Successive percentage change

If a stock rises $10\%$ on Monday and falls $10\%$ on Tuesday, has it returned to its Monday-morning price? No. It is now slightly below where it started. The reason is the multiplicative nature of percent change.

Concept

The multiplicative rule. A percent change of $p\%$ multiplies the quantity by $\big(1 + \tfrac{p}{100}\big)$. (Negative $p$ for a decrease.) When changes happen in sequence, you multiply all the factors.

For two changes $p\%$ then $q\%$:

$\text{final} = \text{initial} \times \left(1 + \frac{p}{100}\right) \left(1 + \frac{q}{100}\right).$

Example 1. $+10\%$ then $-10\%$ of $\mathbb{?}\,100$.

• $100 \times 1.10 = 110$.
• $110 \times 0.90 = 99$.

Net: $-1\%$ from $\mathbb{?}\,100$.

Why is it not zero? Because the $-10\%$ is applied to the new amount $\mathbb{?}\,110$, which is bigger than $\mathbb{?}\,100$. The drop "off" the higher number is more rupees than the rise "off" the lower number.

Three-step changes. Just multiply three factors. E.g., $+5\%, +10\%, -20\%$:

$1.05 \times 1.10 \times 0.80 = 0.924,$

a net $-7.6\%$.

Discount chains. "Successive discounts of $20\%$ then $10\%$ on $\mathbb{?}\,500$".

• After $20\%$ off: $\mathbb{?}\,400$.
• After further $10\%$ off: $\mathbb{?}\,360$.
• Effective discount: $\dfrac{500 - 360}{500} = 28\%$.

If only one discount were offered, $28\%$ off the original $\mathbb{?}\,500$ would also give $\mathbb{?}\,360$. So two successive discounts $20\%$ and $10\%$ are equivalent to one discount of $28\%$ (not $30\%$).

Increase, then increase. "Price rose $10\%$ this year and $10\%$ next year". Total increase:

• $1.10 \times 1.10 = 1.21$.
• A $21\%$ rise.

Note: $20\% + 20\% = 40\%$ would be wrong. The correct net is $44\%$ ($1.2 \times 1.2 = 1.44$).

Formula for equivalent single change. Two changes $p$ and $q$ are equivalent to one change of

$p + q + \frac{pq}{100}.$

For $p = 20, q = 10$: $20 + 10 + \tfrac{200}{100} = 32$. So $+20\%$ then $+10\%$ equals one increase of $32\%$. For $p = 20, q = -10$ (a $20\%$ rise then a $10\%$ fall): $20 - 10 - \tfrac{200}{100} = 8$. Net $+8\%$.

Worked examples

Example 1. A salary of $\mathbb{?}\,30{,}000$ is increased by $10\%$ and then by $20\%$. Find the final salary.

• $30{,}000 \times 1.10 \times 1.20 = 30{,}000 \times 1.32 = \mathbb{?}\,39{,}600$.

Example 2. A bag's MP is $\mathbb{?}\,800$. Successive discounts of $25\%$ and $10\%$ are offered. Find SP.

• $800 \times 0.75 \times 0.90 = 800 \times 0.675 = \mathbb{?}\,540$.

Example 3. A price rose $20\%$ then fell $20\%$. Net change?

• $1.20 \times 0.80 = 0.96$. Net $-4\%$.

Example 4. A class's strength grew $25\%$ in year 1, $40\%$ in year 2, $-10\%$ in year 3. What is the overall percent change from start to end?

• $1.25 \times 1.40 \times 0.90 = 1.575$. Net $+57.5\%$.

Try it yourself

1. A $\mathbb{?}\,500$ item is discounted $10\%$ then $20\%$. Find SP.
2. A salary of $\mathbb{?}\,25{,}000$ is increased $8\%$ then $12\%$. Find final.
3. Net % change after $-30\%$ and $+30\%$?
4. Successive discounts of $10\%$, $20\%$, $30\%$ on $\mathbb{?}\,1000$. Find SP.
5. Population of a town went up $20\%$, then $20\%$, then down $25\%$. Net change?
6. Find equivalent single discount of $15\%$ and $20\%$ in sequence.
7. After a $25\%$ rise and a $20\%$ fall on $\mathbb{?}\,400$, the price is?
8. Is "$40\%$ off then $10\%$ off" cheaper or more expensive than "$10\%$ off then $40\%$ off"? Justify.

Activity / Insight

Stock chart shock. Pretend $\mathbb{?}\,100$ is invested. Apply daily percentage changes of $+5\%, -5\%, +5\%, -5\%, +5\%, -5\%$. After six days, is it back at $\mathbb{?}\,100$? Compute carefully , the answer is below $100$. This is the percent-change lesson that has caught many investors off guard.