Percentage Basics: Percent of a Number, Comparisons and Real-Life Percentages
Percentage Basics: Percent of a Number, Comparisons and Real-Life Percentages
Percentage is the language of half the NTPC Maths paper: profit, interest, data and ratio questions are all percentage questions underneath. As a stand-alone topic NTPC asks percent-of-a-number, "A is r% more than B" reversals, elections, salary spending and literacy-type composition questions. Every one of them is solved with the fraction table below and a single multiplication.
1. Percent, Fractions and Multiplying Factors
x% = x/100 x% of y = xy/100 = y% of x A as a percent of B = (A/B) × 100
| Percent | Fraction | Percent | Fraction | Percent | Fraction |
|---|---|---|---|---|---|
| 50% | 1/2 | 12.5% | 1/8 | 6.25% | 1/16 |
| 33⅓% | 1/3 | 11 1/9% | 1/9 | 5% | 1/20 |
| 25% | 1/4 | 10% | 1/10 | 4% | 1/25 |
| 20% | 1/5 | 9 1/11% | 1/11 | 37.5% | 3/8 |
| 16⅔% | 1/6 | 8⅓% | 1/12 | 62.5% | 5/8 |
| 14 2/7% | 1/7 | 7 9/13% | 1/13 | 87.5% | 7/8 |
Multiplying factor: increase by r% means × (100 + r)/100; decrease by r% means × (100 − r)/100. A 15% rise is × 1.15, a 6% fall is × 0.94.
x% of y = y% of x: 18% of 50 = 50% of 18 = 9. Use it whenever one number is friendly.
2. "More Than" and "Less Than" Reversals
If A is r% more than B, then B is [r/(100 + r)] × 100 % less than A
If A is r% less than B, then B is [r/(100 − r)] × 100 % more than A
| A more than B by | B less than A by | A less than B by | B more than A by |
|---|---|---|---|
| 10% | 9 1/11% | 10% | 11 1/9% |
| 20% | 16⅔% | 20% | 25% |
| 25% | 20% | 25% | 33⅓% |
| 33⅓% | 25% | 33⅓% | 50% |
| 50% | 33⅓% | 50% | 100% |
| 100% | 50% | 40% | 66⅔% |
✗ A is 25% more than B, so B is 25% less than A | ✓ The base changes: B is 25/125 = 20% less than A
Percentage point vs percent: a rate rising from 20% to 25% rises by 5 percentage points but by 25% of itself.
3. Frameworks for the Three Standard Stories
| Story | Chain to write | Equation |
|---|---|---|
| Election | Total voters → voted → valid → candidate shares | Margin = (winner % − loser %) of valid votes |
| Examination | Pass mark = p% of maximum | Score + "failed by" = pass mark; score − "more than pass" = pass mark |
| Spending | Percent heads multiply the remainder; rupee heads subtract | What is left = savings |
In a spending chain, spending "30% of the remaining" leaves 0.7 × what was left (the amount spent is 0.3 × that remainder), not 30% of the original salary. NTPC 2025 papers mixed one percent head (rent) with several fixed rupee heads (food, travel) and the savings; then the rupee heads plus savings equal the remaining percent of salary.
हिंदी नोट: "A, B से r% अधिक है" तो B, A से r/(100 + r) × 100 प्रतिशत कम होता है, क्योंकि आधार (base) बदल जाता है। चुनाव के प्रश्नों में अंतर हमेशा वैध मतों का प्रतिशत होता है।
Exam Pointer: Verified NTPC patterns: percent of a number (January 2017 CBT-2, August 2025 UG CBT-1), "A is r% more than B" reversals (March 2026 Graduate CBT-1), elections with turnout and invalid votes (two August 2025 UG CBT-1 shifts), salary with a rent percentage and fixed rupee heads (two June 2025 Graduate CBT-1 shifts) and literacy composition of a population (September 2025 UG CBT-1). Pass-mark and fast fraction-conversion questions had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Percent of a number and the number from a percent
[PYQ: NTPC UG CBT-1 12-Aug-2025 Shift-1 | NTPC CBT-2 19-Jan-2017 Shift-1]
EXAM LEVEL
Q. 35% of a number is 1,050. Find 60% of the number.
Skip finding the number: 60% = 1,050 × 60/35 = 1,050 × 12/7 = 1,800. (The number itself is 3,000.)
Answer: 1,800
EXAMATLAS LEVEL
Q. If 40% of (A + B) = 60% of (A − B), then B is what percent of A?
0.4A + 0.4B = 0.6A − 0.6B, so B = 0.2A. B is 20% of A. Equivalently A is 5 times B, that is 400% more than B, the trap option.
Answer: 20%
Pattern 2: "A is r% more or less than B" and chained comparisons
[PYQ: NTPC Graduate CBT-1 24-Mar-2026 Shift-3 | NTPC 2020-21 cycle]
EXAM LEVEL
Q. A's income is 25% more than B's. By what percent is B's income less than A's?
Take B = 100, A = 125. B is 25 less than A on a base of 125: 25/125 × 100 = 20%.
Answer: 20%
EXAMATLAS LEVEL
Q. A is 20% more than B, and B is 25% less than C. A is what percent of C, and by what percent is C more than A?
Take C = 100. Then B = 75 and A = 1.2 × 75 = 90, so A is 90% of C. C exceeds A by 10 on a base of 90: 10/90 × 100 = 11 1/9%. Answering 10% for the second part ignores the change of base.
