Want to ace Percentage Questions with Solutions without getting stuck on long formulas? You are in the right place!
Percentage is one of the most important topics in quantitative aptitude. If you master this single concept, topics like Profit & Loss, Interest, and Data Interpretation become ten times easier.
In this guide, you will find carefully chosen aptitude questions along with clear, step-by-step solutions and fast mental-math shortcuts. Whether you are a beginner brushing up on the basics or preparing for an upcoming competitive exam or job interview, these questions will help you solve problems quickly and with zero confusion.
Let’s dive in and boost your math skills!
A percentage represents a fraction of 100. To convert any fraction \frac{a}{b} into a percentage, multiply the fraction by 100%:
\text{Percentage} = \left(\frac{a}{b} \times 100\right)%
Standard Calculation:
\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%
The phrase “x\% of Y” means multiplying the quantity Y by the fractional value of x\%, where x\% = \frac{x}{100}:
\text{Value} = \left(\frac{x}{100}\right) \times Y
Standard Calculation:
\text{Value} = \frac{15}{100} \times 240 = \frac{3}{20} \times 240 = 3 \times 12 = 36
Shortcut / Mental Math Technique (Split Method):
Break 15% into two easy benchmarks: 10% + 5%
10% of 240 = 24
5% of 240 = 24 / 2 = 12
Total: 15% of 240 = 24 + 12 = 36
When a percentage part of an unknown number is known:
Part = (Percentage / 100) × Unknow number
Therefore, to find the original number:
Unknown number = (Part × 100) / Percentage
Standard Calculation:
Let the unknown number be N.
(35 / 100) × N = 28
N = (28 × 100) / 35
Divide both numerator (28) and denominator (35) by their common factor 7:
N = (4 × 100) / 5
N = 400 / 5 = 80
Shortcut / Unitary Method:
35% of the number = 28
Divide by 7 on both sides:
5% of the number = 28 / 7 = 4
To get 100%, multiply 5% by 20:
100% of the number = 4 × 20 = 80
Percentage Increase = (Absolute Increase / Original Value) * 100%
where Absolute Increase = Final Value – Original Value
Calculation Steps:
1. Absolute Increase = 540 – 450 = 90 units
2. Percentage Increase = (90 / 450) * 100%
3. Simplifying the fraction: 90 / 450 = 1 / 5
4. (1 / 5) * 100% = 20%
Shortcut / Ratio Method:
Write the ratio of Original to Final value:
450 : 540 = 5 : 6
Increase = 6 – 5 = 1 unit on the original base of 5 units.
Percentage Increase = 1 / 5 = 20%
Percentage Decrease = (Absolute Decrease / Original Value) * 100%
where Absolute Decrease = Original Value – Reduced Value
Calculation Steps:
1. Absolute Decrease = 2500 – 2150 = Rs. 350
2. Percentage Decrease = (350 / 2500) * 100%
3. Simplify: 350 / 25 = 14%
Shortcut Technique:
1% of original price (Rs. 2,500) = Rs. 25
To find what percent Rs. 350 represents:
350 / 25 = (350 * 4) / 100 = 1400 / 100 = 14%
Verification:
Original Price – 14% of Original Price
= 2500 – (0.14 * 2500)
= 2500 – 350 = Rs. 2,150.
Passing Marks = Marks Obtained + Deficit (Marks failed by)
Maximum Marks = (Passing Marks / Passing Percentage) * 100
Calculation Steps:
1. Passing Marks required = 178 + 22 = 200 marks
2. Since 40% corresponds to the passing score:
40% of Maximum Marks = 200
3. Maximum Marks = (200 / 40) * 100 = 5 * 100 = 500 marks
Shortcut / Unitary Method:
40% = 200 marks
Divide both sides by 4:
10% = 50 marks
Multiply both sides by 10 to get 100%:
100% = 50 * 10 = 500 marks
Verification:
40% of 500 = (40 / 100) * 500 = 200 marks.
Marks obtained = 200 – 22 = 178 marks.
