Percentages, Ratio and Averages — Quick Revision

One-page CSAT revision of percentages, ratio and averages: fraction table, successive change, percentage points, combining ratios, alligation, partnership and averages.

SyllabusCSAT · Basic numeracy (numbers and their relations, orders of magnitude, etc.) — Class X level

By UPSC Notes Editorial Team 2 min read Updated Oct 2026 #quick revision#csat#percentages

The formulas behind most CSAT arithmetic and data-interpretation questions, on one page.

Fraction–Percentage Table

Fraction	%	Fraction	%
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
3/8	37.5	1/12	8.33

Percentages

Situation	Formula
Percentage change	$\dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100$
Successive changes $a\%$, $b\%$	$a + b + \dfrac{ab}{100}$
Same % up and down	Net loss of $\dfrac{a^2}{100}\%$
Price up $r\%$, same spending	Cut consumption by $\dfrac{r}{100 + r} \times 100\%$
A is $r\%$ more than B	B is $\dfrac{r}{100 + r} \times 100\%$ less than A
4% → 5%	+1 percentage point = +25 per cent

Ratio and Proportion

Situation	Method
Divide in $a : b : c$	One part $= \text{Total} \div (a + b + c)$
Combine $A : B$ and $B : C$	Equalise B with the LCM
Compare $a : b$ and $c : d$	Cross-multiply: $ad$ vs $bc$
Proportion $a : b = c : d$	$ad = bc$
Partnership	Share ∝ capital × time
Alligation	Cheaper : dearer $= (d - m) : (m - c)$

Averages

Situation	Method
Any average	Sum = average × number
Replacement	New = old ± $n$ × change
Two groups	Weighted average $= \dfrac{n_1 a_1 + n_2 a_2}{n_1 + n_2}$
Evenly spaced numbers	Average = middle term = (first + last) ÷ 2
Average speed, equal distances	$\dfrac{2xy}{x + y}$

High-yield facts

$x\%$ of $y$ = $y\%$ of $x$.
Equal percentage rise and fall never cancel — net loss of $a^2/100$ per cent.
In a ratio $a : b$, the first share is $\dfrac{a}{a + b}$ of the whole.
A weighted average lies closer to the larger group’s average.

Traps

Four classic errors

Using the new value as the base of a percentage change.
Confusing percentage points with per cent.
Averaging the averages of unequal groups.
Changing the fixed component in a mixture problem.