Percentage/Ratio/Average
Percentage, ratio and average are one topic wearing three names — each is a different way of comparing a part to a whole. Learn the conversions between them and most questions solve in one line.
Mental model
Percentage, ratio and average are one idea seen from three sides — a part compared with a whole. Convert percentages into fractions on sight (25% = 1/4, 12.5% = 1/8, 16.67% = 1/6) and the arithmetic collapses into cancelling. For ratios, always introduce the multiplier k and write the quantities as 3k and 5k; that single habit removes almost all ratio errors. For averages, use the deviation method: guess a base, add the average of the deviations. And remember the asymmetry trap: a rise of x% followed by a fall of x% never returns to the original value.
Associations
A short trigger on the left, everything it pulls with it on the right.
A percentage is a ratio with a fixed denominator of 100.
25% is just 25:100, which simplifies to 1:4. Converting between percentage and ratio is nothing more than expressing the same relationship with a different denominator — memorising a handful of common conversions (1/4=25%, 1/3≈33%, 1/8=12.5%) speeds up every downstream calculation.
Average times count equals total — the one identity that solves most average problems.
If the average of 5 numbers is 20, their total is 100. Adding a sixth number that changes the average to 22 means the new total is 132, so the sixth number is 32. Nearly every 'average changes when a new value is added' question reduces to this one substitution.
Things to remember
The night-before list. Short enough to recall cold.
- 1Common fraction-percentage pairs worth knowing cold: 1/2=50%, 1/3≈33.3%, 1/4=25%, 1/5=20%, 1/8=12.5%, 1/10=10%.
- 2A ratio a:b can be scaled to any equivalent form ka:kb without changing the underlying relationship.
- 3Weighted average gives more influence to values with a larger weight — it is not the same as the simple average of the values.
- 4A percentage increase followed by an equal percentage decrease does not return to the original value — the base has changed between the two steps.
In detail
- A successive rise of x% and fall of x% gives a net LOSS of (x squared)/100 percent — for example plus 10% then minus 10% leaves 99, a 1% loss.
- Net effect of two successive percentage changes a and b = a + b + (a x b)/100, using a negative sign for a decrease.
- Average = sum of observations / number of observations; therefore sum = average x count, which is how most 'new member joins' problems are solved.
- If A is x% more than B, then B is (100x)/(100+x) percent less than A — the two percentages are not equal.
- In ratio problems, express the parts as multiples of a common k (3k, 5k) so that the total and each part stay consistent when a quantity is added or removed.
Where people slip
- A 20% increase followed by a 20% decrease does not cancel out — it leaves the quantity 4% below the original, because the second 20% is taken off a larger base.
Check yourself
Nothing here is scored — it will not change your mastery figures.
The average of 5 numbers is 20. A sixth number is added and the new average becomes 22. What is the sixth number?
From the question bank
What is 25% of 240?
From the question bank
If 40% of a number is 120, what is the number?
Books to read
- Quantitative Aptitude for Competitive Examinations — R S Aggarwal
- Fast Track Objective Arithmetic — Arihant
- NCERT Class VIII — Mathematics
Official sources
Hand-checked official domains only — the notes above carry no links of their own.