When comparing two quantities where one is greater by R%, the reverse percentage comparison changes because the reference base (denominator) shifts:
Percentage Less = [R / (100 + R)] * 100%
Calculation Steps:
1. Let B’s income = Rs. 100
2. A’s income is 25% more than B:
A’s income = 100 + 25 = Rs. 125
3. Difference between A and B = 125 – 100 = Rs. 25
4. Percentage Less = (Difference / A’s income) * 100%
Percentage Less = (25 / 125) * 100% = (1 / 5) * 100% = 20%
Expenditure = Price * Consumption.
If Expenditure is constant, Price and Consumption are inversely proportional.
When price decreases by R%:
Percentage Increase in Consumption = [R / (100 – R)] * 100%
Calculation Steps:
1. Let original price = Rs. 100/kg and original consumption = 100 kg.
Original Expenditure = 100 * 100 = Rs. 10,000.
2. Reduced price = 100 – 20 = Rs. 80/kg.
3. New Consumption = Expenditure / New Price = 10,000 / 80 = 125 kg.
4. Increase in Consumption = 125 – 100 = 25 kg.
5. Percentage Increase = (25 / 100) * 100% = 25%.
Shortcut / Fraction Benchmark Method:
* Price decrease of 20% = -1/5
* To keep product constant, consumption must increase by: 1 / (5 – 1) = +1/4
* 1/4 = 25%
When a quantity undergoes successive percentage changes of a% and b%, the net percentage change is given by:
Net Change% = a + b + (a * b) / 100
(Increase is positive, decrease is negative).
Calculation Steps:
1. Here, a = +20% and b = -10%.
2. Net Change% = 20 + (-10) + [20 * (-10)] / 100
3. Net Change% = 10 – (200 / 100) = 10 – 2 = +8%
Since the result is positive, it represents an 8% increase.
Alternative Base 100 Method:
1. Let initial salary = Rs. 100
2. After 20% increase: 100 + 20 = Rs. 120
3. After 10% decrease on Rs. 120: 120 – (10% of 120) = 120 – 12 = Rs. 108
4. Net change = 108 – 100 = +Rs. 8 on Rs. 100 = 8% increase.
When any quantity is first increased by x% and then decreased by the same x% (or vice versa), there is always a net decrease given by:
Net Percentage Loss/Decrease = (x^2 / 100)%
Calculation Steps:
1. Net Change% = (+20) + (-20) + [(20 * -20) / 100]
2. Net Change% = 0 – (400 / 100) = -4%
A negative result signifies a net decrease of 4%.
Alternative Base 100 Method:
1. Let initial price = Rs. 100
2. After 20% increase: 100 + 20 = Rs. 120
3. 20% decrease on Rs. 120: 120 – (0.20 * 120) = 120 – 24 = Rs. 96
4. Net change from original 100: 96 – 100 = -4, which is a 4% decrease.
Verification:
Multiplier: 1.20 * 0.80 = 0.96.
Net Change = (0.96 – 1) * 100% = -4%.
Population growth over successive periods follows the compound interest growth model:
Population after n years = P * [1 + (R / 100)]^n
where P = present population, R = growth rate per annum, and n = number of years.
Calculation Steps:
1. P = 180,000, R = 10%, n = 2 years
2. Population = 180,000 * [1 + (10 / 100)]^2
3. Population = 180,000 * (11 / 10) * (11 / 10)
4. Population = 180,000 * (121 / 100)
5. Population = 1,800 * 121 = 217,800
Shortcut / Step-by-Step Addition Method:
* Year 1: 180,000 + 10% (18,000) = 198,000
* Year 2: 198,000 + 10% (19,800) = 217,800
Verification:
Successive percentage increase of 10% and 10%:
Net Increase = 10 + 10 + (10 * 10)/100 = 21%.
180,000 * 1.21 = 217,800.
When a value depreciates at R% per annum for n years:
Present Value (P) = Original Value (V) * [1 – (R / 100)]^n
To find the Original Value:
V = P / [1 – (R / 100)]^n
Calculation Steps:
1. Present Value P = 162,000, R = 10%, n = 2 years
2. Multiplier for 10% depreciation = 1 – (10/100) = 9/10 = 0.90
3. 162,000 = V * (9/10) * (9/10)
4. 162,000 = V * (81 / 100)
5. V = (162,000 * 100) / 81
6. Since 162 / 81 = 2:
V = 2,000 * 100 = Rs. 200,000
Shortcut / Ratio Method:
* 10% depreciation = 10 -> 9 each year.
