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Given:

By increasing 10 students, total expenditure increased by Rs.80 

By increasing 10 students, average decreased by Rs.5

Formula Used:

Average = Total expenditure/Number of students

Calculation:

Let the initial average be x

The total expenditure of 50 students = 50 × x = 50x

The total expenditure of 60 students = 50x + 80

The new average = x - 5

According to the question;

⇒ (50x + 80)/60 = (x - 5)

⇒ 50x + 80 = 60x - 300

⇒ 380 = 10x

⇒ x = 38

The total expenditure of 50 students = 50 × 38 = Rs.1900

∴ The total expenditure of 50 students is Rs.1900

Given:

The cost price of 1 kg rice and 10 kg of sugar is Rs.260

After increasing, new cost price is  Rs.352

Calculation:

Let the price of 1 kg rice be x

Let the price of 1 kg of sugar be y

⇒ x + 10y = 260  

⇒ x = 260 - 10y

The increased price of 1 kg of rice = (100 + 20)/100 × x = 1.20x

The increased price of 1 kg of sugar = (100 + 40)/100 = 1.40y

⇒ 1.20x + 10(1.4y) = 352 

⇒ 1.2 (260 - 10y) + 14y = 352

⇒ 312 - 12y + 14y = 352

⇒ 2y = 40

⇒ y = 20

The value of 1 kg of rice = 260 - 10 × 20 = Rs.60

∴ The price of 1kg of rice is Rs.60

Given:

a - b = 3

b - c = 12

a + b + c = 240

Calculation:

⇒ a = 3 + b

⇒ c = b - 12

⇒ a + b + c = 240

⇒ 3 + b + b + b - 12 = 240

⇒ 3b = 249

⇒ b = 83

⇒ a = 86

Then, a + b = 83 + 86 = 169

∴ The value of a + b is 169

Calculation:

⇒ 13/16 = 0.8125

⇒ 11/16 = 0.6875

⇒ 13/21 = 0.6190

⇒ 11/19 = 0.5789

∴ The greatest number is 13/16

Given:

Average marks of total 200 students = 40

Average marks of passed students = 45

Average marks of failed students = 20

Formula Used:

Average = Total marks/Number of students

Calculation:

The sum of marks of 200 students = 40 × 200 = 8000

Let the number of passed students be x

The number of failed students = 200 - x

According to the question;

⇒ 45 × x + (200 - x) × 20 = 8000

⇒ 45x + 4000 - 20x = 8000

⇒ 25x = 4000

⇒ x = 160

∴ The number of passed students is 160

Formula Used:

Circumference of a circle = 2πr

Here, r = radius

Calculation:

According to the question;

⇒ 2r = 2πr - 180

⇒ 2r(π - 1) = 180

⇒ 2r(22/7 - 1) = 180

⇒ 2r × 15/7 = 180

⇒ r = 42 cm

∴ The radius of the circle is 42 cm

Given:

Person X travel with speed of 5 km/h, then he late for the destination by 45 minutes.

Person X travel with speed of 6 km/h, then he reaches the destination before by 25 minutes.

Formula Used:

Distance = Speed × Time

Calculation:

Let the distance be D

Let the original Speed be S.

When person X travel with speed of 5 km/h,

⇒ D/5 - D/S = 45/60

⇒ D(1/5 - 1/S) = 3/4

⇒ D = 3 × 5S/4(S - 5)

When person X travel with speed of 6 km/h

⇒ D/S - D/6 = 25/60

⇒ D(1/S - 1/6) = 5/12

⇒ [3 × 5S/4(S - 5)] × [(6 - S)/6S] = 5/12

⇒ (6 - S)/2(S - 5) = 1/3

⇒ 18 - 3S = 2S - 10

⇒ 5S = 28

⇒ S = 28/5

The distance = 3 × 5 × (28/5) × 4(28/5 - 5) = 35 km

∴  The distance between his destination from his starting point  is 35 km 

Shortcut Trick 

By using the formula, S₁ × S₂ × (total change in time)/(S₁ - S₂)

⇒ (5 × 6 × 70)/[60 × (6 - 5)]  = 35 km

∴  The distance between his destination from his starting point  is 35 km

Given:

Divisor = 8 × Quotient

Divisor = 5  × Remainder

Remainder = 24

Formula Used:

Dividend = Divisor × Quotient + Remainder

Calculation:

Divisor =  5 × 24 = 120

Quotient = 120/8 = 15

Dividend = 120 × 15 + 24 = 1824

∴ The dividend is 1824

Calculation:

Let the fraction be x/y.

The numerator = x × (100 + 30)/100

The denominator = y × (100 - 20)/100

The new fraction = x × (100 + 30)/y × (100 - 20)

The new fraction = 13x/8y

The difference  = 13x/8y -  x/y = 5x/8y

The required percentage = 5x/8y/x/y  × 100 = 62.5%

∴ The required percentage is 62.5%

Given:

Area of ΔABC = 156 cm2

BD = BC/2

Calculation:

Area of ΔABC = √3/4 × (BC)²

Area of ΔABC = √3/4 × (2BD)2

Area of ΔABC = 4 × √3/4 × (BD)2

Area of ΔABC = 4 × Area of ΔBDE

Area of ΔBDE = 1/4 × Area of ΔABC  = 1/4 × 156 = 39 cm2

∴ Area of ΔBDE is  39 cm2