Say Tim gets $50 every week as pocket money. To his delight, his father increases his allowance to $90. So, how much did Tim’s allowance increase (in percentage)?

Things change with time and often, we are interested in finding out how much that change was. Also, usually, it’s the percent change that we are interested in. (Why? We will answer that later.)

Before we look at the formula for finding percent change, let’s look at an example to gain some intuition for what it actually is.

Well, first we’ll find out the change in dollar terms. That would be equal to –

$\$ 90 - \$ 50 = \$ 20$

Now, this increase came upon his old pocket money (base) of $50. So to find the percent change we need to find what percent of the base that change was.

So what percent of $50 is $40?

We know –

$\text{percentage} = \frac{\text{value}}{\text{base}} \times 100 \, \%$

In our case here, the values is $\hspace{0.2em} 40 \hspace{0.2em}$ and the base is $\hspace{0.2em} 50 \hspace{0.2em}$. So,

$\begin{align*} \text{percentage} &= \frac{40}{50} \times 100 \, \% \\[1.3em] &= 80 \% \end{align*}$

And we have found the percent change in Tim’s allowance – an increase of $\hspace{0.2em} 40 \% \hspace{0.2em}$.

So, percentage change (or percent change) is the change in a value expressed as a percent of the initial (old) value.

If we combine the steps from the previous example, we get the formula for percentage change.

If you are wondering what those vertical lines are (enclosing “old value” at the bottom) – they are symbols for absolute value. Simply put, they turn negative values into positive and leave positive values unchanged.

Also, remember – when calculating percent change, we always take the old value as the base (denominator). Think of it this way — it’s the old value that undergoes change. And it’s only natural that we measure the change relative to the original, the old value.

Alright, now let’s use the formula to solve a couple of example problems.

Example

When Maya’s cat, Molly, was born, it weighed 3.5 ounces. Two weeks later, she weighed 7.35 ounces. What was the percent change in her weight?

Solution

All we need to do is identify and plug the values in the formula. So, the required percent change is –

$\begin{align*} & \frac{V_2 - V_1}{|V_1|} \times 100 \, \% \\[1.5em] = \, & \frac{7.35 \text{ oz} - 3.5 \text{ oz}}{|3.5 \text{ oz}|} \times 100 \, \% \\[1.5em] = \, & \frac{3.85 \text{ oz}}{3.5 \text{ oz}} \times 100 \, \% \\[1.5em] = \, & 1.1 \times 100 \, \% \\[1.5em] = \, & 110 \% \end{align*}$

That’s it.

Example

The average number of children born daily at a maternity clinic decreased from 40 in 2016 to 38 in 2018. Find the percentage change in the number of daily births.

Solution

Again, let’s plug in the values in the formula for percent change.

$\begin{align*} & \frac{V_2 - V_1}{|V_1|} \times 100 \, \% \\[1.5em] = \, & \frac{38 - 40}{|40|} \times 100 \, \% \\[1.5em] = \, & \frac{-2}{40} \times 100 \, \% \\[1.5em] = \, & -5 \% \end{align*}$

Now here, we have a negative answer. So what does it mean?

A negative value of percent change implies that there is a decrease in the original value.

In the present example, the number of babies born at the clinic decreased from 40 to 38.

Note – If the question asks you to find the “percent decrease“, you should drop that negative sign because the question is already talking about the decrease.

For example, in the present case, the “percent change” is $\hspace{0.2em} 5 \% \hspace{0.2em}$ but the “percent decrease” is $\hspace{0.2em} -5 \% \hspace{0.2em}$.

Say, the price of a stock falls from $50 to $40. That’s a decrease of $\hspace{0.2em} 20 \% \hspace{0.2em}$. What percentage increase is required to reverse this change?

One might assume that to reverse a $\hspace{0.2em} 20 \% \hspace{0.2em}$ decrease, you need a $\hspace{0.2em} 20 \% \hspace{0.2em}$ increase. That’s not correct, though.

In the present example, a $\hspace{0.2em} 20 \% \hspace{0.2em}$ increase would take the price from $40 to $48 (not to $\hspace{0.2em} \$50 \hspace{0.2em}$).

But why, you ask?