* Over 2 years: (10)^2 -> (9)^2, which is 100 : 81.
* 81 units = Rs. 162,000 -> 1 unit = Rs. 2,000.
* 100 units = 100 * 2,000 = Rs. 200,000.
Verification:
Year 1: 200,000 – 10% (20,000) = 180,000
Year 2: 180,000 – 10% (18,000) = 162,000. (Matches present value)
In a two-candidate election with no invalid votes:
Total Votes = 100%
Loser’s Share = 100% – Winner’s Share
Winning Margin (Majority) = Winner’s Share – Loser’s Share
Calculation Steps:
1. Winner’s Share = 58%
2. Loser’s Share = 100% – 58% = 42%
3. Winning Margin = 58% – 42% = 16% of total votes
4. We are given that 16% of Total Votes = 2,880
5. Total Votes = (2,880 / 16) * 100
6. 2,880 / 16 = 180
7. Total Votes = 180 * 100 = 18,000 votes
Shortcut / Unitary Method:
Difference = 58% – 42% = 16%
16% = 2,880
1% = 2,880 / 16 = 180
100% = 180 * 100 = 18,000 votes
Verification:
Winner’s votes = 58% of 18,000 = 10,440
Loser’s votes = 42% of 18,000 = 7,560
Difference = 10,440 – 7,560 = 2,880. (Matches problem statement)
Valid Votes = Total Enrolled * (Voters Polled Fraction) * (Valid Fraction)
Majority = (Winner% of Valid – Loser% of Valid) * Valid Votes
Calculation Steps:
1. Let the total enrolled voters = x
2. Votes polled = 90% of x = 0.90x
3. Valid votes = 90% of polled votes = 0.90 * 0.90x = 0.81x
4. Winner got 54% of valid votes, so Loser got 100% – 54% = 46% of valid votes.
5. Majority margin in valid votes = 54% – 46% = 8% of valid votes.
6. Margin = 0.08 * 0.81x = 0.0648x
7. Given: 0.0648x = 1,620
8. x = 1,620 / 0.0648 = 1,620,000 / 64.8 = 25,000
Shortcut / Chain Ratio Method:
* Enrolled -> Polled: 10 -> 9
* Polled -> Valid: 10 -> 9
* Valid votes = 81 units out of 100 units enrolled.
* Majority = 8% of 81 units = 6.48 units.
* 6.48 units = 1,620
* 1 unit = 1,620 / 6.48 = 250
* 100 units = 250 * 100 = 25,000
Verification:
Enrolled = 25,000
Polled (90%) = 22,500
Valid (90% of 22,500) = 20,250
Winner (54%) = 10,935
Loser (46%) = 9,315
Difference = 10,935 – 9,315 = 1,620.
When percentages are spent sequentially “of the remaining”, each expenditure reduces the remaining balance multiplicatively:
Remaining Balance = Total Income * (1 – r1/100) * (1 – r2/100) * (1 – r3/100)
Calculation Steps:
1. Let monthly income = I
2. After house rent (20%): Remaining = 1 – 0.20 = 80% = 4/5
3. After food (25% of remaining): Remaining = 1 – 0.25 = 75% = 3/4
4. After education (10% of balance): Remaining = 1 – 0.10 = 90% = 9/10
5. Total savings = I * (4/5) * (3/4) * (9/10)
6. Simplifying the product: (4/5) * (3/4) * (9/10) = (3 * 9) / (5 * 10) = 27 / 50
7. Given: I * (27 / 50) = 16,200
8. I = (16,200 * 50) / 27
9. Since 162 / 27 = 6:
I = 600 * 50 = Rs. 30,000
Shortcut / Fraction Method:
Remaining ratio = (4/5) * (3/4) * (9/10) = 27 / 50.
27 units = Rs. 16,200 -> 1 unit = Rs. 600.
Original Income (50 units) = 50 * 600 = Rs. 30,000.
Verification:
Income = 30,000
Rent (20%) = 6,000 -> Remaining = 24,000
Food (25% of 24,000) = 6,000 -> Remaining = 18,000
Education (10% of 18,000) = 1,800 -> Final Savings = 18,000 – 1,800 = 16,200.
