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Relative Difference

Say you have a number (let's call it your reference number), and you want to compare it with a new number. One way we can represent this comparison is with a ratio called relative difference. Here's how to calculate relative difference:

Let's do an example.

You started with $9$ flowers, so your reference value is $9.$

After your mother adds $3$ flowers to your original $9$ flowers, you end up with: $$3+9=12 \text{ flowers}$$ So $12$ is your new value.

Now, we can substitute these reference and new values into our relative difference equation! $$\begin{align} \text{relative difference} &= \dfrac{\text{new - reference}}{\text{reference}}\\ &=\dfrac{12-9}{9}\\ &=\dfrac{3}{9}\\ &=\dfrac{1}{3}\\ \end{align}$$

So, the relative difference between your original $9$ flowers and your final $12$ flowers is $\frac{1}{3}.$

(Video) Relative Difference or Percentage Difference

by Math Mammoth

This video by Math Mammoth explains how to compare values using percents.

$\large{\text{Summary}}$

The style of problems you will encounter will be along the lines:

How many percent more, less, taller, smaller, bigger, etc. is one thing than another?

The formula we use is: $$\frac{\text{difference between the two things}}{\text{reference}}$$ Then, we convert this to a percent.

If this seems confusing, don't worry. A few examples will make it clearer.

Example 1

A loaf of bread weighs $660\text{ g}$. Another weighs $800\text{ g}$. How many percent heavier is the second than the first?

First, we want to calculate the difference between the two weights. $$800-660=140$$

Next, we pick the reference weight. Notice the problem says "than the first". This tells us we want to know difference compared to the first weight. So the first weight is our reference. We can find the percent to be: $$\frac{140}{660}=21.\overline{21}\%$$

So we can say that the second loaf weighs about $21\%$ more than the first loaf.

Example 2

Class $1$ has an average of $80$ points. Class $2$ has an average of $76$ points. How many percent worse did class $2$ do compared to class $1$?

Again, start by finding the difference. $$80-76=4$$

It wants the difference compared to class $1$. Therefore, they're our reference. $$\frac{4}{80}=0.05=5\%$$ Class $2$ did $5\%$ worse than Class $1$.

Calculating Relative Difference $\\$

Relative difference compares one value to another. It is helpful in determining the percentage increase or decrease from one value to another. This concept is similar to calculating percent change and percent error, which you'll learn about later. First, let's dive into relative difference.

The formula for relative difference is: $$\frac{\text{Difference}}{\text{Reference Value}}$$ Remember to convert the value into a percentage!

Examples

Example 1 $\\$ The lacrosse team had $17$ members last year. This year, they gained some new members and currently have $24$ members. What is the relative difference?

We are comparing the change in members with the original (last year's) team members. $$\frac{24-17}{17}=\frac{7}{17}\approx 0.412 = 41.2 \%$$

There has been a $41.2\%$ increase in team members.

Example 2 $\\$ It has been a turbulent February—probably thanks to the impact of climate change. It was $60$ degrees Fahrenheit two days ago and today it is $41$ degrees Fahrenheit. What is the relative difference in temperature? $$\frac{60-41}{60} =\frac{21}{60} = 0.35=35$$

There is a $35\%$ decrease in temperature in the past $2$ days.

Try the problem below!

Relative Difference: Images and Examples

To find the relative difference between two values, divide the difference by the original value: $$\frac{\text{difference}}{\text{original value}}$$ Convert this number to a percentage. If the value increased, we say there is a $x$ percentage increase. If the value decreased, we say there is a $x$ percentage decrease. $x$ is the number we calculated.

Now, let's look an example.

At Millie's Donuts, $6$ donuts are sold at the price of $\$3.50$.

For their $5$ year anniversary promotion, the store is selling $8$ donuts at the same price.

Question 1

What is the relative difference in the amount of donuts you can buy with $\$3.50$?

We are comparing the relative difference from $6$ to $8$ donuts. $$\frac{\text{difference}}{\text{original value}}=\frac{8-6}{6}\approx 0.333 = 33.3\%$$

There is a $33.3\%$ increase in the amount of donuts you can buy with $\$3.50$.

If you got a funky number, it might be good to double check your conversion from a decimal to a percent!

Question 2

What is the relative difference in the unit price (price per donut)?

Let's calculate the original and sales unit prices.

• Original unit price: $\frac{3.50}{6}$
• Sales unit price: $\frac{3.50}{8}$

Now, we can calculate the relative difference: $$\begin{align} &\frac{\text{difference}}{\text{original value}}\\ &=\frac{\frac{3.50}{6}-\frac{3.50}{8}}{\frac{3.50}{6}} \\ &=\frac{8\times 3.5 -6\times 3.5}{48}\times \frac{6}{3.5}\\ &=\frac{2\times 3.5}{48}\times \frac{6}{3.5}\\ &=\frac{2}{8}\\ &=25\% \end{align}$$ There is a $25\%$ decrease in unit price!

If you don't want to do all the math with fractions, you can also find the approximate values of the unit prices and calculate with those!

Now, try the problem below!

$$\frac{\text{difference}}{\text{original value}}=\frac{1.5-1}{1.5}\approx 33\%$$