Successive percentage change, two ways

It’s one of the most common traps in SSC and banking papers:

A shirt’s price is increased by 20% and then decreased by 20%. What is the net change?

The instinct is “no change”: +20 and −20 cancel, right? They don’t, because the second 20% is taken on a bigger number. Successive change goes by many names in exam papers (overall, effective or net percentage change), but it’s the same idea every time.

Method 1: the formula

If a value changes by $x\%$ and then by $y\%$, the net change is:

$$\text{Net change} = x + y + \frac{xy}{100}\ \%$$

Use a minus sign for a decrease, and keep that sign when you multiply.

Back to the shirt, with $x = +20$ and $y = -20$:

$$20 + (-20) + \frac{20 \times (-20)}{100} = -4\%$$

So the price ends up 4% lower. Check with ₹100: up 20% → ₹120, down 20% of 120 = ₹24 → ₹96. ✔

Method 2: the ratio method

The formula is great, but with awkward percentages like $37\tfrac{1}{2}\%$ it gets messy. The faster route is to turn each percentage into a fraction, then into a ratio.

Step 1: percentage → fraction

You should know these by heart:

Percentage	Fraction	Percentage	Fraction
50%	$\tfrac{1}{2}$	$12\tfrac{1}{2}\%$	$\tfrac{1}{8}$
$33\tfrac{1}{3}\%$	$\tfrac{1}{3}$	$11\tfrac{1}{9}\%$	$\tfrac{1}{9}$
25%	$\tfrac{1}{4}$	10%	$\tfrac{1}{10}$
20%	$\tfrac{1}{5}$	$9\tfrac{1}{11}\%$	$\tfrac{1}{11}$
$16\tfrac{2}{3}\%$	$\tfrac{1}{6}$	$8\tfrac{1}{3}\%$	$\tfrac{1}{12}$
$14\tfrac{2}{7}\%$	$\tfrac{1}{7}$	$6\tfrac{1}{4}\%$	$\tfrac{1}{16}$

Multiples work too: $40\% = \tfrac{2}{5}$ and $37\tfrac{1}{2}\% = 3 \times 12\tfrac{1}{2}\% = \tfrac{3}{8}$.

Step 2: fraction → ratio (initial : final)

The denominator is the starting value. Add the numerator for an increase, subtract it for a decrease:

Change	Fraction	Initial : Final
+40%	$\tfrac{2}{5}$	5 : 7
−20%	$\tfrac{1}{5}$	5 : 4
+25%	$\tfrac{1}{4}$	4 : 5
$+37\tfrac{1}{2}\%$	$\tfrac{3}{8}$	8 : 11

Step 3: multiply the ratios

Stack the ratios and multiply straight down. The result is your overall initial : final.

Example. A worker’s salary goes up 40%, then down 20%.

$$\begin{array}{rcl} +40\% & 5 : 7 \\ -20\% & 5 : 4 \\ \hline & 25 : 28 \end{array}$$

It went from 25 to 28, a rise of 3 on 25:

$$\frac{3}{25} \times 100 = 12\% \text{ increase}$$

Check with the formula: $40 - 20 - \frac{40 \times 20}{100} = +12\%$. ✔ And with real money: ₹100 → ₹140 → ₹112.

Where it shines. A price rises $37\tfrac{1}{2}\%$, then falls 20%:

$$\begin{array}{rcl} +37\tfrac{1}{2}\% & 8 : 11 \\ -20\% & 5 : 4 \\ \hline & 40 : 44 \end{array} \qquad \frac{4}{40} \times 100 = 10\% \text{ increase}$$

No decimals, no $\tfrac{xy}{100}$, done in your head.

Method 2½: multipliers for the actual value

When the question asks for the final amount rather than the percentage, use the same fractions as multipliers. A +20% change keeps 120%, which is $\tfrac{6}{5}$. A +25% change keeps 125%, which is $\tfrac{5}{4}$.

A town of 3,600 grows 20%, then 25%. What is the new population?

$$3600 \times \frac{6}{5} \times \frac{5}{4} = 5400$$

This works for any number of changes in a row. Just keep multiplying.

Where else it shows up

Situation	x	y	Net
Price +20%, then +30%	20	30	+56%
Population −10%, then +20%	−10	20	+8%
Length +20%, breadth +20% (area)	20	20	+44%
Compound interest, 10% for 2 years	10	10	21%

The last two rows are the same formula wearing different clothes. Area is length × breadth, and 2-year compound interest is “10% on top of 10%”.

Practice

Try each one with both methods.

1. Salary is cut by 25%, then raised by 20%. Net change? (−10%)
2. A rectangle’s length rises 10% and breadth falls 10%. Area change? (−1%)
3. A price rises $57\tfrac{1}{7}\%$, then $16\tfrac{2}{3}\%$. Net change? (7 : 11 and 6 : 7 → 42 : 77, so $83\tfrac{1}{3}\%$)