📖 Introduction
Mastering percentages is about more than simply calculating increases and decreases. In competitive exams, percentage concepts are often combined with profit and loss, mixtures, population changes, income analysis, examination marks, price changes, and work efficiency to create questions that require both speed and careful interpretation.
BrainQuro Percentage Questions Part 6 brings you 10 carefully verified questions covering Questions 51–60, designed around practical and exam-oriented situations. Each problem includes a shortcut solution to help you recognize percentage relationships quickly and reduce unnecessary calculations.
This practice set is especially useful for aspirants preparing for IBPS PO, SBI PO, RBI Assistant, LIC AAO, SSC CGL, RRB, Insurance Exams, and other competitive examinations.
Try solving each question independently before checking the shortcut method. The goal is not just to get the correct answer, but to develop the ability to identify the fastest possible approach under exam pressure.
Table of Contents
Question 51
📖 Question
A person spends 35% of his monthly income on rent, 20% of the remaining on groceries, and 25% of the further remaining on transportation, depositing the rest into a savings account. If his total annual savings amount to Rs. 1,87,200, what is his monthly income in Rupees?
⚡ Shortcut Solution
Let the monthly income be ₹X.
Step 1: Rent
He spends 35% on rent, so remaining:
100% − 35% = 65%
Remaining income = 65% of X
Step 2: Groceries
He spends 20% of the remaining income:
20% of 65% = 13% of X
So remaining after groceries:
65% − 13% = 52%
Step 3: Transportation
He spends 25% of the further remaining:
25% of 52% = 13% of X
So savings:
52% − 13% = 39% of X
Step 4: Calculate monthly savings
Annual savings = ₹1,87,200
Monthly savings:
₹1,87,200 ÷ 12 = ₹15,600
Therefore:
39% of X = ₹15,600
X = 15,600 × 100/39
X = ₹40,000
✅ Final Answer:
₹40,000 per month
Question 52
📖 Question
In an election between two candidates, 15% of the registered voters did not cast their votes. Out of the votes cast, 500 votes were declared invalid. The winning candidate secured 55% of the valid votes and defeated the opponent by 800 votes. Find the total number of registered voters.
⚡ Shortcut Solution
Let the total number of registered voters be N.
Step 1: Votes actually cast
15% did not vote, so 85% cast their votes.
Votes cast = 85% of N = 0.85N
Out of these, 500 votes were invalid.
So valid votes = 0.85N − 500
Step 2: Use the winning margin
The winner secured 55% of valid votes.
Therefore, the opponent secured 45%.
Difference = 55% − 45% = 10% of valid votes
Given winning margin = 800:
10% of valid votes = 800
Valid votes = 800 × 100/10 = 8,000
Step 3: Find total registered voters
Votes cast = Valid votes + Invalid votes
= 8,000 + 500
= 8,500
Since 85% of registered voters = 8,500:
N = 8,500 × 100/85
N = 10,000
✅ Final Answer
10,000 registered voters
Question 53
📖 Question
A vessel contains 300 liters of a milk-water mixture with 40% water. ‘x’ liters of pure milk is added to the mixture, reducing the water concentration to 25%. Then, ‘y’ liters of pure water is added to the resulting mixture, increasing the water concentration back to 40%. Find the value of (x + y).
⚡ Shortcut Solution
Let the initial mixture be 300 litres with 40% water.
Step 1: Initial quantities
Water = 40% of 300 = 120 L
Milk = 300 − 120 = 180 L
Step 2: Add x litres of pure milk
Water remains 120 L, while total mixture becomes:
300 + x
Water concentration becomes 25%:
120/(300 + x) = 25/100
300 + x = 120 × 4
300 + x = 480
Therefore,
x = 180 L
Step 3: Add y litres of pure water
After adding 180 L milk:
Total mixture = 480 L
Water = 120 L
Now add y litres of pure water.
