Fractions, percentage, ratio and average
Percentage means per hundred, a ratio compares two quantities, and an average is total divided by number. Most NET items need one formula and quick mental arithmetic.
Fractions to percentages: learn these
| Fraction | Per cent | Fraction | Per cent |
|---|---|---|---|
| 1/2 | 50% | 1/7 | 14.29% (approx.) |
| 1/3 | 33.33% | 1/8 | 12.5% |
| 1/4 | 25% | 1/9 | 11.11% |
| 1/5 | 20% | 1/11 | 9.09% |
| 1/6 | 16.67% | 1/12 | 8.33% |
Worked example: successive percentage change
The price of a book rises by 20% and then falls by 20%. What is the net percentage change?
- Use net change = a + b + ab/100, with a = +20 and b = −20.
- Net = 20 − 20 + (20 × −20)/100 = −400/100 = −4.
- Check with ₹100: 100 → 120 → 120 − 24 = 96, a fall of 4.
A net fall of 4%. Shortcut: an equal rise and fall of x% always gives a net fall of x²/100 per cent.
Worked example: ratio of incomes and expenses
The incomes of A and B are in the ratio 3 : 4 and their expenses in the ratio 5 : 7. If each saves ₹2,000, find their incomes.
- Let the incomes be 3x and 4x, and the expenses 5y and 7y.
- Savings: 3x − 5y = 2,000 and 4x − 7y = 2,000.
- Multiply the first by 4 and the second by 3: 12x − 20y = 8,000 and 12x − 21y = 6,000.
- Subtract: y = 2,000. Then 3x = 2,000 + 5 × 2,000 = 12,000, so x = 4,000.
- Incomes: 3x = ₹12,000 and 4x = ₹16,000. Check: 12,000 − 10,000 = 2,000 and 16,000 − 14,000 = 2,000.
₹12,000 and ₹16,000. Shortcut: in an MCQ, test the options; the right pair leaves equal savings with expenses in the ratio 5 : 7.
Worked example: combined average
The average mark of 30 students in Section A is 60 and of 20 students in Section B is 70. Find the average of all 50 students.
- Total for A = 30 × 60 = 1,800; total for B = 20 × 70 = 1,400.
- Combined total = 3,200 for 50 students.
- Average = 3,200 ÷ 50 = 64.
- Quick check: the averages differ by 10 and the group sizes are 3 : 2, so the answer lies 2/5 of the way from 60 to 70: 60 + 4 = 64.
64. Shortcut: a combined average always lies between the two averages, closer to the larger group.
NET asks one-step percentage and ratio problems, successive changes, and combined or changed averages, often hidden inside a short table. Speed comes from knowing fraction-percentage equivalents by heart.
Typical MCQ: “If A's salary is 25% more than B's, by what per cent is B's salary less than A's?” 20%. Use r ÷ (100 + r) × 100 = 25 ÷ 125 × 100 = 20%.
Typical MCQ: “The average of 5 numbers is 20. When one number is removed, the average becomes 18. Find the removed number.” 28. The total was 100; the remaining four total 72; so the removed number is 100 − 72 = 28.
x% of y = y% of x. An equal rise and fall of x% loses x²/100 per cent. If A is r% more than B, B is r/(100 + r) less than A.