According to the principle of Set Theory (Venn Diagram):
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Percentage passing in at least one subject:
Passed in at least one = Passed in Math + Passed in English – Passed in Both
Failed in both = 100% – Passed in at least one.
Calculation Steps:
1. Passed in at least one subject = 65% + 55% – 30% = 90%
2. Percentage of students who failed in both subjects = 100% – 90% = 10%
3. Let total students = N.
4. 10% of N = 40
5. N = (40 * 100) / 10 = 400 students.
Shortcut / Venn Breakdown:
* Passed only Math = 65% – 30% = 35%
* Passed only English = 55% – 30% = 25%
* Passed both = 30%
* Total passed in at least one = 35% + 25% + 30% = 90%
* Failed both = 100% – 90% = 10% -> 10% = 40 -> 100% = 400.
Verification:
Math only = 35% of 400 = 140
English only = 25% of 400 = 100
Both = 30% of 400 = 120
Failed both = 10% of 400 = 40
Sum = 140 + 100 + 120 + 40 = 400 students.
Work directly with the failure sets:
Failed in at least one subject = Failed(Physics) + Failed(Chemistry) – Failed(Both)
Passed in both subjects = 100% – (Failed in at least one subject).
Calculation Steps:
1. Percentage failing in at least one subject = 42% + 52% – 17%
= 94% – 17% = 77%
2. Percentage passing in both subjects = 100% – 77% = 23%
3. Given that 46 candidates passed in both subjects:
23% of Total Candidates = 46
4. Total Candidates = (46 / 23) * 100 = 2 * 100 = 200 candidates.
Shortcut / Unitary Method:
23% = 46
1% = 46 / 23 = 2
100% = 2 * 100 = 200.
Verification:
Failed Physics only = 42% – 17% = 25% -> 50 students
Failed Chemistry only = 52% – 17% = 35% -> 70 students
Failed both = 17% -> 34 students
Passed both = 23% -> 46 students
Total = 50 + 70 + 34 + 46 = 200.
During evaporation, only water content changes; the solid pulp (dry matter) remains constant in absolute weight:
Initial Pulp Weight = Final Pulp Weight
Calculation Steps:
1. In fresh watermelon (weight = 40 kg):
Water = 90% -> Pulp = 100% – 90% = 10%
Weight of Pulp = 10% of 40 kg = 0.10 * 40 = 4 kg.
2. In the dried watermelon:
Water = 20% -> Pulp = 100% – 20% = 80%
3. Let the new total weight be W kg.
The 4 kg of pulp now represents 80% of this new total weight:
80% of W = 4 kg
4. W = (4 / 80) * 100 = (4 / 4) * 5 = 5 kg.
Shortcut / Constant Quantity Ratio:
Pulp fraction changes from 10% to 80% (an increase of 8 times).
Since Pulp Weight = Total Weight * Pulp Percentage = Constant:
Total weight must decrease by a factor of 8:
New Weight = 40 kg / 8 = 5 kg.
Verification:
In 5 kg of dried watermelon:
Pulp = 80% of 5 kg = 4 kg.
Water = 20% of 5 kg = 1 kg.
Total = 4 + 1 = 5 kg. Pulp matches initial 4 kg.
When pure water is added, the quantity of milk remains unchanged:
Initial Milk Volume = Final Milk Volume
Calculation Steps:
1. In 60 liters of solution:
Water = 20% of 60 = 12 liters
Milk = 80% of 60 = 48 liters
2. In the new mixture, water is 25%, meaning milk is 100% – 25% = 75%.
3. Let the new total volume be V liters.
75% of V = 48 liters
(3/4) * V = 48
V = (48 * 4) / 3 = 16 * 4 = 64 liters
4. Water added = New Total Volume – Initial Volume
Water added = 64 – 60 = 4 liters.
Shortcut / Ratio Method:
* Initial ratio (Milk : Water) = 80 : 20 = 4 : 1
* Final ratio (Milk : Water) = 75 : 25 = 3 : 1
* Make Milk equal in both ratios (LCM of 4 and 3 is 12):
Initial = 12 : 3 (Total = 15 units)
Final = 12 : 4 (Total = 16 units)
* 15 units = 60 liters -> 1 unit = 4 liters.