New water = 120 + y
New total = 480 + y
Water concentration becomes 40%:
(120 + y)/(480 + y) = 40/100
120 + y = 192 + 0.4y
0.6y = 72
y = 120 L
Step 4: Find x + y
x + y = 180 + 120
✅ Final Answer
300 litres
Question 54
📖 Question
The population of a city increases by 25% in the first year, decreases by 20% in the second year, decreases by 20% in the third year, and increases by ‘r’% in the fourth year. If the population at the end of the fourth year is equal to the initial population at the start, find the value of r.
⚡ Shortcut Solution
Let the initial population be P.
Step 1: Apply the first three changes
After the 1st year:
P × 125/100 = 1.25P
After the 2nd year, it decreases by 20%:
1.25P × 80/100 = P
After the 3rd year, it again decreases by 20%:
P × 80/100 = 0.8P
Step 2: Apply the fourth-year increase
The fourth-year increase is r%.
So final population:
0.8P × (1 + r/100)
Given that the final population equals the initial population:
0.8P × (1 + r/100) = P
Cancel P:
0.8 × (1 + r/100) = 1
1 + r/100 = 1.25
r/100 = 0.25
✅ Final Answer
r = 25%
Question 55
📖 Question
An individual’s income and expenditure were in the ratio 5 : 3. In the following year, his income increased by 20% and his savings increased by 20%. Find the percentage increase in his expenditure.
⚡ Shortcut Solution
Let the initial income and expenditure be 5x and 3x respectively.
Step 1: Initial savings
Savings = Income − Expenditure
= 5x − 3x
= 2x
Step 2: New income
Income increases by 20%:
New income = 5x × 120/100 = 6x
Savings also increase by 20%:
New savings = 2x × 120/100 = 2.4x
Step 3: Find new expenditure
New expenditure = New income − New savings
= 6x − 2.4x
= 3.6x
Step 4: Percentage increase in expenditure
Initial expenditure = 3x
New expenditure = 3.6x
Increase = 3.6x − 3x = 0.6x
Percentage increase:
0.6x/3x × 100 = 20%
✅ Final Answer
20% increase
📚 Recommended Reading
- Previous Percentage Practice Set
- Next Percentage Questions Practice Set
- Profit and Loss Questions with Answers
- Ratio and Proportions
Question 56
📖 Question
In an examination, 65% of the total candidates passed in English, 55% passed in Mathematics, and 30% passed in both. If 120 candidates failed in both subjects, find the total number of candidates who appeared for the exam.
⚡ Shortcut Solution
Let the total number of candidates be N.
Step 1: Find the percentage who passed at least one subject
Passed in English = 65%
Passed in Mathematics = 55%
Passed in both = 30%
Using the inclusion-exclusion principle:
Passed in at least one subject
= 65% + 55% − 30%
= 90%
Therefore, candidates who failed in both:
100% − 90% = 10%
Step 2: Find total candidates
Given 10% of candidates = 120
N = 120 × 100/10
N = 1,200
✅ Final Answer
1,200 candidates
Question 57
📖 Question
A salesman receives a commission of 8% on total sales up to Rs. 50,000 and an additional commission of ‘p’% on sales exceeding Rs. 50,000. If his total sales are Rs. 1,20,000 and his total earnings are Rs. 11,000, find the value of p.
⚡ Shortcut Solution
Let the additional commission rate be p%.
Step 1: Commission on first ₹50,000
Commission = 8% of ₹50,000
= ₹4,000
Step 2: Commission on remaining sales
Total sales = ₹1,20,000
Excess sales = ₹1,20,000 − ₹50,000
= ₹70,000
Total earnings = ₹11,000
So, additional commission earned:
₹11,000 − ₹4,000 = ₹7,000
Thus,
p% of ₹70,000 = ₹7,000
p = (7,000/70,000) × 100
p = 10%
✅ Final Answer
p = 10%
Question 58
📖 Question
Alloy X contains 20% silver and Alloy Y contains 50% silver. If 40 kg of Alloy X is mixed with ‘m’ kg of Alloy Y, the resulting mixture contains 40% silver. Find the value of m.
