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Given:

Total amount which is divided among A, B and C = Rs. 6,300

X : Y = 7 : 5 and Y : Z = 4 : 3

Calculation:

X : Y = 7 : 5     ----(1)

Y : Z = 4 : 3     ----(2)

Multiply by 4 in equation (1) and multiply by 5 in equation (2)

X : Y : Z = 28 : 20 : 15

Ratio of X : Y : Z = 28x : 20x : 15x

According to the question

28x + 20x + 15x = 6,300

⇒ 63x = 6,300

⇒ x = 6,300/63

⇒ x = 100

∴ Share of Y = 20 × 100 = Rs. 2,000

Given:

Profit percentage = 20%

Concept:

If there is a profit of x%, the ratio of selling price to cost price is (100 + x) ∶ 100.

Solution:

The ratio of selling price to cost price = (100 + 20) ∶ 100 = 120 ∶ 100 = 6 ∶ 5

Therefore, the ratio of the selling price to the cost price is $6 \over 5$.

Shortcut Trick

Rate of Interest (R) = 40% ÷ 4 = 10% per annum.

For Compound Interest at 10% for 3 years, the effective rate is 33.1%.

CI = 14000 × 33.1% = 140 × 33.1 = 4634.

∴ The correct answer is Rs 4634.

Alternate Method

Given: Increase in sum = 40%, Time (T₁) = 4 years, Principal (P₂) = Rs 14000, Time (T₂) = 3 years.

Formula Used: SI = (P × R × T) ÷ 100; CI = P[(1 + R/100)^T − 1]

⇒ Let Principal (P₁) = 100

⇒ SI = 40% of 100 = 40

⇒ 40 = (100 × R × 4) ÷ 100

⇒ R = 40 ÷ 4 = 10%

⇒ For CI: P = 14000, R = 10%, T = 3

⇒ CI = 14000 × [(1 + 10/100)³ − 1]

⇒ CI = 14000 × [(11/10)³ − 1]

⇒ CI = 14000 × [1331/1000 − 1]

⇒ CI = 14000 × [331/1000]

⇒ CI = 14 × 331 = 4634

∴ The correct answer is Rs 4634.

Additional Information

Effective Rate for CI

For 10% rate, effective CI rates are: 2 years = 21%, 3 years = 33.1%, 4 years = 46.41%.

SI vs CI Difference

Difference for 3 years = P(R/100)² × (300 + R)/100.

Shortcut Trick

Let the weights of Ram, Shyam, and Mohan be 7x, 8x, and 5x respectively.

Total weight = 7x + 8x + 5x = 20x.

Since the average weight is 80 kg, total weight = 80 × 3 = 240 kg.

20x = 240 ⇒ x = 12 kg.

Average weight of Ram and Shyam = (7x + 8x) ÷ 2 = 7.5x = 7.5 × 12 = 90 kg.

∴ The correct answer is 90 kg.

Alternate Method

Given:

Ratio of weights of Ram, Shyam, and Mohan = 7 : 8 : 5

Average weight of all three persons = 80 kg

Calculations:

⇒ Total weight of Ram, Shyam, and Mohan = Average weight × Number of persons

⇒ Total weight = 80 × 3 = 240 kg

⇒ Sum of ratio parts = 7 + 8 + 5 = 20 parts

⇒ Value of 1 part = 240 ÷ 20 = 12 kg

⇒ Weight of Ram = 7 × 12 = 84 kg

⇒ Weight of Shyam = 8 × 12 = 96 kg

⇒ Total weight of Ram and Shyam = 84 + 96 = 180 kg

⇒ Average weight of Ram and Shyam = 180 ÷ 2 = 90 kg

∴ The correct answer is 90 kg.

Additional Information

Averages

The average provides the central or mean value of a set of quantities. Formula: Average = Sum of all observations ÷ Total number of observations.

Ratio and Proportion

When a total sum is distributed in a given ratio (a : b : c), the value of any specific share is calculated as: Share = (Ratio part ÷ Sum of all parts) × Total Value.