* Water added = 4 – 3 = 1 unit = 4 liters.
Verification:
New total = 60 + 4 = 64 liters.
Water = 12 + 4 = 16 liters.
Water percentage = (16 / 64) * 100% = 25%.
Since only zinc is added, the absolute mass of copper remains constant:
Initial Copper Mass = Final Copper Mass
Calculation Steps:
1. Initial copper mass = 90% of 80 kg = 0.90 * 80 = 72 kg.
2. In the new alloy, copper constitutes 80% of the total mass.
3. Let the new total weight of the alloy be W kg.
80% of W = 72 kg
(4/5) * W = 72
W = (72 * 5) / 4 = 18 * 5 = 90 kg
4. Zinc added = New Total Weight – Initial Weight
Zinc added = 90 – 80 = 10 kg.
Shortcut / Constant Quantity:
Copper = 72 kg is constant.
New Total = 72 / 0.80 = 90 kg.
Added Zinc = 90 – 80 = 10 kg.
Verification:
Total new alloy = 80 + 10 = 90 kg.
Copper = 72 kg.
Copper% = (72 / 90) * 100% = (4 / 5) * 100% = 80%.
Percentage Error = (|True Value – Erroneous Value| / True Value) * 100%
The error must always be evaluated with respect to the True (correct) Value as the base.
Calculation Steps:
1. Choose a convenient number divisible by both denominators (3 and 5), such as LCM(3, 5) = 15.
2. Correct Value = 15 * (5/3) = 25
3. Erroneous Value = 15 * (3/5) = 9
4. Absolute Error = 25 – 9 = 16
5. Percentage Error = (16 / 25) * 100% = 16 * 4 = 64%
Shortcut / Cross-Multiplication Method:
* True fraction = 5/3, Wrong fraction = 3/5
* Cross multiply denominators: True value = 5 * 5 = 25; Wrong value = 3 * 3 = 9
* Error = 25 – 9 = 16
* % Error = (16 / 25) * 100% = 64%
Verification:
(Erroneous / True) = (3/5) / (5/3) = 9/25 = 0.36 = 36% of the true value.
Error = 100% – 36% = 64%.
Area of Rectangle = Length * Breadth.
Since Area is the product of two independently changing dimensions, net percentage change follows the successive percentage change formula:
Net Change% = a + b + (a * b) / 100
where a = +30% and b = -20%.
Calculation Steps:
1. Net Change% = 30 + (-20) + [30 * (-20)] / 100
2. Net Change% = 10 – (600 / 100) = 10 – 6 = +4%
The positive sign indicates a 4% increase.
Alternative Multiplier Method:
* Length Multiplier = 1 + 0.30 = 1.3
* Breadth Multiplier = 1 – 0.20 = 0.8
* New Area = 1.3 * 0.8 * Original Area = 1.04 * Original Area
* Change = (1.04 – 1.00) * 100% = +4% increase.
Verification:
Let L = 10, B = 10 -> Area = 100.
New L = 13, New B = 8 -> New Area = 13 * 8 = 104.
Increase = 104 – 100 = 4%.
Area of Circle = π * r^2.
Since Area is proportional to r^2 (i.e., r * r), an increase in radius affects the area twice successively:
Percentage Increase = 2r + (r^2 / 100)
Calculation Steps:
1. Increase in radius = 20%
2. Net Percentage Increase = 20 + 20 + (20 * 20) / 100
3. Net Increase = 40 + (400 / 100) = 40 + 4 = 44%
Alternative Base 100 / Multiplier Method:
* Multiplier for radius = 1 + 0.20 = 1.2
* Multiplier for area = (1.2)^2 = 1.44
* Percentage increase = (1.44 – 1.00) * 100% = 44%
Verification:
Let original radius = 10 -> Area = 100π.
New radius = 12 -> Area = 144π.
Increase = 144π – 100π = 44π.
% Increase = (44π / 100π) * 100% = 44%.
Money saved due to price reduction = Reduction% * Total Expenditure.