⚡ Shortcut Solution
Let the quantity of Alloy Y be m kg.
Step 1: Silver in Alloy X
Alloy X = 40 kg
Silver = 20%
Silver = 40 × 20/100 = 8 kg
Step 2: Silver in Alloy Y
Alloy Y = m kg
Silver = 50%
Silver = 0.5m kg
Step 3: Use the final silver concentration
Total mixture = 40 + m kg
Silver = 8 + 0.5m kg
The resulting mixture contains 40% silver:
(8 + 0.5m)/(40 + m) = 40/100
8 + 0.5m = 16 + 0.4m
0.1m = 8
m = 80 kg
✅ Final Answer
80 kg
Question 59
📖 Question
Due to a 25% increase in the price of sugar, a family reduces its consumption by ‘k’% so that their expenditure on sugar increases by only 10%. Find the value of k.
⚡ Shortcut Solution
Let the original price of sugar be ₹100 per unit and the original consumption be 100 units.
Step 1: Original expenditure
Original expenditure = 100 × 100 = ₹10,000
Step 2: New price
Price increases by 25%:
New price = 100 × 125/100 = ₹125
Expenditure increases by only 10%:
New expenditure = 10,000 × 110/100 = ₹11,000
Step 3: Find new consumption
New consumption = 11,000 ÷ 125 = 88 units
So consumption decreases from 100 to 88 units:
Decrease = 100 − 88 = 12 units
Percentage decrease:
12/100 × 100 = 12%
✅ Final Answer
k = 12%
Question 60
📖 Question
In a village with a total population of 10,000, 60% of the population is literate. Out of the literate population, 40% are females. If the number of illiterate males in the village is 1,400, find the percentage of illiterate males relative to the total male population of the village.
⚡ Shortcut Solution
Step 1: Find the literate population
Total population = 10,000
Literate = 60%
Literate population = 10,000 × 60/100 = 6,000
Therefore, illiterate population:
10,000 − 6,000 = 4,000
Step 2: Find literate males
40% of the literate population are females:
Literate females = 6,000 × 40/100 = 2,400
So, literate males:
6,000 − 2,400 = 3,600
Step 3: Find total male population
Given illiterate males = 1,400
Total males = Literate males + Illiterate males
= 3,600 + 1,400
= 5,000
Step 4: Find the required percentage
Illiterate males = 1,400
Total males = 5,000
Percentage = 1,400/5,000 × 100
= 28%
✅ Final Answer
28%
🎯 Challenge Question
📖 Question
A company has 40% of its employees working in Production and the remaining employees working in Administration and Sales. The number of employees in Administration is 25% less than the number in Sales. During a restructuring process, 20% of Production employees and 30% of Administration employees are transferred to another branch, while 10% of Sales employees are newly recruited.
After these changes, the company has 4,680 employees across these three departments. What was the company’s original number of employees?
📌 Information
- Production = 40% of original employees
- Administration + Sales = 60%
- Administration is 25% less than Sales
- 20% of Production employees are transferred
- 30% of Administration employees are transferred
- Sales employees increase by 10%
- Final total employees = 4,680
- Find the original number of employees
💡 Challenge: Try solving this without using a conventional equation. Look for a percentage-parts relationship.
⭐ Shortcut Trick of the Day
Successive Percentage Changes: Think in Multipliers
When a value undergoes several percentage changes, don’t calculate each change separately. Convert every change into a multiplier.
| Change | Multiplier |
|---|---|
| +20% | × 1.20 |
| −20% | × 0.80 |
| +25% | × 1.25 |
| −25% | × 0.75 |
| +10% | × 1.10 |
| −10% | × 0.90 |
For example:
Increase by 20%, decrease by 25%, then increase by 10% 1.20×0.75×1.10=0.99
So the final value is 99% of the original, meaning there is an overall 1% decrease.
🧠 BrainQuro Speed Rule
Successive changes → Convert to multipliers → Multiply → Compare with 100%.
This is usually much faster and safer than repeatedly calculating the percentage change in long steps
