The perimeter of rectangle = 2(l + b), where l and b are length and breadth respectively.

Let the ratio of l ∶ b = 5x ∶ 2x

Perimeter of rectangle = 2(5x + 2x) 

⇒ 14x = 70

⇒ x = 5

Hence, length = 5x = 25 cm and breadth= 2x = 10 cm

Area of rectangle = l × b = 25 × 10 = 250 cm²

Given:

The given number = 957x5y6z3

Concept used:

Divisibility rule of 3: The sum of all digits of a number is divisible by 3 then the number is divisible by 3

Divisibility rule of 11: The provided integer is divisible if the difference between the sum of the digits at odd places and the sum of the digits at even places is 0 or divisible by 11

Calculation:

957x5y6z3 is divisible by 33

⇒ 957x5y6z3 is divisible by 3 and 11

⇒ (9 + 5 + 7 + x + 5 + y + 6 + z + 3) is divisible by 3

⇒ (35 + x + y + z) is divisible by 3 .................... (1)

Now check which option will be satisfied.

If x + y + z = 24 then (35 + x + y + z) = 35 + 24 = 59 → Not divisible by 3

If x + y + z = 27 then (35 + x + y + z) = 35 + 27 = 62 → Not divisible by 3

If x + y + z = 25 then (35 + x + y + z) = 35 + 25 = 60 → Divisible by 3

If x + y + z = 26 then (35 + x + y + z) = 35 + 26 = 61 → Not divisible by 3

So, only one option is being satisfied.

∴ The maximum value of (x + y + z) is 25

Alternate Method

According to the divisibility rule of 11

⇒ 9 + 7 + 5 + 6 +3 = 5 + x + y + z

x + y + z = 25

Shortcut Trick

Traditional Method:

Given:

M : N = 8 : 15

N : O = 5 : 9

O : P = 18 : 19

Calculation:

N : O = 5 : 9 

We multiply the above ratio by 2, we get

N : O = 10 : 18

And we have O : P = 18 :19

So, N : O : P = 10 : 18 : 19 

M : N = 8 : 15

We multiply this by 2

⇒ M : N = 16 : 30

And N : O : P = 10 : 18 : 19 

We multiply this by 3

⇒ N : O : P = 30 : 54 : 57

Now, we get

M : N : O : P = 16 : 30 : 54 : 57 

∴ The equivalent proportional of M : N : O : P is 16: 30 : 54 : 57

Given:

Number of days taken by 100 men to make 60 chairs working 2 hours a day = 4 days

Chairs to be made in 10 days working 4 hours a day = 120

Formula used:

(M1 × D1 × H1)/W1 = (M2 × D2 × H2)/W2

Where, M1, M2 = Number of men D1, D2 = Number of days, H1, H2 = Number of hours, W1, W2 = Work Done

We will use only the terms that are given in the question

Calculation:

According to the question,

M1 = 100, D₁ = 4 days, H₁ = 2 hours and W₁ = 60 chairs

D2 = 10 days, H₂ = 4 hours and W₂ = 120 chairs

(M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂         

⇒ (100 × 4 × 2)/60 = (M₂ × 10 × 4)/120

⇒ 100 × 8 × 120 = M₂ × 40 × 60

⇒ M₂ = 40

∴ 40 men are required to make 120 chairs in 10 days working 4 hours a day.

Shortcut Trick

Efficiency ratio = 3 : 1

Total efficiency = 4

Work done = 4 × 36 = 144 min

Time taken by slower pipe = 144/1 = 144 min

∴ Time taken to fill the tank by the slower pipe is 144 min.

Traditional method:

Let the time taken by pipe A to fill the tank be x min

⇒ Time taken by pipe B to fill the tank will be x/3 min

⇒ Part of tank filled by pipe A in 1 min = 1/x

⇒ Part of tank filled by pipe B in 1 min = 3/x

⇒ Part of tank filled by pipe A and B together in 1 min = (1/x + 3/x) = 4/x       ----(1)

Also, given that A and B together can fill the tank in 36 min

So, part of tank filled by A and B together in 1 min = 1/36      ----(2)

So, from equation 1 and 2

⇒ 4/x = 1/36

⇒ x = 144 min

∴ Time taken to fill the tank by the slower pipe is 144 min.

Given:

The average weight of 20 boys was marked as 89.4 kg.

Later it was found that instead of 87 kg, a value has been marked as 78 kg.

Concept used:

Total = Mean (Average) × number of entities

Calculation:

Total weight of 20 boys = 20 × 89.4 = 1788 kg

Correct total weight = 1788 - 78 + 87 = 1797 kg

Hence, the correct average weight = 1797 ÷ 20 = 89.85 kg

∴ The correct average weight is 89.85 kg.

Given:

Diameter of sphere = 6 cm

Radius of sphere = 3 cm

Length of wire, H = 36m = 3600 cm

Formula:

Volume of cylinder = πr²H

Calculation:

Let radius of wire be R cm

According to question,

Volume of sphere = Volume of Cylinder

Given

AB = 6 cm, BC = 7 cm and AC = 8 cm

Concept

Internal angle bisector theorem

The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

In ∆ABC, AE bisects ∠A.

(BE/EC) = (AB/AC)

BE/(7 - BE) = 6/8

8 BE = 42 - 6BE

14 BE = 42

BE = 42/14

BE = 3 cm

∴ The length of BE is 3 cm.

Given:

Principal = Rs. 12,000

Rate% he borrowed = 2%

Rate% he lends = 17/4%

Time = 2 years

Formula used:

S.I = (P × R × T)/100

Where S.I = Simple Interest, P = Principal, R = Rate%, T = Time

Calculation:

According to the question:

Interest he pays;

S.I = (12,000 × 2 × 2)/100

⇒ S.I = 480

Interest he gets;

S.I = (12,000 × 17/4 × 2)/100 = 1020

Now,

Difference in the transaction = 1020 – 480 = 540

Gain in 1 year = 540/2 = 270

∴ His gain in the transaction per year is Rs. 270

Shortcut Trick

Difference in rate = 17/4% - 2% = 9/4%

Gain = 9/4% of 12000

⇒ (9/4 × 12000)/100 = 270

∴ His gain in the transaction per year is Rs. 270

Given 

A merchant purchases 15 articles @ Rs. 140 each

A merchant purchases other 10 articles @ Rs. 120 each

Selling Price of each article = Rs. 143 

Formula used 

Profit percent = (Selling price - Cost Price)/Cost Price × 100

Calculation 

The total Cost Price = 140 × 15 + 120 × 10 = Rs. 3300 

The total Selling Price = 143 × 25 = Rs. 3575 

Profit percentage = (275/3300) × 100 = 8.33% 

∴ The required answer is 8.33%