Weighted Average

Used when combining different groups of different sizes and individual averages into a single overall average. Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Calculation:

According to the question,

Total scored marks by Ayush in all subjects = 88 + 80 + 90 + 82 + 85 = 425

Correct Answer: 4) 425

Given:

Speed without stoppage = 60 km/h

Speed with stoppage = 40 km/h

Formula used:

Speed = Distance/Time

Calculation:

Distance travel by man with the speed of 60 km/h in one hour,

Distance = speed × Time

⇒ Distance = 60 × 1 = 60 km

Distance travel by man with the speed of 40 km/h in one hour,

Distance = speed × Time

⇒ Distance = 40 × 1 = 40 km

Now, the time is taken to cover the extra 20 km with the speed of 60 km/h

Time = Distance/Speed

⇒ Time = 20/60

⇒ Time = 1/3 hour = (1/3) × 60

⇒ Time = 20 minutes

∴ A man stops for 20 minutes per hour.

Stoppage in min/hr = {(F - S)/F ×  60},

Where,

F is without stoppage,

S is with stippage,

Stoppage in min/hr,

⇒ (60 - 40)/60 × 60

⇒ 20/60 × 60 = 20

∴ A man stops for 20 minutes per hour.

Given:

Boys = 100

Girls = 80

Total boys pass = 48%

Total girls pass = 30%

Concept Used:

Failed percentage = (Total failed students/Total students) × 100

Calculation:

Total number of students = 100 + 80

⇒ 180

According to the question, we have

The number of boys who passed the examination = 100 × (48/100)

⇒ 48

The number of boys who failed the examination = 100 - 48

⇒ 52

The number of girls who passed the examination = 80 × (30/100)

⇒ 24

The number of girls who failed the examination = 80 - 24

⇒ 56

The total number of students who failed the examination = 52 + 56

⇒ 108

Now,

The percentage of students who failed the examination = (108/180) × 100

⇒ 60%

∴ The percentage of students who failed the examination is 60%.

Shortcut Trick 

Boys who fail = 100 × (52/100) = 52

Girls who fail =80 × (70/100) = 56

The percentage of the total who failed = [(52 + 56)/(100 + 80)] × 100

⇒ (108/180) × 100

⇒ 60%

∴ The percentage of students who failed the examination is 60%.

Calculation:

Number of boys in B = 30.

Number of girls in E = 55

According to the question:

⇒ 30/55 × 100 = 54.54 ≈ 55 %

∴ Option 2 is the correct answer.

Shortcut Trick

Let Income = 100, then Savings = 8 and Expenditure = 92.

New Income = 116 (16% increase); Savings remains constant = 8.

New Expenditure = 116 − 8 = 108.

Expenditure increase = 108 − 92 = 16.

% Increase = (16 ÷ 92) × 100 ≈ 17.39%.

∴ The correct answer is 17.3%.

Alternate Method

Given:

Initial Savings = 8% of Income

Income increase = 16%

Savings change = 0% (remains the same)

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let initial Income = 100x

⇒ Initial Savings = 8x

⇒ Initial Expenditure = 100x − 8x = 92x

⇒ New Income = 100x + 16% of 100x = 116x

⇒ New Savings = 8x (given as same)

⇒ New Expenditure = 116x − 8x = 108x

⇒ Increase in Expenditure = 108x − 92x = 16x

⇒ Percentage increase = (16x ÷ 92x) × 100

⇒ Percentage increase = (4 ÷ 23) × 100 ≈ 17.39%

∴ The correct answer is 17.3%.

Additional Information

Income Expenditure Relation

The fundamental relation is Income = Expenditure + Savings. Any change in one component must be balanced by the others.

Constant Savings Rule

If savings remain constant, the absolute increase in income is exactly equal to the absolute increase in expenditure (ΔI = ΔE).

Percentage Change Formula

Percentage change = (Final Value − Initial Value) ÷ Initial Value × 100. Always use the original expenditure as the base for calculating its percentage increase.

The pattern follows here is:

The differences are in the increasing order of consecutive prime numbers.

Hence, the correct answer is "99".