This saved amount pays for the additional quantity purchased:
Reduced Price per kg = (Money Saved) / (Extra Quantity)
Original Price = Reduced Price / [1 – (Reduction% / 100)]
Calculation Steps:
1. Total expenditure = Rs. 360
2. Money saved due to 20% reduction = 20% of 360 = Rs. 72
3. This Rs. 72 buys the extra 2 kg of apples.
Reduced Price per kg = 72 / 2 = Rs. 36/kg
4. Since the price was reduced by 20%, Rs. 36 is 80% of the original price:
Original Price = 36 / 0.80 = 360 / 8 = Rs. 45/kg
Shortcut / Inverse Ratio Method:
* Price ratio (Original : Reduced) = 100 : 80 = 5 : 4
* Since expenditure is constant, Quantity ratio = 4 : 5
* Difference in quantity = 5 – 4 = 1 unit
* 1 unit = 2 kg
* Original quantity = 4 units = 8 kg
* Original Price = Rs. 360 / 8 kg = Rs. 45/kg
Verification:
Original: 360 / 45 = 8 kg.
Reduced price (20% off Rs. 45) = Rs. 36/kg.
New quantity: 360 / 36 = 10 kg.
Extra quantity = 10 – 8 = 2 kg. (Matches problem statement)
Difference in percentage scores = Total difference in marks obtained:
(Percentage 2 – Percentage 1) of Maximum Marks = Surplus + Deficit
Calculation Steps:
1. Difference in percentage = 42% – 30% = 12%
2. Difference in marks = 21 – (-15) = 21 + 15 = 36 marks
3. 12% of Maximum Marks = 36
1% of Maximum Marks = 36 / 12 = 3 marks
Maximum Marks = 3 * 100 = 300 marks
4. Passing marks = (30% of 300) + 15 = 90 + 15 = 105 marks
5. Passing percentage = (105 / 300) * 100% = 105 / 3 = 35%
Shortcut / Direct Percentage Method:
1% = 3 marks -> 15 marks = 15 / 3 = 5%.
Since the first student got 30% and failed by 15 marks (which is 5%):
Passing Percentage = 30% + 5% = 35%.
Verification:
Student 2: 42% – 35% = 7% surplus.
7% in marks = 7 * 3 = 21 marks surplus. (Matches problem statement)
Weighted Average of Non-Scholarship Holders:
Overall Non-Scholarship% = (Boys * Non-Scholarship Boys% + Girls * Non-Scholarship Girls%) / Total Students
Calculation Steps:
1. Assume convenient numbers proportional to 3 : 2:
Number of Boys = 300
Number of Girls = 200
Total Students = 300 + 200 = 500
2. Non-scholarship boys = 100% – 20% = 80%
Number of non-scholarship boys = 80% of 300 = 240
3. Non-scholarship girls = 100% – 25% = 75%
Number of non-scholarship girls = 75% of 200 = 150
4. Total non-scholarship students = 240 + 150 = 390
5. Required Percentage = (390 / 500) * 100% = 390 / 5 = 78%
Shortcut / Weighted Average:
Scholarship% = [(3 * 20%) + (2 * 25%)] / (3 + 2)
= (60% + 50%) / 5 = 110% / 5 = 22%
Non-Scholarship% = 100% – 22% = 78%
Verification:
Total scholarship holders = 20% of 300 (60) + 25% of 200 (50) = 110.
Non-scholarship holders = 500 – 110 = 390.
(390 / 500) * 100% = 78%.
Expenditure = Price * Consumption
Multiplier form: E_factor = P_factor * C_factor
Therefore: C_factor = E_factor / P_factor
Percentage Reduction in Consumption = (1 – C_factor) * 100%
Calculation Steps:
1. Let initial Price = 100, initial Consumption = 100 kg.
Initial Expenditure = 100 * 100 = 10,000.
2. New Price = 100 + 25 = 125.
3. Desired Expenditure = 100 + 10% = 110% of 10,000 = 11,000.
4. New Consumption = New Expenditure / New Price
New Consumption = 11,000 / 125 = 88 kg.
