerna.

Riya  gives a lead of 100m.

FORMULA USED:

Speed = Distance/Time.

CALCULATION:

 Riya  runs 3/2 times as fast as Prerna.

Riya = 3/2  Prerna.

Riya/Prerna = 3/2.

So, the ratio  of speeds of Riya and Prerna =  3 : 2 

Let the speed be 3x and 2x

Let the distance covered by Prerna be 'L' 

Then , distance covered by Riya  = 'L' + 100

Since, they both will meet , hence time is same.

⇒ L /2x =( L+100) /3x 

⇒ 3L = 2L  + 200 

⇒ L = 200 m.

Riya will cover the distance = ( 200 +100) = 300m.

Hence, Riya has to run 300m before meet.

Shortcut Trick

Calculation:

If time is constant, then

Ratio of speed = Ratio of distance

Ratio of speed = 3 : 2

Ratio of distance = 3 : 2

So,

1 unit = 100 m

3 units = 300 m

∴ The correct answer is 300 m.

Given:

SP of the article = Rs.3753

When SP = Rs. 1432, earned a profit = 11% more than the loss

Formula Used:

Profit or Loss = SP ∼ CP

Calculation:

Let the cost price of the article be x Rs.

When sold at Rs.3753, profit = 3753 - x

When sold at Rs.1432, loss = x - 1432

According to the question

⇒ 3753 - x = (x - 1432) + 11% of (x - 1432)

⇒ 3753 - x = (x - 1432) + 0.11(x - 1432)

⇒ 3753 - x = 1.11(x - 1432)

⇒ 3753 - x = 1.11x - 1589.52

⇒ 3753 + 1589.52 = 1.11x + x

⇒ 5342.52 = 2.11x

⇒ x = 5342.52 / 2.11

⇒ x = 2532

∴ The cost price of the article is Rs.2532.

Shortcut Trick

GIVEN:

15 boys earn Rs900 in 5 days.

FORMULA USED:

 M1 × D1 × W2 =  M2 × D2 × W1

Where M1, M2 = no. of persons respectively

D1, D2 = No. of days respectively

W1, W2 = Wages earn respectively

CALCULATIONS:

Here , we have

M1 = 15 . D1 = 5 , W1 = 900 

M2 = 20 , D2 = 7 , W2 = ?

⇒ M1 × D1 × W2 = M2 × D2 × W1 

⇒ 15 × 5 × W2 =  20 × 7 × 900 

⇒ W2 = Rs1680.

Hence, 20 boys will earn Rs1,680 in 7 days.

Given:

The ratio of the cost price of an item and the profit gained while selling the item is 4 : 1.

Calculation:

According to the question,

If the cost price is 4 then the profit is 1 on the cost price

So, profit % = (1/4) × 100

⇒ 25%

∴ The profit percentage is 25%.

Given:

Selling price of 12 articles = Cost price of 9 articles

Formula Used:

Loss = Cost price - Selling price

Loss% = (Loss/Cost price) × 100

Calculation:

Let assume that cost price of 1 article = x

So, cost price of 9 articles = 9x

Selling price of 1 article = y

So, selling price of 12 articles = 12y

According to question

⇒ 9x = 12y

⇒ x/y = 12/9

⇒ x/y = 4/3

Loss = Cost price - Selling price

⇒ Loss = 4unit - 3 unit = 1 unit

Loss% = (Loss/Cost price) × 100

⇒ Loss% = 1/4 × 100

⇒ Loss% = 25%

∴ Loss% is 25%

The correct option is 1 i.e. 25%

Given:

Volume = v cm³

Height = h cm

Concept: The formula for the volume of a cone is V = 1/3πr² h

Solution:

⇒ v = 1/3πr² * h