Percentages for CSAT
CSAT percentages made fast — the fraction–percentage table, percentage change, successive changes, percentage points, and timed UPSC Prelims Paper II practice.
SyllabusCSAT · Basic numeracy (numbers and their relations, orders of magnitude, etc.) — Class X level
Prerequisites: Percentages, Ratio and Averages
The Hook: The 4% That Vanished
A shop raises a price by 20% before a festival and then announces a “20% off” sale. Back to the original price? No — the customer still pays 4% less than the original, and the shop has quietly lost money on the item.
CSAT asks exactly this kind of question, with a timer running. Knowing why the 4% vanishes lets you answer in five seconds instead of fifty.
The Core Concept
A percentage is a fraction with denominator 100:
$$x\% = \frac{x}{100}$$1. Percentage change
$$\text{Percentage change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$$The base is always the old value. A rise from 80 to 100 is a 25% increase; a fall from 100 to 80 is a 20% decrease.
2. Successive changes
Two successive changes of $a\%$ and $b\%$ (use negative values for decreases) give a net change of
$$\boxed{a + b + \frac{ab}{100}\ \%}$$For $+20\%$ then $-20\%$: $20 - 20 + \dfrac{(20)(-20)}{100} = -4\%$.
3. Keeping expenditure constant
If a price rises by $r\%$, consumption must fall by
$$\frac{r}{100 + r} \times 100\ \%$$to keep spending unchanged. A 25% price rise needs a $\dfrac{25}{125} \times 100 = 20\%$ cut in consumption.
4. Percent versus percentage points
If a rate moves from 4% to 5%, it has risen by 1 percentage point but by 25 per cent ($\tfrac{1}{4} \times 100$). Data-interpretation questions use this distinction to set traps.
Memory Anchors
The fraction table — learn it once, use it in every CSAT paper:
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/7 | 14.29% |
| 1/3 | 33.33% | 1/8 | 12.5% |
| 1/4 | 25% | 1/9 | 11.11% |
| 1/5 | 20% | 1/10 | 10% |
| 1/6 | 16.67% | 1/11 | 9.09% |
| 1/12 | 8.33% |
Swap trick: $x\%$ of $y$ = $y\%$ of $x$. So 8% of 50 = 50% of 8 = 4.
Common Traps
Practice
Prelims (CSAT)
The price of an article is increased by 20% and then decreased by 20%. What is the net change in its price?
Solution
Net change $= 20 - 20 + \dfrac{20 \times (-20)}{100} = -4\%$. A 4% decrease.
Answer: B
The price of sugar rises by 25%. By what percentage must a family reduce its consumption of sugar so that its expenditure on sugar does not change?
Solution
Required reduction $= \dfrac{25}{100 + 25} \times 100 = 20\%$.
Answer: B
The average of five numbers is 24. When one number is removed, the average of the remaining four becomes 22. What is the number removed?
Solution
Sum of five numbers $= 5 \times 24 = 120$. Sum of the remaining four $= 4 \times 22 = 88$. Removed number $= 120 - 88 = 32$.
Answer: B
If $A : B = 2 : 3$ and $B : C = 4 : 5$, then $A : B : C$ is:
Solution
Make B common: $A : B = 2 : 3 = 8 : 12$ and $B : C = 4 : 5 = 12 : 15$. So $A : B : C = 8 : 12 : 15$.
Answer: B
Quick Revision
| Situation | Approach |
|---|---|
| Percentage change | $\dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100$ |
| Two successive changes | $a + b + \dfrac{ab}{100}$ |
| Same % up then down | Net loss of $\dfrac{a^2}{100}\%$ |
| Price up $r\%$, same spend | Cut consumption by $\dfrac{r}{100+r} \times 100$ |
| 4% → 5% | +1 percentage point = +25 per cent |
Key Takeaways
- The base of a percentage change is always the old value.
- Successive changes multiply — use $a + b + \tfrac{ab}{100}$.
- Memorise the fraction table; it turns percentage questions into mental arithmetic.