5. Reduction in Consumption = 100 – 88 = 12 kg.
6. Percentage Reduction = (12 / 100) * 100% = 12%.
Shortcut / Multiplier Method:
* Price Factor = 1.25 = 5/4
* Expenditure Factor = 1.10 = 11/10
* Consumption Factor = (11/10) / (5/4) = (11/10) * (4/5) = 44/50 = 88/100
* Consumption decreases from 100 to 88 -> 12% decrease.
Verification:
Expenditure change = 1.25 * (1 – 0.12) = 1.25 * 0.88 = 1.10 (exactly 10% increase).
Total Earnings = Fixed Salary + Commission
Commission = Rate * (Total Sales – Exemption Threshold)
Calculation Steps:
1. Commission earned = Total Earnings – Fixed Salary
Commission = 6,800 – 4,000 = Rs. 2,800
2. This commission is 5% on sales over Rs. 10,000:
5% of (Total Sales – 10,000) = 2,800
3. (5 / 100) * (Total Sales – 10,000) = 2,800
Total Sales – 10,000 = (2,800 * 100) / 5
Total Sales – 10,000 = 2,800 * 20 = 56,000
4. Total Sales = 56,000 + 10,000 = Rs. 66,000.
Shortcut / Unitary Method:
5% = Rs. 2,800
100% of excess sales = 2,800 * 20 = Rs. 56,000
Total Sales = 56,000 + 10,000 threshold = Rs. 66,000.
Verification:
Sales above Rs. 10,000 = 66,000 – 10,000 = 56,000.
Commission = 5% of 56,000 = Rs. 2,800.
Total Income = 4,000 + 2,800 = Rs. 6,800. (Matches problem statement)
| Department | Total Employees | % of Female Employees |
|---|---|---|
| IT | 400 | 45% |
| HR | 150 | 60% |
| Finance | 250 | 40% |
| Marketing | 200 | 35% |
What is the overall percentage of female employees across all four departments combined?
Overall Female% = (Total Number of Female Employees / Total Number of All Employees) * 100%
Calculation Steps:
1. Calculate the female count in each department:
* IT: 45% of 400 = 0.45 * 400 = 180
* HR: 60% of 150 = 0.60 * 150 = 90
* Finance: 40% of 250 = 0.40 * 250 = 100
* Marketing: 35% of 200 = 0.35 * 200 = 70
2. Total Female Employees = 180 + 90 + 100 + 70 = 440
3. Total Employees across all departments = 400 + 150 + 250 + 200 = 1,000
4. Overall Percentage = (440 / 1000) * 100% = 44%
Shortcut / Weighted Calculation:
Because the total pool sums conveniently to exactly 1,000:
Overall% = (Sum of individual female counts) / 10 = 440 / 10 = 44%.
Verification:
Total males = (400 – 180) + (150 – 90) + (250 – 100) + (200 – 70)
= 220 + 60 + 150 + 130 = 560.
Total employees = 440 females + 560 males = 1,000.
Female% = 440 / 1000 = 44%.
Net Percentage Change in Total Cost = (Total Increase in Components / Original Total Cost) * 100%
or Weighted Percentage Change = Sum(Weight_i * Rate_i) / Sum(Weights)
Calculation Steps:
1. Let the original costs be:
Raw Material = Rs. 400
Labour = Rs. 300
Overheads = Rs. 100
Total Initial Cost = 400 + 300 + 100 = Rs. 800
2. Calculate changes in each component:
* Raw Material increase = +10% of 400 = +Rs. 40
* Labour increase = +20% of 300 = +Rs. 60
* Overheads decrease = -5% of 100 = -Rs. 5
3. Net Increase in Total Cost = 40 + 60 – 5 = +Rs. 95
4. Percentage Increase = (95 / 800) * 100%
= 95 / 8 = 11.875%
Shortcut / Weighted Average Formula:
Net Change% = [(4 * 10%) + (3 * 20%) + (1 * (-5%))] / (4 + 3 + 1)
= (40% + 60% – 5%) / 8
= 95% / 8 = 11.875%
Verification:
New Cost = (400 + 40) + (300 + 60) + (100 – 5) = 440 + 360 + 95 = Rs. 895.
Percentage Increase = [(895 – 800) / 800] * 100% = (95 / 800) * 100% = 11.875%.