Answer: 90%; 11 1/9%
Pattern 3: Elections: margin, turnout and invalid votes
[PYQ: NTPC UG CBT-1 19-Aug-2025 Shift-2 | NTPC UG CBT-1 7-Aug-2025 Shift-1]
EXAM LEVEL
Q. In an election between two candidates, the winner got 58% of the valid votes and won by 1,920 votes. Find the total number of valid votes.
The loser got 42%, so the margin is 16% of the valid votes. 16% = 1,920 gives 100% = 12,000.
Answer: 12,000
EXAMATLAS LEVEL
Q. In an election between two candidates, 10% of the voters on the list did not vote and 5% of the votes cast were invalid. The winner got 60% of the valid votes and won by 6,840 votes. How many voters were on the list?
Let the list have T voters. Votes cast = 0.9T, valid = 0.95 × 0.9T = 0.855T. The margin is 60% − 40% = 20% of valid votes: 0.2 × 0.855T = 0.171T = 6,840, so T = 40,000. Taking 20% of T directly gives 34,200, the trap option.
Answer: 40,000
Pattern 4: Examinations: pass marks and maximum marks
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A student needs 36% marks to pass. He scores 190 and fails by 26 marks. Find the maximum marks.
Pass mark = 190 + 26 = 216 = 36% of the maximum, so maximum = 216 ÷ 0.36 = 600.
Answer: 600
EXAMATLAS LEVEL
Q. A candidate who scores 30% fails by 30 marks, while another who scores 45% gets 30 marks more than the pass mark. Find the pass percentage.
The gap between the two scores is 15% of the maximum and equals 30 + 30 = 60 marks, so the maximum is 400. Pass mark = 30% of 400 + 30 = 150, which is 150/400 = 37.5%.
Answer: 37.5%
Pattern 5: Salary spending: percent heads and rupee heads
[PYQ: NTPC Graduate CBT-1 18-Jun-2025 Shift-2 | NTPC Graduate CBT-1 17-Jun-2025 Shift-3]
EXAM LEVEL
Q. A man spends 30% of his monthly salary on rent, ₹6,500 on food, ₹3,500 on travel and saves ₹11,000. Find his salary.
Everything except rent is 70% of the salary: 6,500 + 3,500 + 11,000 = ₹21,000 = 70%. Salary = 21,000 ÷ 0.7 = ₹30,000. Check: rent 9,000 + 21,000 = 30,000.
Answer: ₹30,000
EXAMATLAS LEVEL
Q. A person spends 20% of his income on rent, 30% of the remaining on his children's education and 40% of the rest on food. He is left with ₹8,064. Find his income and the amount spent on education.
Each "remaining" multiplies: 0.8 × 0.7 × 0.6 = 0.336 of the income is left. Income = 8,064 ÷ 0.336 = ₹24,000. Education = 30% of (0.8 × 24,000) = 30% of 19,200 = ₹5,760. Adding 20% + 30% + 40% = 90% and calling 10% the savings is the trap.
Answer: Income ₹24,000; education ₹5,760
Pattern 6: Composition of a population (literacy, choices, groups)
[PYQ: NTPC UG CBT-1 3-Sep-2025 Shift-2]
EXAM LEVEL
Q. In a town 55% of the people are men. 40% of the men and 30% of the women are literate. What percent of the town is literate?
Weighted percent: 0.55 × 40 + 0.45 × 30 = 22 + 13.5 = 35.5%.
Answer: 35.5%
EXAMATLAS LEVEL
Q. In a school 60% of the students are boys. 25% of the boys and 40% of the girls opted for science, and 372 students opted for science in all. How many students are in the school, and how many girls did not opt for science?
Science fraction = 0.6 × 0.25 + 0.4 × 0.4 = 0.15 + 0.16 = 0.31. Total = 372 ÷ 0.31 = 1,200. Girls = 480, of whom 60% (288) did not opt for science. Averaging 25% and 40% to 32.5% ignores the unequal group sizes.
Answer: 1,200 students; 288 girls
Pattern 7: Fast percentage-fraction calculation
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find 37.5% of 640 + 12.5% of 480.
37.5% = 3/8 and 12.5% = 1/8. 3/8 × 640 = 240 and 1/8 × 480 = 60. Sum = 300.
Answer: 300
EXAMATLAS LEVEL
Q. Find 16⅔% of 1,806 + 14 2/7% of 1,617 − 11 1/9% of 1,449.
Convert: 16⅔% = 1/6, 14 2/7% = 1/7, 11 1/9% = 1/9. So 1,806/6 + 1,617/7 − 1,449/9 = 301 + 231 − 161 = 371. Each division is exact, which is the signal that the fraction route was intended.
Answer: 371
60-Second Revision
- x% of y = y% of x; learn the 1/2 to 1/16 percent table cold.
- r% more becomes [r/(100 + r)] × 100 % less; r% less becomes [r/(100 − r)] × 100 % more.
- Election margin = difference of percentages × VALID votes; build turnout and invalid votes as factors.
- Pass mark = score + failed-by; two candidates: percentage gap = mark gap.
- Spending chains multiply the remainders; rupee heads + savings = remaining percent.
- Group mixtures: weighted percent, never the simple average.