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Total population increase = 8680 − 8000 = 680.

If all 8000 were males, increase = 8% of 8000 = 640.

Excess increase = 680 − 640 = 40.

This 40 represents the extra 4% (12% − 8%) of the females.

4% of females = 40 ⇒ Total females = (40 ÷ 4) × 100 = 1000.

∴ The correct answer is 1000.

Alternate Method

Given:

Initial population = 8000

Final population = 8680

Male growth rate = 8%

Female growth rate = 12%

Formula Used:

Total Increase = (Growth % of Males × Males) + (Growth % of Females × Females)

Calculations:

⇒ Total increase = 8680 − 8000 = 680

⇒ Overall increase percentage = (680 ÷ 8000) × 100 = 8.5%

⇒ Using alligation method:

⇒ Ratio of Males to Females = (12 − 8.5) : (8.5 − 8)

⇒ Ratio = 3.5 : 0.5 = 7 : 1

⇒ Total ratio parts = 7 + 1 = 8 units

⇒ 8 units = 8000

⇒ 1 unit = 1000

⇒ Number of females = 1 unit = 1000

∴ The correct answer is 1000.

Additional Information

Rule of Alligation

It is a shortcut to find the ratio in which two components are mixed to get a mean value: Ratio = (Higher Value − Mean) : (Mean − Lower Value).

Percentage Change

The percentage change is calculated as: (Final Value − Initial Value) ÷ Initial Value × 100.

Weighted Average

The net change in a population can be seen as a weighted average of the changes in its subgroups (males and females).

Shortcut Trick

Identify the total extra income earned through overtime: 6000 × 3 = ₹18,000.

Since the question asks for the correct monthly income (average), spread this extra income over all 12 months.

Extra increase per month = 18000 ÷ 12 = ₹1,500.

New Average = Old Average + Monthly Increase = 62000 + 1500 = ₹63,500.

∴ The correct answer is ₹63500.

Alternate Method

Given:

Average Monthly Salary = ₹62,000

Overtime earnings = ₹6,000 per month (for 3 months)

Formula Used:

Average = (Total Sum of Observations) ÷ (Total Number of Observations)

Calculations:

⇒ Total Base Salary for the year = 62000 × 12 = ₹7,44,000

⇒ Total Overtime Income = 6000 × 3 = ₹18,000

⇒ Total Annual Income = 7,44,000 + 18,000 = ₹7,62,000

⇒ Correct Monthly Income = 7,62,000 ÷ 12 = ₹63,500

∴ The correct answer is ₹63500.

Additional Information

Average Concept

The arithmetic mean or average is calculated as Σx ÷ n, where Σx is the sum of all values and n is the count.

Net Change in Average

If a total sum increases by 'k', the average of 'n' items increases by k ÷ n. This deviation method saves time in exams.

Weighted Average

When different groups have different averages, the combined average is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Given:

The total income = 1080

The expenditure on clothes = 50° 

Concept:

A circle or pie chart has a total degree = 360° 

Calculation:

Let the expenditure on clothes be X

According to the question,

⇒ X = \({{50\over 360}\ \times 1080}\ =\ 150\)

∴ The required result will be Rs. 150.

CALCULATION:

Average exports = (500 + 750 + 1000 + 1200 + 1100 + 900 + 1300)/7

∴ Average Exports from years Y1 to Y7 = 6750/7 = 964.28

Face value = 480 Rs

Discount = 25% 

So, Selling price = 360 Rs

Loss = 40 Rs

Cost Price = 400 Rs

Loss percent = 40/400 * 100 = 10%

Shortcut Trick

Difference in bridge lengths = 1300 − 1000 = 300 m.

Difference in time taken = 19 − 15 = 4 seconds.

Speed of train = 300 ÷ 4 = 75 m/s.

Total distance in 15 seconds = 75 × 15 = 1125 m.

Length of train = Total distance − Bridge length = 1125 − 1000 = 125 m.

∴ The correct answer is 125 metres.

Alternate Method

Given: Bridge₁ = 1000 m, T₁ = 15 s; Bridge₂ = 1300 m, T₂ = 19 s.

