# What is the difference between equation and formula?

Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference.

Simple example:

mpg = distance traveled in miles / fuel used in gallon


Calculating the MPG above, is that an equation or formula?

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I would say it is a fuel efficiency formula. – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ May 3 '14 at 18:52
Similar to the distance formula: $D=rt$ – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ May 3 '14 at 19:00
According to mathsisfun.com/algebra/equation-formula.html, a formula shows a relationship between 2 or more variables, while an equation doesn't necessarily. – endolith Aug 11 '14 at 15:30

An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.

A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.

Mathematicians have long since realized that when it comes to numbers, certain formulas can be expressed most succinctly as equations. For example, the Pythagorean Theorem $a^2+b^2=c^2$ can be thought of as a formula for finding the length of the side of a right triangle, but it turns out that such a length is always equal to a combination of the other two lengths, so we can express the formula as an equation. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.

In your case, "mpg = distance/gallons" is best understood as "a formula in the form of an equation", which means that in this instance the two words are interchangeable.

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There is also the technical meaning of "formula" as a well-formed arrangement of lexical pieces of a formal language, which I think is possibly more to the point here. In which case equations are a special case of formula. – Malice Vidrine Feb 27 '14 at 22:21
Can you give an example of a formula without an equals sign? – ksoo Apr 9 '14 at 13:15

An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.

Your example is a formula for mpg. But it can become an equation if mpg and one of the other value is given and the remaining value is sought.

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So, is $y=3x+1$ the "formula for a line in the plane" or an equation for a line in the plane? – Arturo Magidin May 10 '11 at 2:22
And if you treat a formula as an equation, solving for one variable to express in terms of other variables, then you have a new formula... – J. M. May 10 '11 at 2:22
@lhf: Suffice it to say, I don't think I agree with your dichotomy. – Arturo Magidin May 10 '11 at 2:25
Better to ask this at the "english stack exchange". I think there are really sensical members there... – Gottfried Helms May 10 '11 at 9:24
@Alexander, so a formula is like a dead equation? :-) – lhf May 10 '11 at 12:02

I'd say an equation is anything with an equals sign in it; a formula is an equation of the form $A={\rm\ stuff}$ where $A$ does not appear among the stuff on the right side.

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I think the second one is a bit different. I think formula is an useful equation or kind of. – Sawarnik Jan 28 '14 at 11:10

Please down vote me if you wish - but I would say these words are really synonyms to each other. They both express that there is some underlying relation between some mathematical expressions.

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Synonyms? $ax^2+bx+c=0$ is a quadratic equation; $x={-b\pm\sqrt{b^2-4ac}\over2a}$ is the quadratic formula. – Gerry Myerson May 10 '11 at 13:28
By your definition, Gerry, the quadratic equation is a formula for zero. – Greg L Jun 12 '12 at 4:35
No downvote. I think that over time the distinction is lost. My math teacher, 35 years ago stated "formulas are used in chemistry, in math we have equations". To this day, the word 'formula' in math seems wrong, but I'd accept it's used commonly. – JoeTaxpayer Oct 10 '13 at 14:48
@JoeTaxpayer Thanks. Your teacher was right, but that does not exclude formulas in mathematics. I myself use both words. – AD. Oct 10 '13 at 17:28
@JoeTaxpayer I my opinion we can use both things as long as we understand each other. – AD. Oct 10 '13 at 22:00

A formula is an equation that shows the relationship between two or more quantities. It would be the rule or instructions that is use to show the relationship between two or more quantities.

An equation is a problem displayed with numerals or symbols with an equals (=) sign included somewhere; usually near the end of the equation. Unless, it is a ratio or division.

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$e=mc^2$ and $f=ma$ are "equations", not normally called "formulas". You wouldn't say the "force formula", but the "force equation". They can have an infinite number of solutions. so i'd say the terms are interchangeable too.

http://en.wikipedia.org/wiki/Force

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You solve an equation, while you evaluate a formula.

By the way, an equation that holds whatever the values of the variables is an identity.

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The answer to this question, in my opinion, comes down to a matter of definition:

equation: A statement formed by placing an equals sign between two numerical or variable expressions.

For example, $11-7=4$, $5x-1=9$, and $y+2=2+y$ are all equations because they all satisfy the definition given.

formula: An equation that states a rule about a relationship.

Here are two useful formulas: $A=lw$, the formula for the area of a rectangle; $P=2l+2w$, the formula for the perimeter of a rectangle.

As you can see, a formula is a special kind of equation, one that states a rule about a relationship and are often used in algebra.

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Note: Not every equation is a formula; but by the above definitions, every formula must be an equation in algebra. – Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ May 3 '14 at 19:04

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