Because the percentage changes (decrease and increase) are acting on two different bases (starting values). The $\hspace{0.2em} 20 \% \hspace{0.2em}$ decrease acted on $\hspace{0.2em} \$50 \hspace{0.2em}$, while the $\hspace{0.2em} 20 \% \hspace{0.2em}$ increase is acting on $\hspace{0.2em} \$40 \hspace{0.2em}$. And

$20 \% \text{ of } 50 \neq 20 \% \text{ of } 40$

**The Correct Solution**

Let’s use our formula to find out the actual percent change required to reverse the change.

The present value ($\hspace{0.2em} \$40 \hspace{0.2em}$) becomes the “old value” and the future value ($\hspace{0.2em} \$50 \hspace{0.2em}$) will be the “new value”. So the required percent change is –

The percent increase/decrease required to reverse an earlier change of x% is –

$\begin{align*} &\frac{50 - 40}{|40|} \times 100 \, \% \\[1.5em] = \, &\frac{10}{40} \times 100 \, \% \\[1.5em] = \, &25 \% \end{align*}$

In case you wanted a simple formula, here it is. The percent change required to reverse an earlier change of $\hspace{0.2em} P\% \hspace{0.2em}$ is given by –

$P_r \hspace{0.2em} = \hspace{0.2em} \frac{-100P}{100 + P} \, \%$

Note – When plugging $\hspace{0.2em} P \hspace{0.2em}$ in to the formula above, do not forget to include its sign (positive for percentage increase and negative for decrease).

In the previous example, we wanted to reverse a decrease of $\hspace{0.2em} 20\% \hspace{0.2em}$. So we'll replace $\hspace{0.2em} P \hspace{0.2em}$ with $\hspace{0.2em} -20 \hspace{0.2em}$ in the formula.

Hence the required percent change equals

$\begin{align*} &\frac{-100 \cdot -20}{100 + (-20)} \, \% \\[1.5em] =\hspace{0.4em}& 25 \, \% \end{align*}$

Remember Tim from the opening example? A $\hspace{0.2em} \$20 \hspace{0.2em}$ increase in his pocket money (from $\hspace{0.2em} \$50 \hspace{0.2em}$ to $\hspace{0.2em} \$70 \hspace{0.2em}$) made him quite happy.

Do you think his father would be as happy if he got a $\hspace{0.2em} \$20 \hspace{0.2em}$ raise in his monthly salary? Very unlikely.

Let me explain why.

$\hspace{0.2em} \$20 \hspace{0.2em}$ is significant compared to $\hspace{0.2em} \$50 \hspace{0.2em}$ – the old pocket money. It’s a $\hspace{0.2em} 40\% \hspace{0.2em}$ increase. But if we assume his dad made $\hspace{0.2em} \$4000 \hspace{0.2em}$ before the raise, a $\hspace{0.2em} \$20 \hspace{0.2em}$ increase is barely noticeable. It’s a $\hspace{0.2em} 0.5\% \hspace{0.2em}$ change. Almost nothing.

This example should help you understand why percent change can often be more useful to know than the magnitude of change.

Because the percent change gives us the change in relation to the initial value, we can generally get some idea of how big of an impact that change would have.

However, I must add that it’s best to know the percent change(s) as well as the associated values (initial and final). Here's again an example (hypothetical) to show why.

Say a medical study involved two groups, each with $\hspace{0.2em} 1000 \hspace{0.2em}$ people with a certain disease. Now, the first group was given a placebo (fake medicine) and $\hspace{0.2em} 1 \hspace{0.2em}$ person recovered completely by the end of the study. The second group was given the trial medicine and $\hspace{0.2em} 2 \hspace{0.2em}$ people showed complete recovery.

So, the number of people who recovered was $\hspace{0.2em} 100 \% \hspace{0.2em}$ more in the second group compared to the first ($\hspace{0.2em} 2 \hspace{0.2em}$ against $\hspace{0.2em} 1 \hspace{0.2em}$).

Does that mean the medicine is a game-changer, or even mildly useful? Not really. Both $\hspace{0.2em} 1 \hspace{0.2em}$ and $\hspace{0.2em} 2 \hspace{0.2em}$ are such small numbers, that virtually nobody recovered in either group. The medicine doesn't appear to be doing anything.

So again, when interpreting data, it’s best to know the percent change(s) as well as the actual values.

And with that, we come to the end of this tutorial on percent change. Until next time.

We use cookies to provide and improve our services. By using the site you agree to our use of cookies. Learn more