Formula Used: Speed = (Length of Train + Length of Bridge) ÷ Time

⇒ Let length of Train be L.

⇒ Speed = (L + 1000) ÷ 15 = (L + 1300) ÷ 19

⇒ 19 × (L + 1000) = 15 × (L + 1300)

⇒ 19L + 19000 = 15L + 19500

⇒ 19L − 15L = 19500 − 19000

⇒ 4L = 500

⇒ L = 125 m

∴ The correct answer is 125 metres.

Additional Information

Train crossing a stationary object with length

When a train crosses a bridge, platform, or tunnel, the total distance covered is the sum of the length of the train and the length of the object.

Train crossing a point object

When a train crosses a pole, man, or tree, the distance covered is equal to the length of the train itself (L = Speed × Time).

Unit Conversion

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

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Let Income = 100 units ⇒ Expenditure = 80, Savings = 20.

Change in Income = +12% of 100 = +12.

Change in Savings = −10% of 20 = −2.

Change in Expenditure = Income Change − Savings Change = 12 − (−2) = 14.

% Increase in Expenditure = (14 ÷ 80) × 100 = 17.5%.

∴ The correct answer is 17.5%.

Alternate Method

Given:

Expenditure = 80% of Income

Income increase = 12%

Savings decrease = 10%

Formula Used:

Income = Expenditure + Savings

Percentage Increase = (Change ÷ Original Value) × 100

⇒ Let original Income = ₹100.

⇒ Original Expenditure = 80% of 100 = ₹80.

⇒ Original Savings = 100 − 80 = ₹20.

⇒ New Income = 100 + 12% of 100 = ₹112.

⇒ New Savings = 20 − 10% of 20 = 20 − 2 = ₹18.

⇒ New Expenditure = New Income − New Savings = 112 − 18 = ₹94.

⇒ Increase in Expenditure = 94 − 80 = ₹14.

⇒ Percentage increase in Expenditure = (14 ÷ 80) × 100

⇒ Percentage increase = (7 ÷ 40) × 100 = 17.5%.

∴ The correct answer is 17.5%.

Additional Information

Income Expenditure Identity

Income is always the sum of Expenditure and Savings. Change in Income (ΔI) = Change in Expenditure (ΔE) + Change in Savings (ΔS).

Percentage Change Formula

% Change = [(Final Value − Initial Value) ÷ Initial Value] × 100. Always use the original value as the base for calculations.

Alligation Method

These problems can be solved using alligation where the ratio of Expenditure to Savings acts as weights for their respective percentage changes to find the net income change.

Given:

Investment of A = Rs. 13,750

Investment of B = Rs. 16,250

Investment of C = Rs. 18,750

B's share in profit = Rs. 5200

Calculation:

Ratio of investment 

A ∶ B ∶ C = 13,750 ∶ 16,250 ∶ 18,750

⇒ A ∶ B ∶ C = 11 ∶ 13 ∶ 15

Ratio of their profit will be same as ratio of their investment

Let profit of A be 11x, B be 13x and C be 15x

As, B's share = 5200

⇒ 13x = 5200

⇒ x = 5200/13 = 400

Total profit = 11x + 13x + 15x = 39x

⇒ Total profit = Rs. (39 × 400)

⇒ Rs. 15,600

∴ Total profit is Rs. 15,600

Given:

Average weight of 8 persons = 40 kg

Average weight of 7 persons = 38 kg

Concept:

Average weight = (Total weight/Number of persons)

Solution:

Total weight of 8 persons = Average weight × Number of persons

⇒ 40 kg × 8 = 320 kg

Total weight of 7 persons = Average weight × Number of persons

⇒ 38 kg × 7 = 266 kg

Weight of the removed person = Total weight of 8 persons - Total weight of 7 persons

⇒ 320 kg - 266 kg = 54 kg

Therefore, the weight of the removed person is 54 kg.

According to the English alphabets & their positional values:

The pattern followed is:

     First alphabet is increased by 1 at odd steps & by 2 at even steps.

     Second alphabet is increased by the difference of 2 at odd steps & decrease by the difference of 2 at even steps.

Hence the correct answer is option 3.

Alternate Method 

Hence the correct answer is option 3.